ON THE DISTRIBUTION OF ZEROS OF THE DERIVATIVE OF SELBERG’S ZETA FUNCTION ASSOCIATED TO FINITE VOLUME RIEMANN SURFACES

Abstract

We study the distribution of zeros of the derivative of the Selberg zeta function associated to a noncompact, finite volume hyperbolic Riemann surface $M$. Actually, we study the zeros of $(Z_{M}H_{M})^{\prime }$, where $Z_{M}$ is the Selberg zeta function and $H_{M}$ is the Dirichlet series component of the scattering matrix, both associated to an arbitrary finite volume hyperbolic Riemann surface $M$. Our main results address finiteness of number of zeros of $(Z_{M}H_{M})^{\prime }$ in the half-plane $\operatorname{Re}(s)<1/2$, an asymptotic count for the vertical distribution of zeros, and an asymptotic count for the horizontal distance of zeros. One realization of the spectral analysis of the Laplacian is the location of the zeros of $Z_{M}$, or, equivalently, the zeros of $Z_{M}H_{M}$. Our analysis yields an invariant $A_{M}$ which appears in the vertical and weighted vertical distribution of zeros of $(Z_{M}H_{M})^{\prime }$, and we show that $A_{M}$ has different values for surfaces associated to two topologically equivalent yet different arithmetically defined Fuchsian groups. We view this aspect of our main theorem as indicating the existence of further spectral phenomena which provides an additional refinement within the set of arithmetically defined Fuchsian groups.

1 Introduction

1.1 Selberg zeta functions for compact Riemann surfaces

In [22], Luo initiated the study of the nontrivial zeros of the derivative $Z_{M}^{\prime }$ of the Selberg zeta function $Z_{M}$ associated to a compact, hyperbolic Riemann surface $M$ , proving analogues of results obtained by Spira [27] and Berndt [4] for the Riemann zeta function. Further refinements of results by Luo were established in [11] and [12]. As is standard in analytic number theory, the nontrivial zeros of $Z_{M}$ , or $Z_{M}^{\prime }$ , are its zeros which do not arise from the poles of the multiplicative factor of the functional equation. In the case of a compact Riemann surface, the nontrivial zeros of $Z_{M}$ and $Z_{M}^{\prime }$ are zeros different from negative integers. Let us summarize the three main results stemming from the aforementioned articles, which are location, vertical distribution, and an asymptotic count for the weighted vertical distribution of the nontrivial zeros of $Z_{M}^{\prime }$ .

In [22] it is shown that $Z_{M}^{\prime }(s)$ has at most a finite number of nontrivial zeros in the half-plane $\text{Re}(s)<1/2$ . This result was strengthened in [24] and [25] where it is proved that $Z_{M}^{\prime }(s)$ has no nontrivial zeros in the half-plane $\text{Re}(s)<1/2$ .

Let $\text{vol}(M)$ denote the hyperbolic volume of $M$ . Let $\ell _{M,0}$ be the length of the shortest closed geodesic on $M$ . Let $m_{M,0}$ denote the number of inconjugate geodesics whose length is $\ell _{M,0}$ . Let $N_{\text{ver}}(T;Z_{M}^{\prime })$ be the number of nontrivial zeros of $Z_{M}^{\prime }(s)$ where $s=\unicode[STIX]{x1D70E}+it$ with $\unicode[STIX]{x1D70E}\geqslant 1/2$ and $0<t<T$ , and let

$$\begin{eqnarray}N_{\text{w}}(T;Z_{M}^{\prime })=\mathop{\sum }_{{Z_{M}^{\prime }(\unicode[STIX]{x1D70E}+it)=0\atop 0<t<T,\unicode[STIX]{x1D70E}>1/2}}(\unicode[STIX]{x1D70E}-1/2)\end{eqnarray}$$

be the weighted vertical distribution with weights equal to distances of zeros to the critical line. Then, building on the results form [22], it is proved in [11] and [12] that

(1) $$\begin{eqnarray}N_{\text{ver}}(T;Z_{M}^{\prime })=\frac{\text{vol}(M)}{4\unicode[STIX]{x1D70B}}T^{2}-\frac{\ell _{M,0}}{2\unicode[STIX]{x1D70B}}T+o(T)\quad \text{as }T\rightarrow \infty ,\end{eqnarray}$$

and

(2) $$\begin{eqnarray}\displaystyle N_{\text{w}}(T;Z_{M}^{\prime }) & = & \displaystyle \frac{T}{2\unicode[STIX]{x1D70B}}\log T+\frac{T}{2\unicode[STIX]{x1D70B}}\left(\frac{1}{2}\ell _{M,0}+\log \left(\frac{\text{vol}(M)(1-e^{-\ell _{M,0}})}{m_{M,0}\ell _{M,0}}\right)-1\right)\nonumber\\ \displaystyle & & \displaystyle +\,o(T)\quad \text{as }T\rightarrow \infty .\end{eqnarray}$$

The study of the zeros of $Z_{M}^{\prime }$ is of particular interest because of the connection with spectral analysis. Recall that if $s$ is a nontrivial zero of $Z_{M}(s)$ , then $\unicode[STIX]{x1D706}=s(1-s)$ is an eigenvalue of an $L^{2}$ -eigenfunction of the hyperbolic Laplacian which acts on the space of smooth functions on $M$ . Common zeros of $Z_{M}$ and $Z_{M}^{\prime }$ are, in fact, zeros of $Z_{M}(s)$ with multiplicity greater than one. Such zeros of $Z_{M}$ correspond to multi-dimensional eigenspaces of the Laplacian. As shown in [22, p. 1143], all zeros of $Z_{M}^{\prime }(s)$ on the line $\text{Re}(s)=1/2$ , except possibly at $s=1/2$ , correspond to multiple zeros of $Z_{M}$ . The problem of obtaining nontrivial bounds for the dimension of eigenspaces of the Laplacian is very difficult; see [17, p. 160]. Thus, it is possible that refined information regarding (1) possibly could shed light on this important, outstanding question.

1.2 Noncompact Riemann surfaces

Let $\mathbb{H}$ denote the hyperbolic upper half-plane. Let $\unicode[STIX]{x1D6E4}\subseteq \text{PSL}(2,\mathbb{R})$ be any Fuchsian group of the first kind acting by fractional linear transformations on $\mathbb{H}$ , and let $M$ be the quotient space $\unicode[STIX]{x1D6E4}\backslash \mathbb{H}$ .

One realization of the spectral analysis of the Laplacian on the surface $M$ is the location of nontrivial zeros of the associated Selberg zeta function $Z_{M}$ , defined for $s\in \mathbb{C}$ with $\text{Re}(s)>1$ by the Euler product

(3) $$\begin{eqnarray}\displaystyle Z_{M}(s)=\mathop{\prod }_{n=0}^{\infty }\mathop{\prod }_{P_{0}\in {\mathcal{H}}(\unicode[STIX]{x1D6E4})}\big(1-e^{-(s+n)\ell _{P_{0}}}\big)=\mathop{\prod }_{n=0}^{\infty }\mathop{\prod }_{P_{0}\in {\mathcal{H}}(\unicode[STIX]{x1D6E4})}\big(1-N(P_{0})^{-(s+n)}\big). & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Here ${\mathcal{H}}(\unicode[STIX]{x1D6E4})$ denotes a complete set of representatives of inconjugate, primitive hyperbolic elements of $\unicode[STIX]{x1D6E4}$ , $P_{0}$ is a primitive hyperbolic element, $\ell _{P_{0}}$ is the hyperbolic length of the geodesic path in the homotopy class determined by $P_{0}$ and the norm $N(P_{0})$ is equal to $\exp (\ell _{P_{0}})$ .

The function $Z_{M}$ possesses a meromorphic continuation to the whole complex plane and satisfies the functional equation $Z_{M}(s)\unicode[STIX]{x1D719}_{M}(s)=\unicode[STIX]{x1D702}_{M}(s)Z_{M}(1-s)$ , where $\unicode[STIX]{x1D719}_{M}(s)$ denotes the determinant of the scattering matrix $\unicode[STIX]{x1D6F7}_{M}(s)$ ,

(4) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D702}_{M}^{\prime }}{\unicode[STIX]{x1D702}_{M}}(s) & = & \displaystyle \text{vol}(M)(s-1/2)\tan (\unicode[STIX]{x1D70B}(s-1/2))-\unicode[STIX]{x1D70B}\nonumber\\ \displaystyle & & \displaystyle \cdot \,\underset{0<\unicode[STIX]{x1D703}(R)<\unicode[STIX]{x1D70B}}{\underset{\left\{R\right\}}{\sum }}\frac{1}{M_{R}\sin \unicode[STIX]{x1D703}}\frac{\cos (2\unicode[STIX]{x1D703}-\unicode[STIX]{x1D70B})(s-1/2)}{\cos \unicode[STIX]{x1D70B}(s-1/2)}\nonumber\\ \displaystyle & & \displaystyle +\,2n_{1}\log 2+n_{1}\left(\frac{\unicode[STIX]{x1D6E4}^{\prime }}{\unicode[STIX]{x1D6E4}}(1/2+s)+\frac{\unicode[STIX]{x1D6E4}^{\prime }}{\unicode[STIX]{x1D6E4}}(3/2-s)\right)=\frac{\unicode[STIX]{x1D702}_{M}^{\prime }}{\unicode[STIX]{x1D702}_{M}}(1-s)\end{eqnarray}$$

and where $\{R\}$ denotes a complete, finite set of inconjugate elliptic elements of $\unicode[STIX]{x1D6E4}$ so that $0<\unicode[STIX]{x1D703}(R)<\unicode[STIX]{x1D70B}$ is the uniquely determined real number such that $R$ is conjugate to the matrix

$$\begin{eqnarray}\left(\begin{array}{@{}cc@{}}\cos \unicode[STIX]{x1D703}(R) & -\sin \unicode[STIX]{x1D703}(R)\\ \sin \unicode[STIX]{x1D703}(R) & \cos \unicode[STIX]{x1D703}(R)\\ \end{array}\right).\end{eqnarray}$$

The scattering determinant $\unicode[STIX]{x1D719}_{M}(s)$ has a decomposition into a product of a general Dirichlet series and Gamma functions. Specifically, we can write

$$\begin{eqnarray}\unicode[STIX]{x1D719}_{M}(s)=\unicode[STIX]{x1D70B}^{n_{1}/2}\left(\frac{\unicode[STIX]{x1D6E4}\left(s-\frac{1}{2}\right)}{\unicode[STIX]{x1D6E4}(s)}\right)^{n_{1}}\overset{\infty }{\underset{n=1}{\sum }}\frac{d(n)}{\mathfrak{g}_{n}^{2s}}\end{eqnarray}$$

where $n_{1}$ is the number of cusps of $M$ , and $\{d(n)\}$ and $\{\mathfrak{g}_{n}\}$ are sequences of real numbers with

$$\begin{eqnarray}0<\mathfrak{g}_{1}<\cdots <\mathfrak{g}_{n}<\mathfrak{g}_{n+1}<\cdots \,;\end{eqnarray}$$

given in terms of Kloosterman sums (see [17, p. 160]). Let us write $\unicode[STIX]{x1D719}_{M}(s)=K_{M}(s)\cdot H_{M}(s)$ where

(5) $$\begin{eqnarray}\displaystyle K_{M}(s) & = & \displaystyle \unicode[STIX]{x1D70B}^{n_{1}/2}\left(\frac{\unicode[STIX]{x1D6E4}\left(s-\frac{1}{2}\right)}{\unicode[STIX]{x1D6E4}(s)}\right)^{n_{1}}e^{c_{1}s+c_{2}},\quad \text{with }\nonumber\\ \displaystyle & & \displaystyle c_{1}=-2\log \mathfrak{g}_{1}\qquad \text{and}\qquad c_{2}=\log d(1),\end{eqnarray}$$

and

(6) $$\begin{eqnarray}\displaystyle H_{M}(s)=1+\overset{\infty }{\underset{n=2}{\sum }}\frac{a(n)}{r_{n}^{2s}}\quad \text{with }r_{n}=\mathfrak{g}_{n}/\mathfrak{g}_{1}>1\qquad \text{and}\qquad a(n)=d(n)/d(1). & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The Dirichlet series expansion for $H_{M}(s)$ converges for all $\text{Re}(s)>1$ . We call the function $H_{M}$ the Dirichlet series portion of the scattering determinant $\unicode[STIX]{x1D719}_{M}$ . In general, the function $H_{M}$ can be expressed as the determinant of a matrix whose entries are general Kloosterman sums; see [17, Theorem 3.4]. The constants $\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$ are explained in terms of the left lower entries of the matrices appearing in the double coset decomposition of $\unicode[STIX]{x1D6E4}$ . Therefore, the constants $\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$ are precisely connected to the Fuchsian group $\unicode[STIX]{x1D6E4}$ and $H_{M}(s)$ is a Dirichlet series carrying the information related to parabolic subgroups of $\unicode[STIX]{x1D6E4}$ .

By nontrivial zeros of $Z_{M}(s)$ we mean all nonreal zeros and real zeros at points $s\in [0,1]$ such that $s(1-s)$ is equal to an eigenvalue of the Laplacian that is less than or equal to $1/4$ . The nontrivial zeros are related to the spectrum of the Laplacian in the sense that, according to [14, Theorem 5.3], the nontrivial zeros of $Z_{M}$ are located at points of the form $1/2\pm ir_{n}$ where $1/4+r_{n}^{2}$ is a discrete eigenvalue of the Laplacian and at points $1-\unicode[STIX]{x1D70C}$ in the half-plane $\text{Re}(s)<1/2$ which are poles of the determinant $\unicode[STIX]{x1D719}_{M}$ of the scattering matrix.

We may conclude that the function $Z_{M}H_{M}$ is “spectrally equivalent” to $Z_{M}$ in the following sense: The function $Z_{M}H_{M}$ can be represented as a general Dirichlet series converging in the half-plane $\text{Re}(s)>1$ and carrying information about the underlying group $\unicode[STIX]{x1D6E4}$ ; it possesses a meromorphic continuation to the entire complex plane satisfying the functional equation

(7) $$\begin{eqnarray}(Z_{M}H_{M})(s)=\unicode[STIX]{x1D702}_{M}(s)K_{M}^{-1}(s)Z_{M}(1-s);\end{eqnarray}$$

and its nontrivial zeros are at points $s=1/2\pm ir_{n}$ , where $1/4+r_{n}^{2}$ is a discrete eigenvalue of the Laplacian and at points $s=\unicode[STIX]{x1D70C}$ in the half-plane $\text{Re}(s)>1/2$ which are zeros of the determinant $\unicode[STIX]{x1D719}_{M}$ of the scattering matrix. Based on this argument, we may, loosely speaking, say that $Z_{M}H_{M}$ carries the same amount of spectral information as $Z_{M}$ does. Besides that, the function $Z_{M}H_{M}$ has no nontrivial zeros in the half-plane $\text{Re}(s)<1/2$ .

The question of studying the zeros of $Z_{M}^{\prime }$ when $M$ is not compact by applying methods presented in this paper begins with one possible technical difficulty stemming from the fact that the function $Z_{M}$ has an infinite number of zeros in the half-plane $\text{Re}(s)<1/2$ , each one of which would produce a negative weight in the weighted counting function $N_{\text{w}}$ . On the other hand, the function $Z_{M}H_{M}$ has no nontrivial zeros in the half-plane $\text{ Re}(s)<1/2$ and carries the same spectral information as $Z_{M}$ does. Therefore, as a result we shall study the zeros of $(Z_{M}H_{M})^{\prime }$ .

As we shall see below, the choice of $Z_{M}H_{M}$ instead of $Z_{M}$ is further justified by the fact that, according to the statement (a) of the Main Theorem, the only zeros of $(Z_{M}H_{M})^{\prime }$ on the critical line (with imaginary part greater than some constant depending upon the group) are the multiple zeros of $Z_{M}$ ; therefore, the study of zeros of $(Z_{M}H_{M})^{\prime }$ is related to the problem of obtaining bounds for the dimension of eigenspaces of discrete eigenvalues of the Laplacian on $M$ .

In conclusion, we followed the guide provided by the technical issues we faced and chose to study the zeros of $(Z_{M}H_{M})^{\prime }$ . To be specific, we viewed the positivity issues described above as important, thus we focused our attention on the zeros of $(Z_{M}H_{M})^{\prime }$ . Nonetheless, the problem of studying the zeros of $Z_{M}^{\prime }$ is both well-posed and remains open. It is quite possible that a successful study of the zeros of $Z_{M}^{\prime }$ when combined with the results of the present paper would yield interesting results. We leave such a study to a motivated reader.

1.3 The main result

The function $H_{M}^{\prime }/H_{M}$ admits the general Dirichlet series expansion

(8) $$\begin{eqnarray}\frac{H_{M}^{\prime }}{H_{M}}(s)=\mathop{\sum }_{i=1}^{\infty }\frac{b(q_{i})}{q_{i}^{s}},\end{eqnarray}$$

where the series on the right converges absolutely and uniformly for $\text{Re}(s)\geqslant \unicode[STIX]{x1D70E}_{0}+\unicode[STIX]{x1D716}>\unicode[STIX]{x1D70E}_{0}\gg 0$ , and where $\{q_{i}\}$ is a nondecreasing sequence of positive real numbers consisting of all finite products of numbers $r_{n}^{2}>1$ . Obviously, $q_{2}>q_{1}=\inf q_{i}=(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ . Furthermore,

$$\begin{eqnarray}b(q_{1})=-a(2)\log q_{1}=-2(d(2)/d(1))\log (\mathfrak{g}_{2}/\mathfrak{g}_{1}).\end{eqnarray}$$

Let $\ell _{M,0}$ be the length of a shortest closed geodesic, or systole, on $M$ . With our notation from above, let

(9) $$\begin{eqnarray}A_{M}=\min \big\{e^{\ell _{M,0}},\left(\mathfrak{g}_{2}/\mathfrak{g}_{1}\right)^{2}\big\}.\end{eqnarray}$$

Here, we have dropped the subscript $M$ on $(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ in order to ease the notation; however, it is clear that $(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ depends on $M$ . Let $m_{M,0}$ denote the number of inconjugate closed geodesics on $M$ with length $\ell _{M,0}$ . If $e^{\ell _{M,0}}\neq (\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ , let

(10) $$\begin{eqnarray}a_{M}=\left\{\begin{array}{@{}cc@{}}\displaystyle \frac{m_{M,0}\ell _{M,0}}{1-e^{-\ell _{M,0}}}; & \text{if }e^{\ell _{M,0}}<\left(\mathfrak{g}_{2}/\mathfrak{g}_{1}\right)^{2}\\ b(\left(\mathfrak{g}_{2}/\mathfrak{g}_{1}\right)^{2}); & \text{if }e^{\ell _{M,0}}>\left(\mathfrak{g}_{2}/\mathfrak{g}_{1}\right)^{2}\end{array}\right\}.\end{eqnarray}$$

If $e^{\ell _{M,0}}=(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ , let

(11) $$\begin{eqnarray}a_{M}=\frac{m_{M,0}\ell _{M,0}}{1-e^{-\ell _{M,0}}}+b(\left(\mathfrak{g}_{2}/\mathfrak{g}_{1}\right)^{2}).\end{eqnarray}$$

Observe that $a_{M}$ is the sum of the two terms which appear in the two cases in (10), not the arithmetic average as one would expect from elementary Fourier analysis.

With all this, the main result of this article is the following.

Main Theorem.

Let $\unicode[STIX]{x1D6E4}\subseteq \text{PSL}_{2}(\mathbb{R})$ be any Fuchsian group of the first kind acting by fractional linear transformations on $\mathbb{H}$ , and let $M$ be the quotient space $\unicode[STIX]{x1D6E4}\backslash \mathbb{H}$ . Let $Z_{M}(s)$ be the associated Selberg zeta function, and $H_{M}(s)$ be the Dirichlet series portion of the determinant of the associated scattering matrix.

1. (a) There are a finite number of nontrivial zeros of $(Z_{M}H_{M})^{\prime }(s)$ in the half-plane $\text{Re}(s)<1/2$ . In addition, there exist some $t_{0}>0$ such that any zero of $(Z_{M}H_{M})^{\prime }(s)$ on the line $\operatorname{Re}(s)=1/2$ with property $|\operatorname{Im}(s)|>t_{0}$ arises from a multiple zero of $Z_{M}(s)$ .

2. (b) Let us define the vertical counting function

   $$\begin{eqnarray}N_{\text{ver}}(T;(Z_{M}H_{M})^{\prime })=\#\{\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70E}+it\,\big|\,(Z_{M}H_{M})^{\prime }(\unicode[STIX]{x1D70C})=0\text{ with }0<t<T\}.\end{eqnarray}$$
   Then
   $$\begin{eqnarray}\displaystyle N_{\text{ver}}(T;(Z_{M}H_{M})^{\prime }) & = & \displaystyle \frac{\text{ vol}(M)}{4\unicode[STIX]{x1D70B}}T^{2}-\frac{T}{2\unicode[STIX]{x1D70B}}\left(\log A_{M}+2n_{1}\log 2+2\log \mathfrak{g}_{1}\right)\nonumber\\ \displaystyle & & \displaystyle +\,o(T),\quad \text{as }T\rightarrow \infty .\nonumber\end{eqnarray}$$
   In particular, if $M$ is co-compact, then (1) holds true.

3. (c) Let us define the weighted vertical counting function

   $$\begin{eqnarray}N_{\text{w}}(T;(Z_{M}H_{M})^{\prime })=\mathop{\sum }_{{(Z_{M}H_{M})^{\prime }(\unicode[STIX]{x1D70E}+it)=0\atop 0<t<T\,\,\text{ and}\,\,\unicode[STIX]{x1D70E}>1/2}}(\unicode[STIX]{x1D70E}-1/2).\end{eqnarray}$$
   Then
   $$\begin{eqnarray}\displaystyle N_{\text{w}}(T;(Z_{M}H_{M})^{\prime }) & = & \displaystyle \left(\frac{n_{1}}{2}+1\right)\frac{T\log T}{2\unicode[STIX]{x1D70B}}+\frac{T}{2\unicode[STIX]{x1D70B}}\left(\log \frac{\text{ vol}(M)A_{M}^{1/2}}{|a_{M}|}-1\right)\nonumber\\ \displaystyle & & \displaystyle +\,\frac{T}{2\unicode[STIX]{x1D70B}}\left(\log \left(\frac{\mathfrak{g}_{1}}{\unicode[STIX]{x1D70B}^{n_{1}/2}|\,d(1)|}\right)-\frac{n_{1}}{2}\right)+o(T),\nonumber\\ \displaystyle & & \displaystyle \quad \text{as }T\rightarrow \infty .\nonumber\end{eqnarray}$$
   In particular, if $M$ is co-compact, then (2) holds true.

As stated in the Main Theorem, the above asymptotic formulas specialize in the case $M$ is compact to give the main results in [11, 12, 22, 24] and [25]. More precisely, in [24] and [25] it is proved that $Z_{M}^{\prime }$ in the compact case possesses no nonreal zeros in the half-plane $\text{Re}(s)<1/2$ , a statement which we believe to hold true for $(Z_{M}H_{M})^{\prime }$ .

Similar results for the zeros of higher derivatives of $Z_{M}H_{M}$ are presented in a later section. In addition, corollaries of the main theorem, analogous to results from [21], are derived.

Since the proof of the Main Theorem is rather technical, let us present here a summary of the ideas involved in its proof.

Part (a) of the Main Theorem is proved in two parts. First, we employ the functional equation for the Selberg zeta function together with a bound for the growth of the logarithmic derivative $D_{M}(s)=Z_{M}^{\prime }(s)/Z_{M}(s)$ in the right half of the critical strip in order to deduce that $\text{Re}(Z_{M}H_{M})^{\prime }(s)\neq 0$ for sufficiently large $\text{Im}(s)$ in the half-plane $\text{Re}(s)<1/2$ . However, this method has a critical line as its boundary. Therefore, in order to show that all (but eventually finitely many) multiple zeros of $Z_{M}$ on the critical line are also zeros of $(Z_{M}H_{M})^{\prime }$ , we employ the Hadamard product representation of the completed Selberg zeta function, which was proved in [7], and conduct careful analysis of the imaginary part of the logarithmic derivative of $(Z_{M}H_{M})^{\prime }$ .

Parts (b) and (c) of the Main Theorem are proved by an application of Littlewood’s theorem [29, p. 132] to the function

(12) $$\begin{eqnarray}X_{M}(s):=\frac{A_{M}^{s}}{a_{M}}(Z_{M}H_{M})^{\prime }(s)\end{eqnarray}$$

followed by a careful technical analysis of the integrals obtained. There are two main difficulties appearing in the noncompact case. The first one is to control the growth of $D_{M}(s)$ inside the critical strip, which is resolved by an application of Theorem 5 below. What remains is the second technical point, which is to study the growth of $\arg X_{M}(\unicode[STIX]{x1D70E}+iT)$ , for large $T$ and $\unicode[STIX]{x1D70E}\in (a,\unicode[STIX]{x1D70E}_{0})$ , where $a\in (0,1/2)$ is an arbitrary constant. In order to address this problem, we prove a Phragmen–Lindelöf type bound for $(Z_{M}H_{M})(s)$ inside the strip $-\unicode[STIX]{x1D70E}_{2}\leqslant -1\leqslant \text{Re}(s)\leqslant \unicode[STIX]{x1D70E}_{0}$ and the Lindelöf type bound for $(Z_{M}H_{M})(s)$ for $\text{Re}(s)$ close to $1/2$ . These bounds are necessary in order to apply the generalized Backlund equivalent for the Lindelöf hypothesis (see § 2.5) which will yield a sharp bound for $(Z_{M}H_{M})^{\prime }(s)$ near the critical line. The resulting estimate enables one to apply Jensen’s theorem and deduce that $\arg X_{M}(\unicode[STIX]{x1D70E}+iT)=o(T)$ , as $T\rightarrow \infty$ .

1.4 Properties of the invariant $A_{M}$

Aspects of the spectral analysis of the Laplacian acting on smooth functions on a hyperbolic Riemann surface can be measured by studying the zeros of the Selberg zeta function. As discussed above, one equivalently can study the zeros of $Z_{M}H_{M}$ . Therefore, by slight extension, the zeros of $(Z_{M}H_{M})^{\prime }$ provide another measure of the spectral analysis of the Laplacian. In this regard, the quantity $(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ is a new spectral invariant. In addition, our Main Theorem asserts that for any given surface, the spectral analysis depends on the comparison of $e^{\ell _{M,0}}$ and $(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ .

In § 7, we consider various arithmetic groups and compare $e^{\ell _{M,0}}$ to $(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ . More precisely, we prove the following proposition.

Proposition 1.

1. (i) For all surfaces $M=\unicode[STIX]{x1D6E4}\setminus \mathbb{H}$ , where $\unicode[STIX]{x1D6E4}$ is a congruence subgroup or principal congruence subgroup of the group $\text{PSL}(2,\mathbb{Z})$ we have that $A_{M}=e^{\ell _{M,0}}$ .

2. (ii) For the surface $M_{5}$ corresponding to the arithmetic group $\unicode[STIX]{x1D6E4}_{0}^{+}(5)$ we have that $A_{M_{5}}=(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}=((1+\sqrt{5})/2)^{2}$ . For the surface $M_{6}$ corresponding to the arithmetic group $\unicode[STIX]{x1D6E4}_{0}^{+}(6)$ , which has the same signature as $\unicode[STIX]{x1D6E4}_{0}^{+}(5)$ , we have that $A_{M_{6}}=e^{\ell _{M_{6},0}}=2$ .

3. (iii) There exists a surface $M$ where $e^{\ell _{M,0}}=(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ .

With respect to statements (i) and (ii) of the above proposition we find it very interesting that, in the sense of our Main Theorem, not all arithmetic surfaces, even those with the same topological signature, have the same behavior.

Also in § 7, in order to prove statement (iii) of Proposition 1 we argue that if one considers a degenerating family of hyperbolic Riemann surfaces within the moduli space of surfaces of fixed topological type, one eventually has the inequality $e^{\ell _{M,0}}<(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ near the boundary. As a result, if one begins with congruence group and degenerates the corresponding surface, one will ultimately encounter a surface where $e^{\ell _{M,0}}=(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ . More generally, however, it seems as if moduli space can be separated into sets defined by the sign of $e^{\ell _{M,0}}-(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ where most, but not all, arithmetic surfaces are in the component where $e^{\ell _{M,0}}-(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}>0$ , and the Deligne–Mumford boundary lies in the component where $e^{\ell _{M,0}}-(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}<0$ .

We could not explicitly construct a surface where $e^{\ell _{M,0}}-(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}=0$ , even though we prove that such surfaces exist.

1.5 A comparison of counting functions

In [14, Theorem 2.22], D. Hejhal establishes the asymptotic behavior of the weighted vertical distribution of zeros of $\unicode[STIX]{x1D719}_{M}$ within the critical strip. In our notation, the zeros of $\unicode[STIX]{x1D719}_{M}$ within the critical strip coincide with the zeros of the Dirichlet series $H_{M}$ , so then [14, Theorem 2.22] establishes the asymptotic behavior of the weighted vertical counting function $N_{\text{w}}(T;H_{M})$ .

Let $M$ be any finite volume hyperbolic Riemann surface. We claim there exists a co-compact hyperbolic Riemann surface $\widetilde{M}$ such that $\text{vol}(M)=\text{vol}(\widetilde{M})$ , $\ell _{M,0}=\ell _{\widetilde{M},0}$ and $m_{M,0}=m_{\widetilde{M},0}$ , which we argue as follows. In the case when the number $n_{1}$ of cusps of the surface $M$ is even, we choose the surface $\widetilde{M}_{1}$ to be any co-compact surface with genus $g_{\widetilde{M}}=g_{M}+n_{1}/2$ and the same structure of elliptic points as $M$ , hence $\text{vol}(M)=\text{vol}(\widetilde{M}_{1})$ . If the number of cusps of the surface $M$ is odd, we choose the surface $\widetilde{M}_{1}$ to be any co-compact surface with genus $g_{\widetilde{M}}=g_{M}+(n_{1}-1)/2$ such that it has the same structure of elliptic points as $M$ , plus one additional elliptic point of order 2. By the Gauss–Bonnet formula, $\text{vol}(M)=\text{vol}(\widetilde{M}_{1})$ . We then deform the surface $\widetilde{M}_{1}$ in moduli space so that its shortest geodesic has the length equal to $\ell _{M,0}$ and the number of inconjugate geodesics of length $\ell _{M,0}$ is $m_{M,0}$ .

In § 9.2, we show that one can compare [14, Theorem 2.22] with generalization of the part (c) of the Main Theorem to higher derivatives in order to establish a simple asymptotic relation between $N_{\text{w}}(T;(Z_{M}H_{M})^{(k)})$ , $N_{\text{w}}(T;Z_{\widetilde{M}}^{(k)})$ for $k\geqslant 1$ and $N_{\text{w}}(T;H_{M})$ . Namely, we prove the following theorem.

Theorem 2. Let $M$ be a finite volume hyperbolic Riemann surface such that, in the notation of (9), $A_{M}=\exp (\ell _{M,0})$ . Then, for all integers $k\geqslant 1$

(13) $$\begin{eqnarray}N_{\text{w}}(T;(Z_{M}H_{M})^{(k)})=N_{\text{w}}(T;Z_{\widetilde{M}}^{(k)})+N_{\text{w}}(T;H_{M})+o(T)\quad \text{as }T\rightarrow \infty .\end{eqnarray}$$

We find it very interesting that, in the case when $M$ is such that $\exp (\ell _{M,0})<(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ , the coefficients of the first two terms, namely $T\log T$ and $T$ , in the asymptotic development of the counting function $N_{\text{w}}(T;(Z_{M}H_{M})^{(k)})$ for all $k\geqslant 1$ coincide with known results, namely Hejhal’s theorem and (2).

1.6 Further comments

Weyl’s law in its classical form evaluates the lead asymptotic behavior of the vertical counting function $N_{\text{ver}}(T;Z_{M})$ for compact $M$ . As far as is known, the expansion in $T$ involves $\text{vol}(M)$ and no other information associated to the uniformizing group $\unicode[STIX]{x1D6E4}$ . If $M$ is noncompact, the generalization of Weyl’s law addresses the asymptotic behavior of

(14) $$\begin{eqnarray}\#\{\unicode[STIX]{x1D706}_{j,M}<1/4+T^{2}\}-\frac{1}{4\unicode[STIX]{x1D70B}}\int _{-T}^{T}\unicode[STIX]{x1D719}_{M}^{\prime }/\unicode[STIX]{x1D719}_{M}(1/2+ir)\,dr\end{eqnarray}$$

where $\unicode[STIX]{x1D706}_{j,M}$ is the eigenvalue of an $L^{2}$ eigenfunction of the Laplacian on $M$ . The asymptotic expansion of (14) is recalled below (see (68)) and, as in the compact case, all terms in the expansion involve elementary quantities associated to the uniformizing group $\unicode[STIX]{x1D6E4}$ .

In § 9.1, we express the function in (14) in terms of $N_{\text{ver}}(T;Z_{M}H_{M})$ , obtaining an expression which involves the constant $\mathfrak{g}_{1}$ . As a result, we accept the appearance of the term $\mathfrak{g}_{2}/\mathfrak{g}_{1}$ in our Main Theorem as being an appropriate generalization of a version of Weyl’s law.

In a different direction, if one considers a degenerating family of finite volume hyperbolic Riemann surfaces, then it was shown in [15] that the asymptotic behavior of the associated sequence of vertical counting functions $N_{\text{ver}}(T;Z_{M})$ has lead asymptotic behavior, for fixed $T$ , which involves the lengths of the pinching geodesics; see [15, Theorem 5.5]. As a result, we do not view the appearance of the invariant $\ell _{M,0}$ in (1) and (2) as a new feature when using Weyl’s laws to understand refined information associated to the uniformizing group $\unicode[STIX]{x1D6E4}$ .

However, we find the appearance of the constants $A_{M}$ and $a_{M}$ , as defined in (9), (10) and (11) to be intriguing, specially since values of $A_{M}$ and $a_{M}$ are different for certain arithmetic groups of the same signature. In particular, for any given surface $M$ , we do not know if there are conditions which determine the value taken by $A_{M}$ . Consequently, we conclude that the study of the vertical counting function $N_{\text{ver}}(T;(Z_{M}H_{M})^{\prime })$ contains a term which provides new information associated to $\unicode[STIX]{x1D6E4}$ which we do not see as being previously detected. In other words, if we are allowed to view the vertical counting function $N_{\text{ver}}(T;(Z_{M}H_{M})^{\prime })$ as another measure of the spectral analysis of the Laplacian on $M$ , then our Main Theorem shows the existence of refined information, namely $A_{M}$ with its conditional definition (9), about the uniformizing group $\unicode[STIX]{x1D6E4}$ .

1.7 Computations for the modular group

After the completion of this article, W. Luo brought to our attention the unpublished article [23] from 2008 in which the author undertakes a related study in the case when $\unicode[STIX]{x1D6E4}=\text{ PSL}(2,\mathbb{Z})$ . There are a number of important differences between the results in the present paper and those in [23], which we now discuss.

In [23], as the title of the article states, the author studies the zeros of the derivative of the zeta function $Z_{M}(s)/\unicode[STIX]{x1D701}_{\mathbb{Q}}(2s)$ where $M=\text{ PSL}(2,\mathbb{Z})\backslash \mathbb{H}$ . If we restrict our analysis to the case when $\unicode[STIX]{x1D6E4}=\text{ PSL}(2,\mathbb{Z})$ , then the function whose derivative we study is $Z_{M}(s)\unicode[STIX]{x1D701}_{\mathbb{Q}}(2s-1)/\unicode[STIX]{x1D701}_{\mathbb{Q}}(2s)$ . Since the article [23] studies a different function than in the present article, one would expect that the statements of the main results are different, as, indeed, is the case. More importantly, however, the asymptotic expansions obtained in [23] has an error term of $O(T)$ , whereas our error term is $o(T)$ , which is significant since the coefficient of the $T$ term contains the quantity $A_{M}$ , which we view as a new spectral invariant. Finally, we note that the article [23] studies the single group $\unicode[STIX]{x1D6E4}=\text{ PSL}(2,\mathbb{Z})$ .

The approach presented in [23] raise the question if a similar modification of the Selberg zeta function $Z_{M}$ can occur in the general setting of the present paper. To do so, one can write the function $H_{M}(s)$ as a ratio $P_{M}(s)/Q_{M}(s)$ of two entire functions of order at most two and then consider $Z_{M}(s)/Q_{M}(s)$ . It is true that the function $Z_{M}(s)/Q_{M}(s)$ has nontrivial zeros only at zeros of $Z_{M}$ stemming from the discrete eigenvalues of the Laplacian. However, the Phillips–Sarnak conjecture/philosophy then asserts that for generic $M$ , the quotient $Z_{M}(s)/Q_{M}(s)$ would have a finite number of nontrivial zeros, hence can be written as a polynomial times Gamma-type functions. Aside from this assertion, while focusing solely from the point of view of application of techniques developed in this paper, the function $Z_{M}(s)/Q_{M}(s)$ is suitable in the sense that it possesses no nonreal zeros in the half-plane $\text{Re}(s)<1/2$ . However, there are two reasons why the investigation of $Z_{M}(s)H_{M}(s)$ is more natural. Firstly, the spectral information carried by $P_{M}(s)$ , namely, the zeros of the scattering determinant, is lost when considering $Z_{M}(s)/Q_{M}(s)$ . More importantly, unless the explicit expression for the scattering determinant is known, it is very difficult to determine $Q_{M}(s)$ explicitly and hence express $Z_{M}(s)/Q_{M}(s)$ in terms of the information related to the underlying group $\unicode[STIX]{x1D6E4}$ , whereas $Z_{M}(s)H_{M}(s)$ has a general Dirichlet series representation in terms of the group information.

1.8 Outline of the paper

This article is organized as follows. In § 2, we establish notation and recall necessary results from the literature. The zero-free region for $(Z_{M}H_{M})^{\prime }$ , as stated in part (a) of the Main Theorem, will be proved in § 3. Various lemmas leading up to the proof of parts (b) and (c) of the main Theorem will be given in § 4, the proof of parts (b) and (c) will be completed in § 5, and in § 6 we state and prove several corollaries of the Main Theorem. The examples of congruence groups and “moonshine” groups will be given in § 7. In § 8, we prove results analogous to our Main Theorem for higher derivatives of $Z_{M}H_{M}$ . Finally, in § 9, we give various concluding remarks.

2 Background material

2.1 Counting functions

Let $F$ denote either a general Dirichlet series with a critical line; $F$ itself may be the derivative of another general Dirichlet series. We assume that $F$ is normalized to be convergent in the half-plane $\text{Re}(s)>1$ with critical line $\text{Re}(s)=1/2$ . We define the vertical counting function of $F$ as

$$\begin{eqnarray}N_{\text{ver}}(T;F)=\mathop{\sum }_{{F(\unicode[STIX]{x1D70E}+it)=0\atop 0<t<T,0<\unicode[STIX]{x1D70E}<1}}1\end{eqnarray}$$

and the weighted vertical counting function of $F$ as

$$\begin{eqnarray}N_{\text{w}}(T;F)=\mathop{\sum }_{{F(\unicode[STIX]{x1D70E}+it)=0\atop 0<t<T,1/2<\unicode[STIX]{x1D70E}<1}}(\unicode[STIX]{x1D70E}-1/2).\end{eqnarray}$$

2.2 Additional identities

The logarithmic derivative $D_{M}(s):=\frac{Z_{M}^{\prime }}{Z_{M}}(s)$ of the Selberg zeta function may be expressed, for $\operatorname{Re}(s)>1$ as the absolutely convergent series

(15) $$\begin{eqnarray}D_{M}(s)=\mathop{\sum }_{P\in {\mathcal{H}}(\unicode[STIX]{x1D6E4})}\frac{\unicode[STIX]{x1D6EC}(P)}{N(P)^{s}},\quad \text{where }\unicode[STIX]{x1D6EC}(P):=\frac{\log N(P_{0})}{1-N(P)^{-1}}.\end{eqnarray}$$

Dirichlet series representation of the logarithmic derivative of the function $Z_{M}H_{M}$ is given by the following lemma.

Lemma 3. There exists a constant $\unicode[STIX]{x1D70E}_{0}^{\prime }\geqslant 1$ such that for all $s\in \mathbb{C}$ with $\operatorname{Re}(s)\geqslant \unicode[STIX]{x1D70E}_{0}^{\prime }+\unicode[STIX]{x1D716}>\unicode[STIX]{x1D70E}_{0}^{\prime }$ , we have that

(16) $$\begin{eqnarray}\frac{(Z_{M}H_{M})^{\prime }}{(Z_{M}H_{M})}(s)=\underset{P\in {\mathcal{H}}(\unicode[STIX]{x1D6E4})}{\sum }\frac{\unicode[STIX]{x1D6EC}(P)}{N(P)^{s}}+\mathop{\sum }_{i=1}^{\infty }\frac{b\left(q_{i}\right)}{\mathop{q}_{i}^{s}}.\end{eqnarray}$$

In addition, the series converge absolutely and uniformly on every compact subset of the half-plane $\operatorname{Re}(s)>\unicode[STIX]{x1D70E}_{0}^{\prime }$ .

Proof. Equation (16) follows immediately from (8) and (15). We may take $\unicode[STIX]{x1D70E}_{0}^{\prime }$ to be equal to $\unicode[STIX]{x1D70E}_{0}$ , which was defined in § 1.4.◻

Lemma 4. The derivative of the function $Z_{M}H_{M}$ satisfies the functional equation

(17) $$\begin{eqnarray}(Z_{M}H_{M})^{\prime }(s)=f_{M}(s)\unicode[STIX]{x1D702}_{M}(s)K_{M}^{-1}(s)\widetilde{Z}_{M}(1-s)Z_{M}(1-s),\end{eqnarray}$$

where

(18) $$\begin{eqnarray}f_{M}(s):=\text{vol}(M)(1/2-s)\left(\tan \unicode[STIX]{x1D70B}(1/2-s)\right)\end{eqnarray}$$

and

(19) $$\begin{eqnarray}\widetilde{Z}_{M}(s):=\frac{1}{f_{M}(s)}\left(\frac{\unicode[STIX]{x1D702}_{M}^{\prime }}{\unicode[STIX]{x1D702}_{M}}(s)-\frac{K_{M}^{\prime }}{K_{M}}(1-s)-\frac{Z_{M}^{\prime }}{Z_{M}}(s)\right).\end{eqnarray}$$

Proof. Straightforward computation stemming from (7).◻

2.3 An integral representation for $D_{M}(s)$

In this section, we recall results from [3] on the growth of the logarithmic derivative $D_{M}(s)$ and its derivatives $D_{M}^{(k)}(s)$ for $s=1/2+\unicode[STIX]{x1D70E}+iT$ , as $T\rightarrow \pm \infty$ , for $\unicode[STIX]{x1D70E}\in (0,1/2)$ .

Theorem 5. [3]

For $s=1/2+\unicode[STIX]{x1D70E}+iT$ , $0<\unicode[STIX]{x1D70E}<1/2$ and every nonnegative integer $k$ , we have the asymptotic bound

(20) $$\begin{eqnarray}\displaystyle D_{M}^{(k)}(s) & = & \displaystyle O\left(\min \left\{\frac{|T|}{\unicode[STIX]{x1D70E}^{k+1}\log |T|},|T|^{1-2\unicode[STIX]{x1D70E}}\log ^{k-2\unicode[STIX]{x1D70E}}|T|\right.\right.\nonumber\\ \displaystyle & & \displaystyle \left.\left.\cdot \,\,\underset{j=0,\ldots ,k}{\max }\left\{\frac{1}{\unicode[STIX]{x1D70E}^{j+1}\log ^{j+1}|T|},\log \left|\frac{T}{\unicode[STIX]{x1D70E}}\right|\right\}\right\}\right),\end{eqnarray}$$

as $|T|\rightarrow \infty$ .

2.4 The completed function $\unicode[STIX]{x1D6EF}_{M}$

In this section, we recall the notation and results from [7]. The notation of [7] is adjusted to our setting; we take $k=0$ , dimension $d=1$ and $\unicode[STIX]{x1D70F}^{\ast }=n_{1}$ .

The completed function $\unicode[STIX]{x1D6EF}_{M}$ associated to the Selberg zeta function is defined by

$$\begin{eqnarray}\unicode[STIX]{x1D6EF}_{M}(s)=\unicode[STIX]{x1D6EF}_{I}(s)\unicode[STIX]{x1D6EF}_{M,\text{hyp}}(s)\unicode[STIX]{x1D6EF}_{M,\text{par}}(s)\unicode[STIX]{x1D6EF}_{M,\text{ell}}(s)\end{eqnarray}$$

where $\unicode[STIX]{x1D6EF}_{M,\text{hyp}}(s)=Z_{M}(s)$ is the Selberg zeta function and the remaining functions are associated to the identity, parabolic, and elliptic elements in the underlying uniformizing group. The logarithmic derivative of the identity term $\unicode[STIX]{x1D6EF}_{I}$ is given by

(21) $$\begin{eqnarray}-\frac{1}{2s-1}\frac{\unicode[STIX]{x1D6EF}_{I}^{\prime }(s)}{\unicode[STIX]{x1D6EF}_{I}(s)}=\frac{\text{vol}(M)}{2\unicode[STIX]{x1D70B}}\frac{\unicode[STIX]{x1D6E4}^{\prime }}{\unicode[STIX]{x1D6E4}}(s);\end{eqnarray}$$

see [7, Remark 3.1.3]. The function $\unicode[STIX]{x1D6EF}_{M,\text{ell}}(s)$ is computed in [7, Corollary 2.3.5]; using Stirling’s formula, one can show that

(22) $$\begin{eqnarray}\frac{1}{2s-1}\frac{\unicode[STIX]{x1D6EF}_{M,\text{ell}}^{\prime }(s)}{\unicode[STIX]{x1D6EF}_{M,\text{ell}}(s)}=O\left(\frac{1}{\left|t\right|}\log \left|t\right|\right)\end{eqnarray}$$

for any $s=\unicode[STIX]{x1D70E}+it$ , $\unicode[STIX]{x1D70E}\leqslant 1/2$ , as $|t|\rightarrow \infty$ .

The function $\unicode[STIX]{x1D6EF}_{M,\text{par}}(s)$ is described in [7, Definition 3.1.4]. For our purposes, it suffices to relate $\unicode[STIX]{x1D6EF}_{M,\text{par}}(s)$ to the scattering determinant $\unicode[STIX]{x1D719}_{M}(s)$ , so that we obtain an expression for $Z_{M}H_{M}(s)$ . The following computations derive such an expression for $Z_{M}H_{M}(s)$ .

Let $\{p_{1},\ldots ,p_{N_{0}}\}$ denote the set of poles of $\unicode[STIX]{x1D719}_{M}$ lying in $(\!1/2,1]$ , counted with multiplicities; let $q_{1},\ldots ,q_{N_{1}}$ denote the set of real zeros of $\unicode[STIX]{x1D719}_{M}$ larger than $1/2$ and let $\{q_{n}\}_{n>N_{1}}$ denote the set of zeros of $\unicode[STIX]{x1D719}_{M}$ with positive imaginary parts, counted with multiplicities. In the notation of [7, Definition 3.2.2], we set ${\mathcal{P}}_{M}\equiv 1$ if $n_{1}=0$ , otherwise we define ${\mathcal{P}}_{M}(s):=f_{1}(s)f_{2}(s)$ where

$$\begin{eqnarray}f_{1}(s):=\mathop{\prod }_{n=1}^{N_{1}}\left(1+\frac{s-1/2}{q_{n}-1/2}\right)\exp \left[\frac{1}{2}\left(\frac{s-1/2}{q_{n}-1/2}\right)^{2}\right]\end{eqnarray}$$

and

$$\begin{eqnarray}\displaystyle f_{2}(s) & := & \displaystyle \mathop{\prod }_{n\geqslant N_{1}+1}\left(1+\frac{s-1/2}{q_{n}-1/2}\right)\left(1+\frac{s-1/2}{\overline{q_{n}}-1/2}\right)\nonumber\\ \displaystyle & & \displaystyle \times \,\exp \left[\frac{1}{2}\left(\frac{s-1/2}{q_{n}-1/2}\right)^{2}+\frac{1}{2}\left(\frac{s-1/2}{\overline{q_{n}}-1/2}\right)^{2}\right].\nonumber\end{eqnarray}$$

The infinite product which defines $f_{2}$ converges uniformly on compact subsets of $\mathbb{C}$ and defines an entire function of finite order.

Lemma 6. For all $s\in \mathbb{C}$ , the product $(\unicode[STIX]{x1D6EF}_{M}{\mathcal{P}}_{M})(1-s)$ can be expressed as

(23) $$\begin{eqnarray}\displaystyle \left(\unicode[STIX]{x1D6EF}_{M}{\mathcal{P}}_{M}\right)(1-s) & = & \displaystyle (Z_{M}H_{M})(s)\cdot \unicode[STIX]{x1D6EF}_{I}(s)\cdot \unicode[STIX]{x1D6EF}_{M,\text{ell}}(s)\cdot \frac{\unicode[STIX]{x1D70B}^{n_{1/2}}d(1)}{\unicode[STIX]{x1D719}_{M}(1/2)}\mathfrak{g}_{1}^{-s-1}\nonumber\\ \displaystyle & & \displaystyle \cdot \,\left(s-\frac{1}{2}\right)^{(1/2)\text{ Tr}(I_{n_{1}}-\unicode[STIX]{x1D6F7}_{M}(1/2))-n_{1}}\nonumber\\ \displaystyle & & \displaystyle \cdot \,\,\unicode[STIX]{x1D6E4}(s)^{-n_{1}}\overset{N_{0}}{\underset{m=1}{\prod }}\left(\frac{s-p_{m}}{1/2-p_{m}}\right)\!.\end{eqnarray}$$

Proof. From the functional equation [7, (3.2.4), p. 123], we have, for all $s\in \mathbb{C}$

(24) $$\begin{eqnarray}\left(\unicode[STIX]{x1D6EF}_{M}{\mathcal{P}}_{M}\right)(1-s)=\left(\unicode[STIX]{x1D6EF}_{M}{\mathcal{P}}_{M}\right)(s)\mathfrak{g}_{1}^{2s-1}\overset{N_{0}}{\underset{m=1}{\prod }}\left(\frac{s-p_{m}}{1-s-p_{m}}\right)\frac{1}{\unicode[STIX]{x1D719}_{M}(1/2)}\unicode[STIX]{x1D719}_{M}(s).\end{eqnarray}$$

On the other hand, by [7, Corollary 2.4.22] it is easy to see that

$$\begin{eqnarray}\displaystyle \left(\unicode[STIX]{x1D6EF}_{M,\text{par}}{\mathcal{P}}_{M}\right)(s) & = & \displaystyle (s-1/2)^{(1/2)\text{ Tr}I_{n_{1}}-\unicode[STIX]{x1D6F7}_{M}(1/2)}\mathfrak{g}_{1}^{-s}\left(\frac{1}{\unicode[STIX]{x1D6E4}(s+1/2)}\right)^{n_{1}}\nonumber\\ \displaystyle & & \displaystyle \cdot \,\overset{N_{0}}{\underset{m=1}{\prod }}\left(1+\frac{s-1/2}{p_{m}-1/2}\right).\nonumber\end{eqnarray}$$

We now write $\unicode[STIX]{x1D719}_{M}$ as

$$\begin{eqnarray}\unicode[STIX]{x1D719}_{M}(s)=\unicode[STIX]{x1D70B}^{n_{1}/2}\mathfrak{g}_{1}^{-2s}d(1)(s-1/2)^{-n_{1}}\left(\frac{\unicode[STIX]{x1D6E4}(s+1/2)}{\unicode[STIX]{x1D6E4}(s)}\right)^{n_{1}}H_{M}(s).\end{eqnarray}$$

The result follows through direct and straightforward computations involving the definition of $\unicode[STIX]{x1D6EF}_{M}$ together with (24).◻

2.5 On generalized Backlund equivalent for the Lindelöf hypothesis

An important ingredient in the proof of the Main Theorem is a bound on the growth of the function $Z_{M}H_{M}$ on the critical line $\text{ Re}(s)=1/2$ . We obtain the bound using a slight modification of [10, Proposition 2], which we now state.

Proposition 7. Let $f(s)$ be a meromorphic function for all $s\in \mathbb{C}$ which is holomorphic in the region $|\operatorname{Im}(s)|\geqslant t_{0}>0$ , for some fixed $t_{0}$ . Let $P(t):\mathbb{R}\rightarrow \mathbb{R}$ be nondecreasing function such that $P(t)\geqslant 2$ . Let $N(\unicode[STIX]{x1D70E},f,T)$ denote the number of zeros $\unicode[STIX]{x1D70C}$ of $f$ in the region $\operatorname{Re}(\unicode[STIX]{x1D70C})>\unicode[STIX]{x1D70E}$ ; $0\leqslant \operatorname{Im}(\unicode[STIX]{x1D70C})\leqslant T$ .

Assume there exist constants $\unicode[STIX]{x1D70E}_{0}>1/2$ and $\unicode[STIX]{x1D714}>0$ such that for $\unicode[STIX]{x1D70E}_{0}-\unicode[STIX]{x1D714}\leqslant \operatorname{Re}(s)\leqslant \unicode[STIX]{x1D70E}_{0}+\unicode[STIX]{x1D714}$ we have

$$\begin{eqnarray}|f(s)|\geqslant c>0\qquad \text{and},\qquad (f^{\prime }/f)(s)=o(P(t))\quad \text{as}\quad t=\text{Im}(s)\rightarrow \infty .\end{eqnarray}$$

Furthermore, assume that $|f(s)|>0$ for $\operatorname{Re}(s)\geqslant \unicode[STIX]{x1D70E}_{0}+\unicode[STIX]{x1D714}$ and that for some fixed number $D>0$ we have

$$\begin{eqnarray}f(s)=O((P(t))^{D})\quad \text{ as }t=\text{ Im}(s)\rightarrow \infty ,\text{ uniformly for }\operatorname{Re}(s)\geqslant 2-3\unicode[STIX]{x1D70E}_{0}.\end{eqnarray}$$

Then if the estimate $N(\unicode[STIX]{x1D70E},f,T+1)-N(\unicode[STIX]{x1D70E},f,T)=o(P(T))$ holds true for all $\unicode[STIX]{x1D70E}\geqslant 1/2$ as $T\rightarrow \infty$ , then $f(1/2+it)=O_{\unicode[STIX]{x1D716}}((P(t))^{\unicode[STIX]{x1D716}})$ as $t\rightarrow \infty$ .

There are slight differences between our statement above and [10, Proposition 2]. Firstly, we assume the function $f$ depends on a single complex variable $s$ , not necessarily a member of a family of meromorphic functions. Secondly, the author of [10] assumes that $f$ has a finite number of poles which lie in a compact set. A review of the proof [10, Proposition 2] reveals that the argument is based on Landau’s theorem (see [10, Lemma 8]) and Hadamard’s three circles theorem (see [10, Lemma 9]). These classical results are applied to the function $f(s)$ in the neighborhood $|s-s_{0}|\leqslant 2(\unicode[STIX]{x1D70E}_{0}-1/2-\unicode[STIX]{x1D6FF})$ of the point $s_{0}=\unicode[STIX]{x1D70E}_{0}+iT$ for sufficiently large $T$ . The proof given in [10] carries through without any changes whatsoever under the assumptions we state above.

We refer the reader to [10] for the proof and various interesting generalizations of Proposition 7.

3 Zeros in a half-plane $\operatorname{Re}(s)<1/2$

In this section, we prove part (a) of the Main Theorem. In fact, we prove more than stated, since our analysis will yield regions where each of the functions $\text{Re}((Z_{M}H_{M})^{\prime })$ and $\text{Im}((Z_{M}H_{M})^{\prime })$ are nonvanishing.

Proposition 8.

1. (a) For $\unicode[STIX]{x1D70E}<1/2$ , there exists $t_{0}>0$ , which may depend on $\unicode[STIX]{x1D70E}$ , such that

   $$\begin{eqnarray}\operatorname{Re}((Z_{M}H_{M})^{\prime }(\unicode[STIX]{x1D70E}+it))\neq 0\quad \text{ for all }t\text{ such that }|t|>t_{0}.\end{eqnarray}$$

2. (b) For every constant $C>0$ and arbitrary $-C<\unicode[STIX]{x1D70E}_{0}^{\prime }<1/2$ there are at most finitely many zeros of $(Z_{M}H_{M})^{\prime }(s)$ inside the strip $-C\leqslant \operatorname{Re}(s)\leqslant \unicode[STIX]{x1D70E}_{0}^{\prime }$ .

Proof. We first present the proof of part (a). By taking the logarithmic derivative of the functional equation (7) we get for $s=\unicode[STIX]{x1D70E}+it$ with $\unicode[STIX]{x1D70E}<1/2$ , the equation

(25) $$\begin{eqnarray}\frac{(Z_{M}H_{M})^{\prime }}{(Z_{M}H_{M})}(\unicode[STIX]{x1D70E}+it)=\frac{\unicode[STIX]{x1D702}_{M}^{\prime }}{\unicode[STIX]{x1D702}_{M}}(\unicode[STIX]{x1D70E}+it)-\frac{K_{M}^{\prime }}{K_{M}}(\unicode[STIX]{x1D70E}+it)-\frac{Z_{M}^{\prime }}{Z_{M}}(1-\unicode[STIX]{x1D70E}-it).\end{eqnarray}$$

From the definition (4) of $\unicode[STIX]{x1D702}_{M}^{\prime }/\unicode[STIX]{x1D702}_{M}$ and $K_{M}$ , one can use Stirling’s formula, together with the bound $0<\unicode[STIX]{x1D703}<\unicode[STIX]{x1D70B}$ , to show that

$$\begin{eqnarray}\displaystyle \text{Re}\left(\frac{\unicode[STIX]{x1D702}_{M}^{\prime }}{\unicode[STIX]{x1D702}_{M}}(\unicode[STIX]{x1D70E}+it)\right) & = & \displaystyle -\text{vol}(M)t+O(\log |t|)\qquad \text{and}\nonumber\\ \displaystyle \frac{K_{M}^{\prime }}{K_{M}}(\unicode[STIX]{x1D70E}+it) & = & \displaystyle O(\log |t|),\nonumber\end{eqnarray}$$

for $\unicode[STIX]{x1D70E}<1/2$ and as $|t|\rightarrow \infty$ . Therefore,

$$\begin{eqnarray}\text{Re}\left(-\frac{(Z_{M}H_{M})^{\prime }}{(Z_{M}H_{M})}(\unicode[STIX]{x1D70E}+it)\right)=\text{vol}(M)t+O(\log |t|)+\text{Re}\left(\frac{Z_{M}^{\prime }}{Z_{M}}(1-\unicode[STIX]{x1D70E}-it)\right)\!.\end{eqnarray}$$

Replacing $\unicode[STIX]{x1D70E}$ by $1/2-\unicode[STIX]{x1D70E}$ in (20) we get, for $\unicode[STIX]{x1D70E}<1/2$

$$\begin{eqnarray}\operatorname{Re}\left(-\frac{(Z_{M}H_{M})^{\prime }}{\left(Z_{M}H_{M}\right)}(\unicode[STIX]{x1D70E}+it)\right)=\text{vol}(M)t+O\left(\frac{|t|}{(1/2-\unicode[STIX]{x1D70E})\log |t|}\right),\end{eqnarray}$$

as $t\rightarrow \pm \infty$ . Therefore, there exists $t_{0}>0$ such that

$$\begin{eqnarray}\operatorname{Re}\left(\frac{(Z_{M}H_{M})^{\prime }}{(Z_{M}H_{M})}(\unicode[STIX]{x1D70E}+it)\right)\neq 0\quad \text{ for all }s=\unicode[STIX]{x1D70E}+it,\text{ with }|t|>t_{0}.\end{eqnarray}$$

On the other hand, the nontrivial zeros of the function $Z_{M}H_{M}$ are either nontrivial zeros $\unicode[STIX]{x1D70C}=\frac{1}{2}\pm ir_{n}$ of $Z_{M}$ or zeros $\unicode[STIX]{x1D70C}$ of $\unicode[STIX]{x1D719}_{M}$ . All except finitely many zeros of $\unicode[STIX]{x1D719}_{M}$ have real part bigger than $1/2$ ; therefore, $Z_{M}H_{M}(\unicode[STIX]{x1D70E}+it)\neq 0$ for $\unicode[STIX]{x1D70E}<1/2$ and $t>t_{0}$ . Therefore, we conclude that $\operatorname{Re}((Z_{M}H_{M})^{\prime }(\unicode[STIX]{x1D70E}+it))\neq 0$ for all $t>t_{0}$ . With all this, the proof of part (a) is complete.

To prove part (b), we employ Lemma 4. Recall the function $\widetilde{Z}_{M}(s)$ which is defined in (19). Let us write $\widetilde{Z}_{M}(s)=1+Z_{M,1}(s)$ . Then

(26) $$\begin{eqnarray}\displaystyle -f_{M}(s)Z_{M,1}(s) & = & \displaystyle \unicode[STIX]{x1D70B}\underset{0<\unicode[STIX]{x1D703}(R)<\unicode[STIX]{x1D70B}}{\underset{\left\{R\right\}}{\sum }}\frac{1}{M_{R}\sin \unicode[STIX]{x1D703}}\frac{\cos (2\unicode[STIX]{x1D703}-\unicode[STIX]{x1D70B})(s-1/2)}{\cos \unicode[STIX]{x1D70B}(s-1/2)}-2n_{1}\log 2\nonumber\\ \displaystyle & & \displaystyle -\,n_{1}\left(\frac{\unicode[STIX]{x1D6E4}^{\prime }}{\unicode[STIX]{x1D6E4}}(1/2+s)+\frac{\unicode[STIX]{x1D6E4}^{\prime }}{\unicode[STIX]{x1D6E4}}(3/2-s)\right.\nonumber\\ \displaystyle & & \displaystyle -\,\left.\frac{\unicode[STIX]{x1D6E4}^{\prime }}{\unicode[STIX]{x1D6E4}}(1/2-s)+\frac{\unicode[STIX]{x1D6E4}^{\prime }}{\unicode[STIX]{x1D6E4}}(1-s)\right)+\frac{Z_{M}^{\prime }}{Z_{M}}(s),\end{eqnarray}$$

where $f_{M}$ is defined in (18).

As in the proof of part (a), we can use Stirling’s formula and (20) to arrive at the bound

$$\begin{eqnarray}Z_{M,1}(\unicode[STIX]{x1D70E}_{1}+it)=O\left(\frac{(|t|\log |t|)^{2-2\unicode[STIX]{x1D70E}_{1}}}{(\unicode[STIX]{x1D70E}_{1}-1/2)|t|}\right)\quad \text{as }|t|\rightarrow \infty\end{eqnarray}$$

for $\unicode[STIX]{x1D70E}_{1}>1/2$ and $(\unicode[STIX]{x1D70E}_{1}+it)\in \mathbb{C}\backslash \cup _{n\in \mathbb{Z}}B_{n}$ where $B_{n}$ are small circles of fixed radius centered at integers. In particular, for $\unicode[STIX]{x1D70E}_{1}>1/2$ and $(\unicode[STIX]{x1D70E}_{1}+it)\in \mathbb{C}\backslash \cup _{n\in \mathbb{Z}}B_{n}$ , function $Z_{M,1}(\unicode[STIX]{x1D70E}_{1}+iT)$ is uniformly bounded in $T$ . Therefore, $\widetilde{Z}_{M}(1-s)=1+o(1)$ , as $\operatorname{Im}(s)\rightarrow \pm \infty$ in the strip $-C\leqslant \operatorname{Re}(s)\leqslant \unicode[STIX]{x1D70E}_{0}^{\prime }<1/2$ , hence $\widetilde{Z}_{M}(1-s)$ has finitely many zeros in this strip.

Since $Z_{M}(s)$ has only finitely many zeros for $\operatorname{Re}(s)>1/2$ , the function $\widetilde{Z}_{M}(1-s)Z_{M}(1-s)$ will have finitely many zeros in the strip $-C\leqslant \operatorname{Re}(s)\leqslant \unicode[STIX]{x1D70E}_{0}^{\prime }<1/2$ . Equation (17) then implies that the set of zeros of $(Z_{M}H_{M})^{\prime }(s)$ in the strip $-C\leqslant \operatorname{Re}(s)\leqslant \unicode[STIX]{x1D70E}_{0}^{\prime }<1/2$ is finite since the factor $\unicode[STIX]{x1D6F7}_{M}(s):=f_{M}(s)\unicode[STIX]{x1D702}_{M}(s)K_{M}^{-1}(s)$ of the functional equation (17) also has at most finitely many zeros in this strip.◻

We note that the zeros of $(Z_{M}H_{M})^{\prime }(s)$ which arise from zeros of $\unicode[STIX]{x1D6F7}_{M}$ are viewed as trivial zeros. They are located in the region $\operatorname{Re}(s)<1/2$ and arise at all negative integers.

The above method of examining zeros of the function $(Z_{M}H_{M})^{\prime }$ has the critical line as its limitation, since the bounds for the logarithmic derivative (20) hold true only in the half-plane $\operatorname{Re}(s)>1/2$ . In order to derive results valid on the critical line we need a representation on the critical line. Such a representation exists for the complete zeta function $\unicode[STIX]{x1D6EF}_{M}(s)$ (see § 2.4).

Proposition 9. There exists a number $t_{0}>0$ such that the following statements hold:

1. (a) $\frac{(Z_{M}H_{M})^{\prime }}{(Z_{M}H_{M})}(\unicode[STIX]{x1D70E}+it)\neq 0$ for all $\unicode[STIX]{x1D70E}<1/2$ and all $|t|>t_{0}$ ;

2. (b) $\frac{(Z_{M}H_{M})^{\prime }}{(Z_{M}H_{M})}(1/2\pm it)\neq 0$ for all $|t|>t_{0}$ , $t\neq r_{n}$ for all $n\geqslant 0$ .

Proof. Let $\unicode[STIX]{x1D70E}<0$ . By Proposition 8, there exists a constant $t_{0}^{\prime }>0$ such that for all $\unicode[STIX]{x1D70E}<0$ and all $|t|>t_{0}^{\prime }$ we have $\frac{(Z_{M}H_{M})^{\prime }}{(Z_{M}H_{M})}(\unicode[STIX]{x1D70E}+it)\neq 0$ . Therefore, it is enough to prove the statement when $0\leqslant \unicode[STIX]{x1D70E}<1/2$ . Without loss of generality, we assume that $t>0$ .

Taking the logarithmic derivatives of the both sides of the equation (23), we get

(27) $$\begin{eqnarray}\displaystyle \frac{1}{2s-1}\frac{(Z_{M}H_{M})^{\prime }}{(Z_{M}H_{M})}(s) & = & \displaystyle -\frac{1}{2s-1}\frac{\unicode[STIX]{x1D6EF}_{I}^{\prime }(s)}{\unicode[STIX]{x1D6EF}_{I}(s)}-\frac{1}{2s-1}\frac{\unicode[STIX]{x1D6EF}_{M,\text{ell}}^{\prime }(s)}{\unicode[STIX]{x1D6EF}_{M,\text{ell}}(s)}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{\log \mathfrak{g}_{1}}{2s-1}+\frac{n_{1}}{2s-1}\frac{\unicode[STIX]{x1D6E4}^{\prime }}{\unicode[STIX]{x1D6E4}}(s)\nonumber\\ \displaystyle & & \displaystyle +\,\frac{n_{1}-\frac{1}{2}\text{ Tr}\left(I_{n_{1}}-\unicode[STIX]{x1D6F7}(\frac{1}{2})\right)}{2(s-1/2)^{2}}-\frac{1}{2s-1}\mathop{\sum }_{m=1}^{N_{0}}\frac{1}{s-p_{m}}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{1}{2s-1}\frac{\left(\unicode[STIX]{x1D6EF}_{M}{\mathcal{P}}_{M}\right)^{\prime }}{\left(\unicode[STIX]{x1D6EF}_{M}{\mathcal{P}}_{M}\right)}(1-s),\end{eqnarray}$$

for all $s\in \mathbb{C}$ different from zeros and poles of $Z_{M}$ and $\unicode[STIX]{x1D719}_{M}$ . Applying [7, formula (3.4.1)] yields

$$\begin{eqnarray}\displaystyle \left(\unicode[STIX]{x1D6EF}_{M}{\mathcal{P}}_{M}\right)(s) & = & \displaystyle e^{Q(s)}(s-1/2)^{2d_{1/4}}\nonumber\\ \displaystyle & & \displaystyle \cdot \,\underset{n\geqslant 0\text{, }r_{n}\neq 0}{\prod }\left(1+\frac{(s-1/2)^{2}}{\mathop{r}_{n}^{2}}\right)\exp \left(-\frac{(s-1/2)^{2}}{\mathop{r}_{n}^{2}}\right)\nonumber\\ \displaystyle & & \displaystyle \cdot \,\,\overset{N_{1}}{\underset{n=1}{\prod }}\left(1-\frac{s-1/2}{\unicode[STIX]{x1D702}_{n}}\right)\exp \left(-\frac{s-1/2}{\unicode[STIX]{x1D702}_{n}}+\frac{(s-1/2)^{2}}{2\unicode[STIX]{x1D702}_{n}^{2}}\right)\nonumber\\ \displaystyle & & \displaystyle \cdot \,\underset{n\geqslant N_{1}+1}{\prod }\left(1+\frac{s-1/2}{\unicode[STIX]{x1D702}_{n}+i\unicode[STIX]{x1D6FE}_{n}}\right)\left(1+\frac{s-1/2}{\unicode[STIX]{x1D702}_{n}-i\unicode[STIX]{x1D6FE}_{n}}\right)\nonumber\\ \displaystyle & & \displaystyle \cdot \,\exp \left(-\frac{2\unicode[STIX]{x1D702}_{n}\left(s-\frac{1}{2}\right)}{\unicode[STIX]{x1D702}_{n}^{2}+\unicode[STIX]{x1D6FE}_{n}^{2}}+\left(s-\frac{1}{2}\right)^{2}\frac{\unicode[STIX]{x1D702}_{n}^{2}-\unicode[STIX]{x1D6FE}_{n}^{2}}{\left(\unicode[STIX]{x1D702}_{n}^{2}+\unicode[STIX]{x1D6FE}_{n}^{2}\right)^{2}}\right),\nonumber\end{eqnarray}$$

for all $s\in \mathbb{C}$ and where the notation is as follows: $\unicode[STIX]{x1D702}_{n}:=\operatorname{Re}(q_{n})$ ; $\unicode[STIX]{x1D6FE}_{n}:=\operatorname{Im}(q_{n})$ , $d_{1/4}$ is the multiplicity of $\unicode[STIX]{x1D706}=1/4$ as an eigenvalue; and $Q(s)=a_{2}(s-1/2)^{2}+a_{1}(s-1/2)+a_{0}$ for some constants $a_{i}$ , $i=0,1,2$ computed in [7]. The constants $a_{1}$ and $a_{2}$ are defined by [7, formulas (3.4.8) and (3.4.9)]. For our purposes it is important to know that $a_{1}$ and $a_{2}$ are real.

We now compute the logarithmic derivative of $(\unicode[STIX]{x1D6EF}_{M}{\mathcal{P}}_{M})(s)$ and substitute the expression into (27). After some elementary calculations, employing the Stirling formula and having in mind (21) and (22) we end up with

(28) $$\begin{eqnarray}\displaystyle \frac{1}{2s-1}\frac{(Z_{M}H_{M})^{\prime }}{(Z_{M}H_{M})}(s) & = & \displaystyle \frac{\text{vol}(M)}{2\unicode[STIX]{x1D70B}}\frac{\unicode[STIX]{x1D6E4}^{\prime }}{\unicode[STIX]{x1D6E4}}(s)+\underset{n\geqslant 0\text{, }r_{n}\neq 0}{\sum }\left(\frac{1}{(s-\frac{1}{2})^{2}+r_{n}^{2}}-\frac{1}{\mathop{r}_{n}^{2}}\right)\nonumber\\ \displaystyle & & \displaystyle +\,a_{2}+\underset{n\geqslant N_{1}+1}{\sum }\left[\frac{\mathop{\unicode[STIX]{x1D702}}_{n}^{2}-\unicode[STIX]{x1D6FE}_{n}^{2}}{\left(\mathop{\unicode[STIX]{x1D702}}_{n}^{2}+\unicode[STIX]{x1D6FE}_{n}^{2}\right)^{2}}\right.\nonumber\\ \displaystyle & & \displaystyle \left.+\,\frac{\unicode[STIX]{x1D6FE}_{n}^{2}-\unicode[STIX]{x1D702}_{n}^{2}+\unicode[STIX]{x1D702}_{n}(s-1/2)}{\big(\big(\unicode[STIX]{x1D702}_{n}-s+1/2\big)^{2}+\unicode[STIX]{x1D6FE}_{n}^{2}\big)\left(\unicode[STIX]{x1D702}_{n}^{2}+\unicode[STIX]{x1D6FE}_{n}^{2}\right)}\right]\nonumber\\ \displaystyle & & \displaystyle +\,\frac{1}{2}\underset{n=1}{\overset{N_{1}}{\sum }}\left(\frac{1}{\unicode[STIX]{x1D702}_{n}^{2}}-\frac{1}{\unicode[STIX]{x1D702}_{n}(\unicode[STIX]{x1D702}_{n}-s+1/2)}\right)+O\left(\frac{\log t}{t}\right),\end{eqnarray}$$

as $t=\operatorname{Im}(s)\rightarrow \infty$ . We now set $s=\unicode[STIX]{x1D70E}+it$ with $t>0$ and $0\leqslant \unicode[STIX]{x1D70E}<1/2$ . By computing the imaginary parts of both sides (28) we get

(29) $$\begin{eqnarray}\displaystyle & & \displaystyle \hspace{-3.99994pt}\text{Im}\left(\frac{1}{2\unicode[STIX]{x1D70E}-1+2it}\frac{(Z_{M}H_{M})^{\prime }}{(Z_{M}H_{M})}(\unicode[STIX]{x1D70E}+it)\right)\nonumber\\ \displaystyle & & \displaystyle \hspace{-3.99994pt}\quad =\frac{\text{vol}(M)}{2\unicode[STIX]{x1D70B}}\cdot \left[\frac{t}{\unicode[STIX]{x1D70E}^{2}+t^{2}}+\mathop{\sum }_{n=1}^{\infty }\frac{t}{(n+\unicode[STIX]{x1D70E})^{2}+t^{2}}\right]\nonumber\\ \displaystyle & & \displaystyle \hspace{-3.99994pt}\qquad +\,\underset{n\geqslant 0\text{, }r_{n}\neq 0}{\sum }\frac{t(1/2-\unicode[STIX]{x1D70E})}{\left((\unicode[STIX]{x1D70E}-1/2)^{2}-t^{2}+r_{n}^{2}\right)^{2}+4t^{2}(\unicode[STIX]{x1D70E}-1/2)^{2}}\nonumber\\ \displaystyle & & \displaystyle \hspace{-3.99994pt}\qquad +\,O\left(\frac{1}{t}\right)+O\left(\frac{\log t}{t}\right)\nonumber\\ \displaystyle & & \displaystyle \hspace{-3.99994pt}\qquad +\,\underset{n\geqslant N_{1}+1}{\sum }\frac{t\unicode[STIX]{x1D702}_{n}(3\unicode[STIX]{x1D6FE}_{n}^{2}-t^{2})-t\unicode[STIX]{x1D702}_{n}^{3}+t(1/2-\unicode[STIX]{x1D70E})(2\unicode[STIX]{x1D6FE}_{n}^{2}-2\unicode[STIX]{x1D702}_{n}^{2}-\unicode[STIX]{x1D702}_{n}(1/2-\unicode[STIX]{x1D70E}))}{\left[((\unicode[STIX]{x1D702}_{n}-\unicode[STIX]{x1D70E}+1/2)^{2}+\unicode[STIX]{x1D6FE}_{n}^{2}-t^{2})^{2}+4t^{2}(\unicode[STIX]{x1D702}_{n}+1/2-\unicode[STIX]{x1D70E})^{2}\right](\mathop{\unicode[STIX]{x1D702}}_{n}^{2}+\unicode[STIX]{x1D6FE}_{n}^{2})}.\end{eqnarray}$$

Since $0\leqslant \unicode[STIX]{x1D70E}<1/2$ we have that $(n+\unicode[STIX]{x1D70E})^{2}<(n+1/2)^{2}$ , for all $n\geqslant 0$ . Therefore

$$\begin{eqnarray}\frac{t}{\unicode[STIX]{x1D70E}^{2}+t^{2}}+\!\mathop{\sum }_{n=1}^{\infty }\frac{t}{(n+\unicode[STIX]{x1D70E})^{2}+t^{2}}>\frac{t}{1/4+t^{2}}+\!\mathop{\sum }_{n=1}^{\infty }\frac{t}{(n+1/2)^{2}+t^{2}}=\frac{\unicode[STIX]{x1D70B}}{2}\tanh (\unicode[STIX]{x1D70B}t).\end{eqnarray}$$

Furthermore, since $0<\unicode[STIX]{x1D702}_{n}<c$ , for some positive constant $c$ and all $n\geqslant 1$ and $\unicode[STIX]{x1D6FE}_{n}\rightarrow \infty$ , as $n\rightarrow \infty$ , by the choice of $\unicode[STIX]{x1D70E}$ we have that

$$\begin{eqnarray}(1/2-\unicode[STIX]{x1D70E})(2\unicode[STIX]{x1D6FE}_{n}^{2}-2\unicode[STIX]{x1D702}_{n}^{2}-\unicode[STIX]{x1D702}_{n}(1/2-\unicode[STIX]{x1D70E}))\geqslant 2\unicode[STIX]{x1D6FE}_{n}^{2}-2\unicode[STIX]{x1D702}_{n}^{2}-\unicode[STIX]{x1D702}_{n}\geqslant 0\end{eqnarray}$$

for all but finitely many $n\geqslant (N_{1}+1)$ . Let $n_{1}\geqslant (N_{1}+1)$ be an integer such that $2\unicode[STIX]{x1D6FE}_{n}^{2}-2\unicode[STIX]{x1D702}_{n}^{2}-\unicode[STIX]{x1D702}_{n}\geqslant 0$ for all $n\geqslant n_{1}$ . For simplicity, we introduce the notation

$$\begin{eqnarray}D(n,\unicode[STIX]{x1D70E},t)=\left((\unicode[STIX]{x1D702}_{n}-\unicode[STIX]{x1D70E}+1/2)^{2}+\unicode[STIX]{x1D6FE}_{n}^{2}-t^{2}\right)^{2}+4t^{2}(\unicode[STIX]{x1D702}_{n}+1/2-\unicode[STIX]{x1D70E})^{2}.\end{eqnarray}$$

From (29), we conclude the existence a constant $C_{1}>0$ and a positive number $t_{0}>t_{0}^{\prime }$ such that for all $t>t_{0}$ ,

(30) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \text{Im}\!\left(\frac{1}{2\unicode[STIX]{x1D70E}-1+2it}\frac{(Z_{M}H_{M})^{\prime }}{(Z_{M}H_{M})}(\unicode[STIX]{x1D70E}+it)\!\right)\nonumber\\ \displaystyle & & \displaystyle ~\displaystyle \quad \geqslant \frac{\text{vol}(M)}{4}\tanh (\unicode[STIX]{x1D70B}t)-C_{1}\frac{\log t}{t}+\underset{|\unicode[STIX]{x1D6FE}_{n}|<t/\sqrt{3}}{\sum }\frac{t\unicode[STIX]{x1D702}_{n}(3\unicode[STIX]{x1D6FE}_{n}^{2}-t^{2})}{D(n,\unicode[STIX]{x1D70E},t)(\mathop{\unicode[STIX]{x1D702}}_{n}^{2}+\unicode[STIX]{x1D6FE}_{n}^{2})}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\underset{N_{1}+1\leqslant n\leqslant n_{1}}{\sum }\frac{(1/2\,-\unicode[STIX]{x1D70E})t(2\mathop{\unicode[STIX]{x1D6FE}}_{n}^{2}\,-\,2\unicode[STIX]{x1D702}_{n}^{2}\,-\,\unicode[STIX]{x1D702}_{n})}{D(n,\unicode[STIX]{x1D70E},t)(\mathop{\unicode[STIX]{x1D702}}_{n}^{2}+\unicode[STIX]{x1D6FE}_{n}^{2})}+\underset{n\geqslant N_{1}+1}{\sum }\frac{-t\mathop{\unicode[STIX]{x1D702}}_{n}^{3}}{D(n,\unicode[STIX]{x1D70E},t)\left(\mathop{\unicode[STIX]{x1D702}}_{n}^{2}+\unicode[STIX]{x1D6FE}_{n}^{2}\right)}.\end{eqnarray}$$

Observe that each term in each summand in (30) is negative. We investigate separately each of the three sums on the right-hand side of (30).

Since $(\unicode[STIX]{x1D702}_{n}-\unicode[STIX]{x1D70E}+1/2)^{2}$ is bounded by some constant, by enlarging $t_{0}$ if necessary, we get that $D(n,\unicode[STIX]{x1D70E},t)\geqslant t^{4}/4$ for all $n$ such that $|\unicode[STIX]{x1D6FE}_{n}|<t/\sqrt{3}$ and all $t>t_{0}$ . Therefore

$$\begin{eqnarray}\displaystyle 0\leqslant \underset{|\unicode[STIX]{x1D6FE}_{n}|<t/\sqrt{3}}{\sum }\frac{t\unicode[STIX]{x1D702}_{n}(t^{2}-3\unicode[STIX]{x1D6FE}_{n}^{2})}{D(n,\unicode[STIX]{x1D70E},t)\left(\mathop{\unicode[STIX]{x1D702}}_{n}^{2}+\unicode[STIX]{x1D6FE}_{n}^{2}\right)}\leqslant \frac{2}{t}\underset{|\unicode[STIX]{x1D6FE}_{n}|<t/\sqrt{3}}{\sum }\frac{2\unicode[STIX]{x1D702}_{n}}{\left(\mathop{\unicode[STIX]{x1D702}}_{n}^{2}+\unicode[STIX]{x1D6FE}_{n}^{2}\right)}=O(1/t), & & \displaystyle \nonumber\end{eqnarray}$$

as $t\rightarrow \infty$ because the series $\sum _{n\geqslant N_{1}+1}2\unicode[STIX]{x1D702}_{n}(\unicode[STIX]{x1D702}_{n}^{2}+\unicode[STIX]{x1D6FE}_{n}^{2})^{-1}$ converges; see [7, Corollary 2.4.17].

For the second and the third sum in (30) we use the elementary inequality $D(n,\unicode[STIX]{x1D70E},t)\geqslant 4t^{2}\unicode[STIX]{x1D702}_{n}^{2}$ to deduce that both sums are $O(1/t)$ as $t\rightarrow \infty$ , hence all sums on the right-hand side of (30) are $O(1/t)$ , as $t\rightarrow \infty$ . Therefore, there exists a constant $C_{2}>0$ , such that for all $t>t_{0}$ , $t\neq r_{n}$ and $0\leqslant \unicode[STIX]{x1D70E}\leqslant 1/2$ one has

(31) $$\begin{eqnarray}\displaystyle \text{Im}\left(\frac{1}{2\unicode[STIX]{x1D70E}-1+2it}\frac{(Z_{M}H_{M})^{\prime }}{(Z_{M}H_{M})}(\unicode[STIX]{x1D70E}+it)\right)\geqslant \frac{\text{vol}(M)}{4}\cdot \tanh \unicode[STIX]{x1D70B}t-\frac{C_{1}\log t+C_{2}}{t}. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Since $\tanh \unicode[STIX]{x1D70B}t=1+O(e^{-\unicode[STIX]{x1D70B}t})$ , as $t\rightarrow \infty$ we conclude that statement (a) holds true.

We now prove part (b). We put $s=1/2+it$ for $t>0$ with $t\neq r_{n}$ in (29) to get

$$\begin{eqnarray}\displaystyle \text{Im}\left(\frac{1}{2it}\frac{\left(Z_{M}H_{M}\right)^{\prime }}{(Z_{M}H_{M})}\left(\frac{1}{2}+it\right)\right) & = & \displaystyle \frac{\text{vol}(M)}{4}\tanh (\unicode[STIX]{x1D70B}t)+O\left(\frac{\log t}{t}\right)\nonumber\\ \displaystyle & & \displaystyle +\,\underset{n\geqslant N_{1}+1}{\sum }\frac{t\unicode[STIX]{x1D702}_{n}(3\unicode[STIX]{x1D6FE}_{n}^{2}-t^{2})-t\unicode[STIX]{x1D702}_{n}^{3}}{D(n,1/2,t)\left(\mathop{\unicode[STIX]{x1D702}}_{n}^{2}+\unicode[STIX]{x1D6FE}_{n}^{2}\right)}.\nonumber\end{eqnarray}$$

Analogously as in the proof of part (a) we deduce that there exist a constant $t_{1}>0$ such that (31) holds true with $\unicode[STIX]{x1D70E}=1/2$ and some constants $C_{1}$ and $C_{2}$ , for all $t>t_{1}$ , $t\neq r_{n}$ . The proof of part (b) is complete.◻

We can now give a proof of part (a) of the Main Theorem.

The function $(Z_{M}H_{M})(s)$ has finitely many nontrivial zeros in the region $\operatorname{Re}(s)<1/2$ . Combining this statement with Proposition 9(a) immediately implies the existence a of constant $t_{0}$ such that $(Z_{M}H_{M})^{\prime }(\unicode[STIX]{x1D70E}+it)\neq 0$ for $\unicode[STIX]{x1D70E}<1/2$ and $|t|>t_{0}$ .

Proposition 9(b) yields that $\frac{(Z_{M}H_{M})^{\prime }}{(Z_{M}H_{M})}(1/2\pm it)\neq 0$ for $|t|>t_{0}$ , $t\neq r_{n}$ for all $n\geqslant 1$ . Therefore, the only zeros of $(Z_{M}H_{M})^{\prime }$ on the line $\text{Re}(s)=1/2$ , with at most a finite number of exceptions, are multiple zeros of $(Z_{M}H_{M})$ , or, equivalently, multiple zeros of $Z_{M}$ .

4 Preliminary lemmas

In this section, we prove some preliminary results needed in the proof of the Main Theorem. Let quantity $A_{M}$ , respectively $a_{M}$ , be defined by (9) and (10), respectively (11) and let $X_{M}(s)$ be defined by (12). Let $P_{00}\in {\mathcal{H}}(\unicode[STIX]{x1D6E4})$ denote the primitive hyperbolic element of $\unicode[STIX]{x1D6E4}$ with the property that $N(P_{00})=e^{\ell _{M,0}}$ ; or equivalently, with the property that $N(P_{00})=\min \{N(P):P\in {\mathcal{H}}(\unicode[STIX]{x1D6E4})\}$ . In the case when $A_{M}=e^{\ell _{M,0}}$ we may write $a_{M}$ in terms of the norm of $P_{00}$ , namely $a_{M}=m_{M,0}\unicode[STIX]{x1D6EC}(P_{00})$ .

Lemma 10. There exist $\unicode[STIX]{x1D70E}_{1}>1$ and a constant $0<c_{\unicode[STIX]{x1D6E4}}<1$ such that for $\unicode[STIX]{x1D70E}=\operatorname{Re}(s)\geqslant \unicode[STIX]{x1D70E}_{1}$ , we have the asymptotic formula

$$\begin{eqnarray}X_{M}(s)=1+O(c_{\unicode[STIX]{x1D6E4}}^{\unicode[STIX]{x1D70E}})\neq 0,\quad \text{as }\unicode[STIX]{x1D70E}\rightarrow +\infty .\end{eqnarray}$$

Proof. From the Euler product definition (3) of $Z_{M}$ and from (6), we have that

(32) $$\begin{eqnarray}Z_{M}(s)=1+O(N(P_{00})^{-\operatorname{Re}(s)})\qquad \text{and}\qquad H_{M}(s)=1+O(r_{2}^{-2\operatorname{Re}(s)})\end{eqnarray}$$

as $\operatorname{Re}(s)\rightarrow \infty$ . Furthermore, by the definition of $A_{M}$ and $a_{M}$

$$\begin{eqnarray}\underset{\{P\}\in {\mathcal{H}}(\unicode[STIX]{x1D6E4})}{\sum }\frac{\unicode[STIX]{x1D6EC}(P)}{N(P)^{s}}+\mathop{\sum }_{i=1}^{\infty }\frac{b\left(q_{i}\right)}{\mathop{q}_{i}^{s}}=\frac{a_{M}}{\mathop{A}_{M}^{s}}(1+O(\mathop{A}_{\unicode[STIX]{x1D6E4},1}^{-\operatorname{Re}(s)})),\end{eqnarray}$$

as $\operatorname{Re}(s)\rightarrow +\infty$ , for some constant $A_{\unicode[STIX]{x1D6E4},1}>1$ . Multiplying the formula (16) by $(Z_{M}H_{M})(s)$ and employing the equation (32) we complete the proof.◻

The following lemma provides the bound for the growth of the function $Z_{M,1}(s)$ ; recall that $Z_{M,1}(s)$ is defined by (26).

Lemma 11. Let $0<a<1/2$ be an arbitrary real number and let $\unicode[STIX]{x1D70E}\geqslant 1-a$ . Then

(33) $$\begin{eqnarray}\displaystyle \log \left|1+Z_{M,1}(\unicode[STIX]{x1D70E}\pm iT)\right| & = & \displaystyle O(|Z_{M,1}(\unicode[STIX]{x1D70E}\pm iT)|)=O(T^{2a-1}\log ^{2a}T),\nonumber\\ \displaystyle & & \displaystyle \text{as }T\rightarrow \infty .\end{eqnarray}$$

Proof. From the bound (20) with $k=0$ and $\unicode[STIX]{x1D70E}\geqslant 1-a$ , we get

$$\begin{eqnarray}\frac{Z_{M}^{\prime }}{Z_{M}}(\unicode[STIX]{x1D70E}\pm iT)=O((T\log T)^{1-2(\unicode[STIX]{x1D70E}-1/2)})=O((T\log T)^{2a}),\quad \text{as }T\rightarrow \infty ,\end{eqnarray}$$

where the implied constant depends only upon $M$ and $a$ . We can then argue in the same manner as in the proof of Proposition 8(b). Namely, applying Stirling’s formula and the above estimate, we get, for $s=\unicode[STIX]{x1D70E}\pm iT$ and $T\geqslant 1$ , the estimate

$$\begin{eqnarray}\left|\frac{1}{f_{M}(s)}\!\left(\frac{\unicode[STIX]{x1D702}_{M}^{\prime }}{\unicode[STIX]{x1D702}_{M}}(s)-\frac{K_{M}^{\prime }}{K_{M}}(1-s)-\frac{Z_{M}^{\prime }}{Z_{M}}(s)\right)-1\right|=O\left(\frac{\log T}{T}+\frac{(T\log T)^{2a}}{T}\right)\!,\end{eqnarray}$$

as $T\rightarrow \infty$ . This implies the bound (33) as claimed.◻

The following lemma is a Phragmén–Lindelöf type bound for $(Z_{M}H_{M})$ . The bound will be used to derive a similar bound for $(Z_{M}H_{M})^{\prime }$ using the Cauchy formula.

Lemma 12. Let $\unicode[STIX]{x1D70E}_{2}\geqslant 1$ be a fixed real number, such that $-\unicode[STIX]{x1D70E}_{2}$ is not a pole of $(Z_{M}H_{M})$ . Then, for an arbitrary $\unicode[STIX]{x1D6FF}>0$

1. (a)

   $$\begin{eqnarray}(Z_{M}H_{M})(\unicode[STIX]{x1D70E}+it)=O_{\unicode[STIX]{x1D6E4}}\left(\exp \left(1/2+\unicode[STIX]{x1D70E}_{2}+\unicode[STIX]{x1D6FF}\right)\text{vol}(M)t\right),\end{eqnarray}$$

2. (b)

   $$\begin{eqnarray}Z_{M}(\unicode[STIX]{x1D70E}+it)=O_{\unicode[STIX]{x1D6E4}}\left(\exp (1/2+\unicode[STIX]{x1D70E}_{2})\text{vol}(M)t\right)\end{eqnarray}$$
   for $t\geqslant 1$ , uniformly in $\unicode[STIX]{x1D70E}\leqslant -\unicode[STIX]{x1D70E}_{2}$ .

Proof. To prove part (a), we apply the Phragmen–Lindelöf theorem to the function

$$\begin{eqnarray}F(s)=(Z_{M}H_{M})(s)\exp \left[\text{vol}(M)(1/2+\unicode[STIX]{x1D70E}_{2}+\unicode[STIX]{x1D6FF})is\right]\end{eqnarray}$$

which is an entire function of finite order at most two in the sector $D:=\{\unicode[STIX]{x1D70B}/4\leqslant \arg (s+\unicode[STIX]{x1D70E}_{2})\leqslant \unicode[STIX]{x1D70B}/2\}$ . Obviously, $(Z_{M}H_{M})(s)=O(1)$ along the line $\arg (s+\unicode[STIX]{x1D70E}_{2})=\unicode[STIX]{x1D70B}/4$ , since $(Z_{M}H_{M})(\unicode[STIX]{x1D70E}+it)=O(1)$ , for $\unicode[STIX]{x1D70E}>\unicode[STIX]{x1D70E}_{1}$ and $t\geqslant 1$ ; see the proof of Lemma 10. Therefore,

$$\begin{eqnarray}\left|F(s)\right|=O\left(1\right)\quad \text{ along the line }\arg (s+\unicode[STIX]{x1D70E}_{2})=\unicode[STIX]{x1D70B}/4.\end{eqnarray}$$

To determine the behavior of the function $F(s)$ along the vertical line $\arg (s+\unicode[STIX]{x1D70E}_{2})=\unicode[STIX]{x1D70B}/2$ , that is, for $s=-\unicode[STIX]{x1D70E}_{2}+it$ , $t\geqslant 0$ , we use the functional equation (7) to get

(34) $$\begin{eqnarray}\displaystyle \left|F(-\unicode[STIX]{x1D70E}_{2}+it)\right| & = & \displaystyle \exp \left(-\left(1/2+\unicode[STIX]{x1D70E}_{2}+\unicode[STIX]{x1D6FF}\right)\text{vol}(M)t\right)\left|\unicode[STIX]{x1D702}(-\unicode[STIX]{x1D70E}_{2}+it)\right|\nonumber\\ \displaystyle & & \displaystyle \times \,\left|K_{M}^{-1}\left(-\unicode[STIX]{x1D70E}_{2}+it\right)\right|\cdot O(1),\end{eqnarray}$$

since $1+\unicode[STIX]{x1D70E}_{2}\geqslant 2$ . It remains to estimate $|\unicode[STIX]{x1D702}_{M}(-\unicode[STIX]{x1D70E}_{2}+it)|$ . Applying (4), [13, formula (4.4), p. 76], Stirling’s formula and the bound $0<\unicode[STIX]{x1D703}<\unicode[STIX]{x1D70B}$ , elementary computations show that

(35) $$\begin{eqnarray}\left|\unicode[STIX]{x1D702}_{M}(-\unicode[STIX]{x1D70E}_{2}+it)\right|=O\left(\exp \left(\text{vol}(M)(1/2+\unicode[STIX]{x1D70E}_{2})t+O(1)\right)\right)\quad \text{as }t\rightarrow +\infty .\end{eqnarray}$$

Formula (6.1.45) from [1], which itself is an application of Stirling’s formula, yields

(36) $$\begin{eqnarray}\left|K_{M}^{-1}\left(-\unicode[STIX]{x1D70E}_{2}+it\right)\right|=O\left(\exp \left(\frac{n_{1}}{2}\log t\right)\right)\quad \text{as }t\rightarrow +\infty .\end{eqnarray}$$

Substituting the bound (36) together with (35) into (34) we get

$$\begin{eqnarray}\left|F(-\unicode[STIX]{x1D70E}_{2}+it)\right|=O\left(\exp \left(-\text{vol}(M)\unicode[STIX]{x1D6FF}t+\frac{n_{1}}{2}\log t+O(\log (t))\right)\right)=o(1),\end{eqnarray}$$

as $t\rightarrow +\infty$ . One now can apply the Phragmen–Lindelöf theorem, which implies that $F(s)=O(1)$ in the sector $D:=\{\unicode[STIX]{x1D70B}/4\leqslant \arg (s+\unicode[STIX]{x1D70E}_{2})\leqslant \unicode[STIX]{x1D70B}/2\}$ and the proof of (a) is complete.

To prove (b), we repeat the proof given above for the function $G(s)=Z_{M}(s)\exp [(1/2+\unicode[STIX]{x1D70E}_{2})\text{vol}(M)is],$ which is an entire function of finite order in the sector $D:=\{\unicode[STIX]{x1D70B}/4\leqslant \arg (s+\unicode[STIX]{x1D70E}_{2})\leqslant \unicode[STIX]{x1D70B}/2\}$ . We omit the details.◻

The following lemma is a Lindelöf type bound for the function $Z_{M}H_{M}$ which will be used to deduce a sharper bound for the function $\arg X_{M}(\unicode[STIX]{x1D70E}+iT)$ , when $\unicode[STIX]{x1D70E}$ is close to $1/2$ .

Lemma 13. For $\unicode[STIX]{x1D716}>0$ and $t\geqslant 1$

$$\begin{eqnarray}(Z_{M}H_{M})(1/2+it)=O(\exp (\unicode[STIX]{x1D716}t))\quad \text{as }t\rightarrow +\infty .\end{eqnarray}$$

Proof. Since

$$\begin{eqnarray}|H_{M}(1/2+it)|=|\unicode[STIX]{x1D719}_{M}(1/2+it)||K_{M}^{-1}(1/2+it)|=O(\exp (n_{1}\log t/2)),\end{eqnarray}$$

as $t\rightarrow +\infty$ , it is enough to prove that $Z_{M}(1/2+it)=O(\exp (\unicode[STIX]{x1D716}t))$ as $t\rightarrow \infty$ .

We apply Proposition 7. In the notation of Proposition 7 we take

$$\begin{eqnarray}f(s)=Z_{M}(s)\quad \text{with }\unicode[STIX]{x1D70E}_{0}=\unicode[STIX]{x1D70E}_{1}+\unicode[STIX]{x1D714}>\unicode[STIX]{x1D70E}_{1}\qquad \text{and}\qquad P(t)=2\exp (t),\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}_{1}$ is defined in Lemma 10. Let us verify that all assumptions of Proposition 7 are fulfilled.

The function $Z_{M}$ is meromorphic function of finite order, with poles at points on the real line; see [14, p. 498]. Hence $Z_{M}(s)$ is holomorphic function for $|\operatorname{Im}(s)|\geqslant t_{0}>0$ , for any $t_{0}>0$ .

From the proof of Lemma 10 it is obvious that $|Z_{M}(s)|\geqslant c>0$ and $Z_{M}^{\prime }/Z_{M}(s)=O(1)$ as $t\rightarrow \infty$ , for $s=\unicode[STIX]{x1D70E}+it$ and with $\unicode[STIX]{x1D70E}_{0}-\unicode[STIX]{x1D714}\leqslant \unicode[STIX]{x1D70E}\leqslant \unicode[STIX]{x1D70E}_{0}+\unicode[STIX]{x1D714}$ . Furthermore, $|Z_{M}(s)|>0$ for $\text{ Re}(s)>\unicode[STIX]{x1D70E}_{0}+\unicode[STIX]{x1D714}$ .

From Lemma 12(b), we have that

$$\begin{eqnarray}Z_{M}(\unicode[STIX]{x1D70E}+it)=O_{\unicode[STIX]{x1D6E4}}(\exp (1/2+3\unicode[STIX]{x1D70E}_{0}-2)\text{vol}(M)t)=O_{\unicode[STIX]{x1D6E4}}(P(t)^{D}),\end{eqnarray}$$

for a fixed $D=(3\unicode[STIX]{x1D70E}_{0}-3/2)\text{vol}(M)$ , uniformly in $\unicode[STIX]{x1D70E}\geqslant 2-3\unicode[STIX]{x1D70E}_{0}$ .

Since $Z_{M}$ has no zeros in the half-plane $\text{ Re}(s)>1/2$ , the Lindelöf condition on the vertical distribution of zeros of $Z_{M}(s)$ in the half-plane $\text{ Re}(s)>1/2$ , as required in Proposition 7, is trivially fulfilled.

Therefore, all the assumptions of Proposition 7 are satisfied, hence $Z_{M}(1/2+it)=O(\exp (\unicode[STIX]{x1D716}t))$ as $t\rightarrow \infty$ .◻

Lemma 14. For an arbitrary $\unicode[STIX]{x1D716}>0$ , $t\geqslant 1$ and $\unicode[STIX]{x1D70E}_{2}$ defined in Lemma 12 we have

$$\begin{eqnarray}(Z_{M}H_{M})^{\prime }(\unicode[STIX]{x1D70E}+it)=\left\{\begin{array}{@{}ll@{}}O(\exp \unicode[STIX]{x1D716}t) & \quad \text{for }\frac{1}{2}\leqslant \unicode[STIX]{x1D70E}\leqslant \unicode[STIX]{x1D70E}_{0}\\ O(\exp (1/2-\unicode[STIX]{x1D70E}+\unicode[STIX]{x1D716})t) & \quad \text{for }-\unicode[STIX]{x1D70E}_{2}\leqslant \unicode[STIX]{x1D70E}<1/2,\end{array}\right.\end{eqnarray}$$

as $t\rightarrow \infty$ .

Proof. The proof involves an application of the Phragmen–Lindelöf theorem to the open sector bounded by the lines $\text{Re}(s)=-\unicode[STIX]{x1D70E}_{2}$ , $\text{ Re}(s)=\frac{1}{2}$ and $\text{ Im}(s)=1$ . The bounds to be used come from Lemma 12, with $\unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D716}$ , and from Lemma 13. A direct application of the Phragmen–Lindelöf theorem yields the bound

(37) $$\begin{eqnarray}(Z_{M}H_{M})(\unicode[STIX]{x1D70E}+it)=O(\exp (1/2-\unicode[STIX]{x1D70E}+\unicode[STIX]{x1D716})t),\end{eqnarray}$$

for $t\geqslant 1$ and $-\unicode[STIX]{x1D70E}_{2}\leqslant \unicode[STIX]{x1D70E}\leqslant 1/2$ . Similarly, for $\unicode[STIX]{x1D70E}_{0}$ defined as in Lemma 12, one can apply the Phragmen–Lindelöf theorem in the open sector bounded by the lines $\text{Re}(s)=\unicode[STIX]{x1D70E}_{0}$ , $\text{Re}(s)=\frac{1}{2}$ and $\text{ Im}(s)=1$ , from which one gets

(38) $$\begin{eqnarray}(Z_{M}H_{M})(\unicode[STIX]{x1D70E}+it)=O(\exp \unicode[STIX]{x1D716}t),\end{eqnarray}$$

for $1/2\leqslant \unicode[STIX]{x1D70E}\leqslant \unicode[STIX]{x1D70E}_{0}$ . The Cauchy integral formula can be applied, from which we have the equation

$$\begin{eqnarray}(Z_{M}H_{M})^{\prime }(s)=\frac{1}{2\unicode[STIX]{x1D70B}i}\int _{C}\frac{(Z_{M}H_{M})(z)}{(z-s)^{2}}\,dz\end{eqnarray}$$

where $C$ is a circle of a small, fixed radius $r<\unicode[STIX]{x1D716}$ , centered at $s=\unicode[STIX]{x1D70E}+it$ . Applying (38) to $(Z_{M}H_{M})(z)$ , when $1/2\leqslant \operatorname{Re}(z)\leqslant \unicode[STIX]{x1D70E}_{0}$ and (37) when $\operatorname{Re}(z)<1/2$ , we deduce that

$$\begin{eqnarray}(Z_{M}H_{M})^{\prime }(\unicode[STIX]{x1D70E}+it)=O\left(\exp ((r+\unicode[STIX]{x1D716})t)/r\right)=O(\exp (2\unicode[STIX]{x1D716}t))\end{eqnarray}$$

for $1/2\leqslant \unicode[STIX]{x1D70E}\leqslant \unicode[STIX]{x1D70E}_{0}$ and $t\geqslant 1$ . This proves the first part of the Lemma when replacing $\unicode[STIX]{x1D700}$ by $\unicode[STIX]{x1D700}/2$ .

In the case when $\unicode[STIX]{x1D70E}<1/2$ , we can use the functional equation for $(Z_{M}H_{M})^{\prime }$ to arrive at the expression

$$\begin{eqnarray}\displaystyle |(Z_{M}H_{M})^{\prime }(-\unicode[STIX]{x1D70E}_{2}+it)| & = & \displaystyle |\unicode[STIX]{x1D702}_{M}(-\unicode[STIX]{x1D70E}_{2}+it)||K_{M}^{-1}(-\unicode[STIX]{x1D70E}_{2}+it)||Z_{M}(1+\unicode[STIX]{x1D70E}_{2}-it)|\nonumber\\ \displaystyle & & \displaystyle \cdot \left|\frac{\unicode[STIX]{x1D702}_{M}^{\prime }}{\unicode[STIX]{x1D702}_{M}}(-\unicode[STIX]{x1D70E}_{2}+it)-\frac{K_{M}^{\prime }}{K_{M}}(-\unicode[STIX]{x1D70E}_{2}+it)\right.\nonumber\\ \displaystyle & & \displaystyle -\,\left.\frac{Z_{M}^{\prime }}{Z_{M}}(1+\unicode[STIX]{x1D70E}_{2}-it)\right|.\nonumber\end{eqnarray}$$

Since $\unicode[STIX]{x1D70E}_{2}\geqslant 1$ , we have $(Z_{M}^{\prime }/Z_{M})(1+\unicode[STIX]{x1D70E}_{2}-it)=O(1),$ as $t\rightarrow +\infty$ . Elementary computations involving the definition of the function $\unicode[STIX]{x1D702}_{M}^{\prime }/\unicode[STIX]{x1D702}_{M}$ and the Stirling formula imply that

$$\begin{eqnarray}\frac{\unicode[STIX]{x1D702}_{M}^{\prime }}{\unicode[STIX]{x1D702}_{M}}(-\unicode[STIX]{x1D70E}_{2}+it)-\frac{K_{M}^{\prime }}{K_{M}}(-\unicode[STIX]{x1D70E}_{2}+it)-\frac{Z^{\prime }}{Z}(1+\unicode[STIX]{x1D70E}_{2}-it)=O(t)\quad \text{ as }t\rightarrow \infty ;\end{eqnarray}$$

in brief, one sees the asymptotic bound by observing that the leading term in the above expression is $\text{vol}(M)(1/2+\unicode[STIX]{x1D70E}_{2}-it)\tan (\unicode[STIX]{x1D70B}(1/2+\unicode[STIX]{x1D70E}_{2}-it))$ . From the bounds (35) and (36) obtained in the proof of Lemma 12, we arrive at the bound

$$\begin{eqnarray}\left|(Z_{M}H_{M})^{\prime }(-\unicode[STIX]{x1D70E}_{2}+it)\right|=O\left(\exp \left(\left(1/2+\unicode[STIX]{x1D70E}_{2}+\unicode[STIX]{x1D716}\right)\text{vol}(M)t\right)\right)\quad \text{as }t\rightarrow \infty .\end{eqnarray}$$

The bound claimed in the statement of the Lemma follows by applying the Phragmen–Lindelöf theorem to the function $(Z_{M}H_{M})^{\prime }$ in the open sector bounded by the lines $\operatorname{Im}(s)=1$ , $\operatorname{Re}(s)=-\unicode[STIX]{x1D70E}_{2}$ and $\operatorname{Re}(s)=1/2$ , keeping in mind that $-\unicode[STIX]{x1D70E}_{2}\leqslant \unicode[STIX]{x1D70E}<1/2$ .◻

5 Vertical and weighted vertical distribution of zeros

In this section, we prove parts (b) and (c) of the Main Theorem.

We fix a large positive number $T$ and choose number $T^{\prime }$ such that $|T^{\prime }-T|=O(1)$ independently of $T$ where no zero of $Z_{M}H_{M}$ has imaginary part equal to $T^{\prime }$ . Let $t_{0}>0$ be a number such that $(Z_{M}H_{M})^{\prime }/(Z_{M}H_{M})$ $(\unicode[STIX]{x1D70E}+it)\neq 0$ for all $\unicode[STIX]{x1D70E}<1/2$ and $|t|>t_{0}$ ; the existence of such $t_{0}$ is established by Proposition 9. Let $\unicode[STIX]{x1D70E}_{0}\geqslant 1$ be a constant chosen so that $\unicode[STIX]{x1D70E}_{0}\geqslant \max \{\unicode[STIX]{x1D70E}_{0}^{\prime },\unicode[STIX]{x1D70E}_{1}\}$ , where $\unicode[STIX]{x1D70E}_{0}^{\prime }$ is defined in Lemma 3 and $\unicode[STIX]{x1D70E}_{1}$ is defined in Lemma 10. Let $0<a<1/2$ be arbitrary.

The function $X_{M}(s)$ , which was defined in (12), is holomorphic in the rectangle $R(a,T^{\prime })$ with vertices $a+it_{0}$ , $\unicode[STIX]{x1D70E}_{0}+it_{0}$ , $\unicode[STIX]{x1D70E}_{0}+iT^{\prime }$ and $a+iT^{\prime }$ . As in [22], we use Littlewood’s theorem from which we get the formula

(39) $$\begin{eqnarray}\displaystyle 2\unicode[STIX]{x1D70B}\!\underset{t_{0}<\unicode[STIX]{x1D6FE}<T^{\prime },\unicode[STIX]{x1D6FD}^{\prime }>a}{\underset{\unicode[STIX]{x1D70C}^{\prime }=\unicode[STIX]{x1D6FD}^{\prime }+i\unicode[STIX]{x1D6FE}}{\sum }}\!\!(\unicode[STIX]{x1D6FD}^{\prime }-a) & = & \displaystyle \int _{t_{0}}^{T^{\prime }}\!\!\log |X_{M}(a+it)|dt-\!\int _{t_{0}}^{T^{\prime }}\log |X_{M}(\unicode[STIX]{x1D70E}_{0}+it)|dt\nonumber\\ \displaystyle & & \displaystyle -\!\int _{a}^{\unicode[STIX]{x1D70E}_{0}}\!\arg X_{M}(\unicode[STIX]{x1D70E}+it_{0})\,d\unicode[STIX]{x1D70E}+\!\int _{a}^{\unicode[STIX]{x1D70E}_{0}}\!\arg X_{M}(\unicode[STIX]{x1D70E}+iT^{\prime })\,d\unicode[STIX]{x1D70E}\nonumber\\ \displaystyle & = & \displaystyle I_{1}-I_{2}-I_{3}+I_{4}.\end{eqnarray}$$

The variable $\unicode[STIX]{x1D70C}^{\prime }$ denotes a zero of $(Z_{M}H_{M})^{\prime }$ , and the integrals $I_{1}$ , $I_{2}$ , $I_{3}$ and $I_{4}$ are defined to be the four integrals in (39), in obvious notation. By Proposition 9, the condition that $\operatorname{Im}(\unicode[STIX]{x1D70C}^{\prime })>t_{0}$ implies that $\operatorname{Re}(\unicode[STIX]{x1D70C}^{\prime })\geqslant 1/2$ , hence the sum on the left-hand side of (39) is actually taken over all zeros of $(Z_{M}H_{M})^{\prime }$ with imaginary part in the interval $(t_{0},T^{\prime })$ .

We investigate integrals $I_{1}$ , $I_{2}$ , $I_{3}$ and $I_{4}$ separately.

Obviously, $I_{3}=O(1)$ as $T\rightarrow \infty$ since, in fact, $I_{3}$ is independent of $T$ . As for $I_{2}$ , the function $\log X_{M}$ is holomorphic and bounded in the infinite strip $\{s\in \mathbb{C}:t_{0}\leqslant \operatorname{Im}(s)\leqslant T^{\prime },\operatorname{Re}(s)\geqslant \unicode[STIX]{x1D70E}_{0}\}$ , hence following the argument from [22] we get that $I_{2}=O(1)$ as $T\rightarrow \infty$ .

The evaluation of $I_{4}$ closely follows the lines of the proof treating the analogous integral in the compact case considered by Garunkštis in [11], the new input being our Lemma 14. In order to show that $I_{4}=o(T)$ , it is sufficient to prove that

(40) $$\begin{eqnarray}\arg X(\unicode[STIX]{x1D70E}+iT^{\prime })=o(T)\quad \text{ for }a\leqslant \unicode[STIX]{x1D70E}\leqslant \unicode[STIX]{x1D70E}_{0}\quad \text{and}\quad \text{as }T\rightarrow \infty .\end{eqnarray}$$

The proof of (40) is very similar to the proof of [11, formula (3.4)], hence we omit the details. It remains to evaluate $I_{1}$ .

5.1 Evaluation of $I_{1}$

We shall break apart further $I_{1}$ by using the functional equation (17) for $(Z_{M}H_{M})^{\prime }$ , the definition (12) of $X_{M}$ , and representation of $\widetilde{Z}_{M}(s)=1+Z_{M,1}(s)$ which was used in the proof of Proposition 8(b). By doing so, we arrive at the expression

$$\begin{eqnarray}\displaystyle I_{1} & = & \displaystyle -\int _{t_{0}}^{T^{\prime }}\log \left|a_{M}A_{M}^{-(a+it)}\right|dt\nonumber\\ \displaystyle & & \displaystyle +\,\int _{t_{0}}^{T^{\prime }}\log |f_{M}(a+it)\unicode[STIX]{x1D702}_{M}(a+it)K_{M}^{-1}(a+it)|dt\nonumber\\ \displaystyle & & \displaystyle +\,\int _{t_{0}}^{T^{\prime }}\log \left|Z_{M}\left(1-(a+it)\right)\right|dt+\int _{t_{0}}^{T^{\prime }}\log \left|1+Z_{M,1}\left(1-(a+it)\right)\right|dt\nonumber\\ \displaystyle & & \displaystyle \qquad =I_{11}+I_{12}+I_{13}+I_{14},\nonumber\end{eqnarray}$$

with the obvious notation for the integrals $I_{11}$ , $I_{12}$ , $I_{13}$ and $I_{14}$ . Clearly, we have that

(41) $$\begin{eqnarray}I_{11}=-T\left(\log |a_{M}|-a\log A_{M}\right)+O(1)\quad \text{ as }T\rightarrow \infty .\end{eqnarray}$$

From the computations on the bottom of [22, p. 1146], we have that

(42) $$\begin{eqnarray}\int _{t_{0}}^{T^{\prime }}\!\!\log |f_{M}(a+it)|dt=T\log T+T(\log \text{vol}(M)-1)+O(\log T)\quad \text{as }T\rightarrow \infty ,\end{eqnarray}$$

hence

(43) $$\begin{eqnarray}\displaystyle I_{12} & = & \displaystyle T\log T+T\left(\log \text{vol}(M)-1\right)+\int _{t_{0}}^{T^{\prime }}\log \left|\unicode[STIX]{x1D702}_{M}(a+it)\right|dt\nonumber\\ \displaystyle & & \displaystyle +\,\int _{t_{0}}^{T^{\prime }}\log \left|K_{M}^{-1}(a+it)\right|dt+O(\log T)\nonumber\\ \displaystyle & = & \displaystyle T\log T+T(\log \text{vol}(M)-1)+I_{121}+I_{122}+O(\log T)\quad \text{as }T\rightarrow \infty ,\end{eqnarray}$$

with obvious notation for $I_{121}$ and $I_{122}$ . Stirling’s formula implies that

$$\begin{eqnarray}\displaystyle & & \displaystyle |K_{M}^{-1}(a+it)|=\unicode[STIX]{x1D70B}^{-n_{1}/2}\exp \left(-c_{1}a-\text{Re}(c_{2})\right)\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\exp \left[n_{1}\left(\frac{1}{2}\log \left|a-1/2+it\right|+O\left(\frac{1}{t}\right)\right)\right]\left(1+O\left(\frac{1}{t^{2}}\right)\right),\nonumber\end{eqnarray}$$

as $t\rightarrow \infty$ , where $c_{1}$ and $c_{2}$ are constants defined in (5). Therefore,

(44) $$\begin{eqnarray}\displaystyle I_{122} & = & \displaystyle \left(-c_{1}a-\text{Re}(c_{2})-\frac{n_{1}}{2}\log \unicode[STIX]{x1D70B}\right)T+\frac{n_{1}}{2}\int _{t_{0}}^{T^{\prime }}\!\!\log |a-1/2+it|+O(\log T)\nonumber\\ \displaystyle & = & \displaystyle \frac{n_{1}}{2}T\log T-T\left(c_{1}a+\text{Re}(c_{2})+\frac{n_{1}}{2}\log \unicode[STIX]{x1D70B}+\frac{n_{1}}{2}\right)+O(\log T)\nonumber\\ \displaystyle & & \displaystyle \quad \text{as }T\rightarrow \infty .\end{eqnarray}$$

As in the proof of Lemma 12, one can use (4) and Stirling’s formula to get

(45) $$\begin{eqnarray}I_{121}=\left(\frac{1}{2}-a\right)\frac{\text{vol}(M)}{2}T^{2}+2n_{1}(\log 2)(a-1/2)T+O(\log T)\quad \text{as }T\rightarrow \infty .\end{eqnarray}$$

By substituting (45) and (44) into (43), we arrive at

(46) $$\begin{eqnarray}\displaystyle I_{12} & = & \displaystyle \left(\frac{1}{2}-a\right)\frac{\text{vol}(M)}{2}T^{2}+\left(\frac{n_{1}}{2}+1\right)T\log T+O(\log T)\nonumber\\ \displaystyle & & \displaystyle +\,T\bigg[2n_{1}\log 2(a-1/2)-c_{1}a+\log \text{vol}(M)-1-\text{Re}(c_{2})\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\frac{n_{1}}{2}\log \unicode[STIX]{x1D70B}-\frac{n_{1}}{2}\bigg]\quad \text{as }T\rightarrow \infty .\end{eqnarray}$$

The integral $I_{13}$ is estimated by applying the Cauchy’s theorem to the function $\log Z_{M}(s)$ within in the rectangle with vertices $1-a-iT^{\prime }$ , $2-iT^{\prime }$ , $2-it_{0}$ and $1-a-it_{0}$ . As in [22], it is easily shown that

$$\begin{eqnarray}I_{13}=-\int _{1-a}^{2}\arg Z_{M}(\unicode[STIX]{x1D70E}-iT^{\prime })\,d\unicode[STIX]{x1D70E}+O(1)=O\left(\underset{1-a\leqslant \unicode[STIX]{x1D70E}\leqslant 2}{\max }\left|\log Z_{M}(\unicode[STIX]{x1D70E}-iT^{\prime })\right|\right).\end{eqnarray}$$

From

$$\begin{eqnarray}\log Z_{M}(\unicode[STIX]{x1D70E}-iT^{\prime })=\log Z_{M}(2-iT^{\prime })-\int _{\unicode[STIX]{x1D70E}-iT^{\prime }}^{2-iT^{\prime }}\frac{Z_{M}^{\prime }}{Z_{M}}(\unicode[STIX]{x1D709})\,d\unicode[STIX]{x1D709},\end{eqnarray}$$

and the bound in (20) we obtain the expression

(47) $$\begin{eqnarray}\displaystyle I_{13} & = & \displaystyle \int _{t_{0}}^{T^{\prime }}\log |Z_{M}(1-a-it)|dt=O((T\log T)^{2-2(1-a)})=O((T\log T)^{2a})\nonumber\\ \displaystyle & & \displaystyle \qquad \text{as }T\rightarrow \infty .\end{eqnarray}$$

Directly from Lemma 11, we have the estimate

(48) $$\begin{eqnarray}\displaystyle I_{14}=\int _{t_{0}}^{T^{\prime }}\log \left|1+Z_{M,1}\left(1-(a+it)\right)\right|dt=O\left((T\log T)^{2a}\right)\quad \text{ as }T\rightarrow \infty . & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Combining (41), (46), (47) and (48) yields

(49) $$\begin{eqnarray}\displaystyle I_{1} & = & \displaystyle \left(\frac{1}{2}-a\right)\frac{\text{vol}(M)}{2}T^{2}+\left(\frac{n_{1}}{2}+1\right)T\log T+O((T\log T)^{2a})+TC_{M,a}\nonumber\\ \displaystyle & & \displaystyle \quad \text{as }T\rightarrow \infty ,\end{eqnarray}$$

where

$$\begin{eqnarray}\displaystyle C_{M,a} & = & \displaystyle (a-1/2)\cdot 2n_{1}\log 2+a(\log A_{M}-c_{1})-\log |a_{M}|\nonumber\\ \displaystyle & & \displaystyle +\,\log \text{vol}(M)-1-\text{Re}(c_{2})-\frac{n_{1}}{2}\log \unicode[STIX]{x1D70B}-\frac{n_{1}}{2}.\nonumber\end{eqnarray}$$

Finally, we have arrived at our estimate for $I_{1}$ .

5.2 Proof of the Main Theorem

Since $0<a<1/2$ , we have that $(T\log T)^{2a}=o(T)$ . We have shown that $I_{2}$ and $I_{3}$ are $O(1)$ as $T\rightarrow \infty$ and that $I_{4}=o(T)$ as $T\rightarrow \infty$ . Hence, by substituting equation (49) into (39) we get

(50) $$\begin{eqnarray}\displaystyle 2\unicode[STIX]{x1D70B}\underset{t_{0}<\unicode[STIX]{x1D6FE}<T^{\prime }}{\underset{\unicode[STIX]{x1D70C}^{\prime }=\unicode[STIX]{x1D6FD}^{\prime }+i\unicode[STIX]{x1D6FE}}{\sum }}(\unicode[STIX]{x1D6FD}^{\prime }-a) & = & \displaystyle \left(\frac{1}{2}-a\right)\frac{\text{vol}(M)}{2}T^{2}\nonumber\\ \displaystyle & & \displaystyle +\,\left(\frac{n_{1}}{2}+1\right)T\log T+TC_{M,a}+o(T)\quad \text{ as }T\rightarrow \infty ,\end{eqnarray}$$

where $C_{M,a}$ is defined above.

Substituting $a/2$ instead of $a$ into (50), subtracting the obtained formulas, and then dividing by $a/2$ yields the statement (b) of the Main Theorem.

As for part (c) of the Main Theorem, we begin with the formula

(51) $$\begin{eqnarray}\underset{0<\unicode[STIX]{x1D6FE}\leqslant T}{\underset{\unicode[STIX]{x1D70C}^{\prime }=\unicode[STIX]{x1D6FD}^{\prime }+i\unicode[STIX]{x1D6FE}}{\sum }}(\unicode[STIX]{x1D6FD}^{\prime }-1/2)=\underset{0<\unicode[STIX]{x1D6FE}<T^{\prime }}{\underset{\unicode[STIX]{x1D70C}^{\prime }=\unicode[STIX]{x1D6FD}^{\prime }+i\unicode[STIX]{x1D6FE}}{\sum }}(\unicode[STIX]{x1D6FD}^{\prime }-a)+(a-1/2)\underset{0<\unicode[STIX]{x1D6FE}<T^{\prime }}{\underset{\unicode[STIX]{x1D70C}^{\prime }=\unicode[STIX]{x1D6FD}^{\prime }+i\unicode[STIX]{x1D6FE}}{\sum }}1\text{.}\end{eqnarray}$$

The first sum on the right-hand side of (51) is estimated by (50). The second sum on the right-hand side of (51) is estimated by part (b) of the Main Theorem, keeping in mind that the difference between the second sum in (51) and the sum in part (b) is the finite number of zeros in the half-plane $\text{ Re}(s)<1/2$ .

With all this, the proof of the Main Theorem is complete.

In the case when the surface is co-compact the statement of the Main Theorem is easily deduced, since, in that case $n_{1}=c_{1}=c_{2}=0$ , $H_{M}=1$ , $A_{M}=\exp (\ell _{M,0})$ and

$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D702}_{M}^{\prime }}{\unicode[STIX]{x1D702}_{M}}(s) & = & \displaystyle \text{vol}(M)(s-1/2)\tan (\unicode[STIX]{x1D70B}(s-1/2))\nonumber\\ \displaystyle & & \displaystyle -\,\unicode[STIX]{x1D70B}\underset{0<\unicode[STIX]{x1D703}(R)<\unicode[STIX]{x1D70B}}{\underset{\left\{R\right\}}{\sum }}\frac{1}{M_{R}\sin \unicode[STIX]{x1D703}}\frac{\cos (2\unicode[STIX]{x1D703}-\unicode[STIX]{x1D70B})(s-1/2)}{\cos \unicode[STIX]{x1D70B}(s-1/2)}.\nonumber\end{eqnarray}$$

6 Corollaries of the Main Theorem

In this section, we deduce three corollaries of our Main Theorem. The results we prove are analogous to [21, Theorem 2 and Theorem 3], with, in their notation, $k=1$ . Similar results may be deduced for the weighted vertical distribution of zeros of the $k$ th derivative, based on the results of § 8, with suitably replaced constants.

Corollary 15. For $\unicode[STIX]{x1D6FF}>1/2$ , let $N_{\text{ver}}(\unicode[STIX]{x1D6FF},T;(Z_{M}H_{M})^{\prime })$ denote the number of zeros $\unicode[STIX]{x1D70C}^{\prime }$ of $(Z_{M}H_{M})^{\prime }$ such that $\operatorname{Re}(\unicode[STIX]{x1D70C}^{\prime })>\unicode[STIX]{x1D6FF}$ and $0<\operatorname{Im}(\unicode[STIX]{x1D70C}^{\prime })<T$ . Then, for an arbitrary $\unicode[STIX]{x1D716}>0$

$$\begin{eqnarray}N_{\text{ver}}\left(\frac{1}{2}+\unicode[STIX]{x1D716},T;(Z_{M}H_{M})^{\prime }\right)<\frac{1}{\unicode[STIX]{x1D716}}N_{\text{w}}(T;(Z_{M}H_{M})^{\prime }).\end{eqnarray}$$

Proof. Trivially, we have the bounds

(52) $$\begin{eqnarray}\displaystyle N_{\text{ver}}\left(\frac{1}{2}+\unicode[STIX]{x1D716},T;(Z_{M}H_{M})^{\prime }\right) & {<} & \displaystyle \frac{1}{1/2+\unicode[STIX]{x1D716}}\underset{\unicode[STIX]{x1D70E}>1/2+\unicode[STIX]{x1D716}\text{, }0<t<T}{\underset{(Z_{M}H_{M})^{\prime }(\unicode[STIX]{x1D70E}+it)=0}{\sum }}\unicode[STIX]{x1D70E}\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{1/2+\unicode[STIX]{x1D716}}\underset{\unicode[STIX]{x1D70E}>1/2+\unicode[STIX]{x1D716}\text{, }0<t<T}{\underset{(Z_{M}H_{M})^{\prime }(\unicode[STIX]{x1D70E}+it)=0}{\sum }}\left(\unicode[STIX]{x1D70E}-\frac{1}{2}\right)\nonumber\\ \displaystyle & & \displaystyle +\,\frac{1/2}{1/2+\unicode[STIX]{x1D716}}N_{\text{ver}}\left(\frac{1}{2}+\unicode[STIX]{x1D716},T;(Z_{M}H_{M})^{\prime }\right)\end{eqnarray}$$

Therefore,

$$\begin{eqnarray}\frac{2\unicode[STIX]{x1D716}}{1+2\unicode[STIX]{x1D716}}N_{\text{ver}}\left(\frac{1}{2}+\unicode[STIX]{x1D716},T;(Z_{M}H_{M})^{\prime }\right)<\frac{2}{1+2\unicode[STIX]{x1D716}}N_{\text{w}}(T;(Z_{M}H_{M})^{\prime }),\end{eqnarray}$$

from which the result immediately follows.◻

Observe that the lead term in the asymptotic expansion in part (b) of the Main Theorem is $O(T^{2})$ , whereas the lead term in the asymptotic expansion in part (c) of the Main Theorem is $O(T\log (T))$ . Consequently, Corollary 15 shows that zeros of $(Z_{M}H_{M})^{\prime }$ are concentrated very close the critical line $\operatorname{Re}(s)=1/2$ . The following corollary further quantifies this observation.

Corollary 16. For any $\unicode[STIX]{x1D6FF}>1/2$ , let $N_{\text{ver}}^{-}(\unicode[STIX]{x1D6FF},T;(Z_{M}H_{M})^{\prime })$ denote the number of nontrivial zeros $\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70E}+it$ of $(Z_{M}H_{M})^{\prime }$ with $\unicode[STIX]{x1D70E}<\unicode[STIX]{x1D6FF}$ and $0<t<T$ . Then, for any constant $\unicode[STIX]{x1D716}>0$ ,

$$\begin{eqnarray}\underset{T\rightarrow \infty }{\lim }\frac{N_{\text{ver}}^{-}(1/2+\unicode[STIX]{x1D716},T;(Z_{M}H_{M})^{\prime })}{N_{\text{vert}}(T;(Z_{M}H_{M})^{\prime })}=1.\end{eqnarray}$$

Proof. Corollary 15 implies that

(53) $$\begin{eqnarray}1\geqslant \frac{N_{\text{ver}}^{-}(1/2+\unicode[STIX]{x1D716},T;(Z_{M}H_{M})^{\prime })}{N_{\text{vert}}(T;(Z_{M}H_{M})^{\prime })}>1-\frac{1}{\unicode[STIX]{x1D716}}\frac{N_{\text{w}}(T;(Z_{M}H_{M})^{\prime })}{N_{\text{vert}}(T;(Z_{M}H_{M})^{\prime })}.\end{eqnarray}$$

From the Main Theorem (b) and (c) we deduce that

$$\begin{eqnarray}\frac{N_{\text{w}}(T;(Z_{M}H_{M})^{\prime })}{N_{\text{vert}}(T;(Z_{M}H_{M})^{\prime })}\rightarrow 0\quad \text{ as }T\rightarrow \infty .\end{eqnarray}$$

Therefore, by passing to the limit as $T\rightarrow \infty$ in (53), the claimed result follows.◻

The following corollary gives estimates of short sums of distances $(\unicode[STIX]{x1D70E}-1/2)$ .

Corollary 17. Let $0<U<T$ . Then,

(54) $$\begin{eqnarray}\displaystyle 2\unicode[STIX]{x1D70B}\underset{\unicode[STIX]{x1D70E}>1/2,T<t\leqslant T+U}{\underset{(Z_{M}H_{M})^{\prime }(\unicode[STIX]{x1D70E}+it)=0}{\sum }}\left(\unicode[STIX]{x1D70E}-\frac{1}{2}\right) & = & \displaystyle \left(\frac{n_{1}}{2}+1\right)U\log (T+U)\nonumber\\ \displaystyle & & \displaystyle +\,\left(\log \frac{\mathfrak{g}_{1}\text{vol}(M)A_{M}^{1/2}}{\unicode[STIX]{x1D70B}^{n_{1}/2}|d(1)a_{M}|}\right)U\nonumber\\ \displaystyle & & \displaystyle +\,o(T)+O(U^{2}/T)\quad \text{ as }T\rightarrow \infty .\end{eqnarray}$$

Proof. The left-hand side of the (54) is equal to $2\unicode[STIX]{x1D70B}(N_{\text{w}}(T+U;(Z_{M}H_{M})^{\prime })-N_{\text{w}}(T;(Z_{M}H_{M})^{\prime })$ , hence part (c) of the Main Theorem yields

(55) $$\begin{eqnarray}\displaystyle 2\unicode[STIX]{x1D70B}\!\underset{\unicode[STIX]{x1D70E}>1/2,T<t\leqslant T+U}{\underset{(Z_{M}H_{M})^{\prime }(\unicode[STIX]{x1D70E}+it)=0}{\sum }}\!\!\left(\!\unicode[STIX]{x1D70E}-\frac{1}{2}\right) & = & \displaystyle \left(\frac{n_{1}}{2}+1\right)\!\!\left(\!T\,\text{log}\!\left(\!1+\frac{U}{T}\right)-U+U\log (T+U)\!\right)\nonumber\\ \displaystyle & & \displaystyle +\!\left(\log \frac{\mathfrak{g}_{1}\text{vol}(M)A_{M}^{1/2}}{\unicode[STIX]{x1D70B}^{n_{1}/2}|d(1)a_{M}|}\right)\!U+o(T)\quad \text{as }T\rightarrow \infty .\end{eqnarray}$$

The elementary observation that $T\log (1+U/T)-U=O(U^{2}/T)$ completes the proof.◻

7 Examples

The Main Theorem naturally leads to the following question: Are there examples of groups $\unicode[STIX]{x1D6E4}$ where $e^{\ell _{M,0}}<(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ as well as groups where $e^{\ell _{M,0}}>(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ ? The purpose of this section is to prove Proposition 1 and present examples of groups in each category. In fact, there are examples of both arithmetic and nonarithmetic groups in each category.

7.1 Congruence subgroups

In this subsection we prove part (i) of Proposition 1.

Let $\unicode[STIX]{x1D6E4}=\overline{\unicode[STIX]{x1D6E4}_{0}(N)}$ be the congruence subgroup defined by the arithmetic condition

$$\begin{eqnarray}\overline{\unicode[STIX]{x1D6E4}_{0}(N)}:=\left\{\left(\begin{array}{@{}cc@{}}a & b\\ c & d\end{array}\right)\in \text{SL}(2,\mathbb{Z}):c\equiv 0\,(\text{mod }N)\right\}\Big/\pm I,\end{eqnarray}$$

where $I$ denotes the identity matrix and $N$ is a positive integer. If $N=p_{1}\cdots p_{r}$ , for distinct primes $p_{1},\ldots ,p_{r}$ ; then, it is proved in [16], that the corresponding surface has $n_{1}=2^{r}$ cusps and the scattering determinant is given by the formula

$$\begin{eqnarray}\unicode[STIX]{x1D711}_{N}(s)=\left[\sqrt{\unicode[STIX]{x1D70B}}\frac{\unicode[STIX]{x1D6E4}(s-1/2)}{\unicode[STIX]{x1D6E4}(s)}\right]^{n_{1}}\left[\frac{\unicode[STIX]{x1D701}_{\mathbb{Q}}(2s-1)}{\unicode[STIX]{x1D701}_{\mathbb{Q}}(2s)}\right]^{n_{1}}\mathop{\prod }_{p\mid N}\left(\frac{1-p^{2-2s}}{1-p^{2s}}\right)^{n_{1}/2},\end{eqnarray}$$

where $\unicode[STIX]{x1D701}_{\mathbb{Q}}$ is the Riemann zeta function. Now, it is easy to show that $(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}=4$ . In the case $N$ is not square-free, an application of [14], formula (4.2), page 536 yields the same conclusion.

All elements of $\overline{\unicode[STIX]{x1D6E4}_{0}(N)}$ have integer entries, so any hyperbolic element has trace whose absolute value is at least equal to 3. Therefore, $e^{\ell _{M,0}}\geqslant u$ where $u$ is a solution to $u^{1/2}+u^{-1/2}=3$ . Solving, we get that $u=((3+\sqrt{5})/2)^{2}>4$ . Therefore, for any such group $\overline{\unicode[STIX]{x1D6E4}_{0}(N)}$ , one has that $e^{\ell _{M,0}}>(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ .

In the case of the principal congruence subgroups $\overline{\unicode[STIX]{x1D6E4}(N)}$ the scattering determinant can be computed using the analysis presented in [14] and [16]. As above, one shows that $(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}=4$ because the Dirichlet series portion of the scattering determinant is shown to be given by ratios of classical Dirichlet series. Furthermore, the matrices in $\overline{\unicode[STIX]{x1D6E4}(N)}$ also have integral entries, so $e^{\ell _{M,0}}\geqslant ((3+\sqrt{(5)})/2)^{2}>4$ .

7.2 Moonshine subgroups

We now prove part (ii) of Proposition 1.

Following [8], we use the term “moonshine group” for any subgroup $\unicode[STIX]{x1D6E4}$ of $\text{PSL}(2,\mathbb{R})$ which satisfies the following two conditions. First, there exists an integer $N\geqslant 1$ such that $\unicode[STIX]{x1D6E4}$ contains $\overline{\unicode[STIX]{x1D6E4}_{0}(N)}$ . Second, $\unicode[STIX]{x1D6E4}$ contains the element $(\begin{smallmatrix}1 & k\\ 0 & 1\end{smallmatrix})$ if and only if $k\in \mathbb{Z}$ .

Following [6, p. 363], let $f$ be a square-free, nonnegative integer, and consider the group

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}_{0}(f)^{+} & := & \displaystyle \left\{e^{-1/2}\left(\begin{array}{@{}cc@{}}a & b\\ c & d\end{array}\right)\in \text{SL}(2,\mathbb{R}):a,b,c,d,\right.\nonumber\\ \displaystyle & & \displaystyle \left.\qquad e\in \mathbb{Z},e\mid f,e\mid a,e\mid d,f\mid c,ad-bc=e\vphantom{\frac{1}{2}}\right\}.\nonumber\end{eqnarray}$$

In [6, Lemma 2.20] it is proved that the parabolic elements of $\unicode[STIX]{x1D6E4}_{0}(f)^{+}$ have integral entries. Therefore, $\unicode[STIX]{x1D6E4}_{0}(f)^{+}$ is a moonshine group. Let $\overline{\unicode[STIX]{x1D6E4}_{0}(f)^{+}}=\unicode[STIX]{x1D6E4}_{0}(f)^{+}/\pm I$ . In [20] it is proved that the Riemann surface $\overline{\unicode[STIX]{x1D6E4}_{0}(f)^{+}}\backslash \mathbb{H}$ for all square-free $f$ has finite volume and one cusp at infinity.

Consider the case when $f=5$ . The scattering matrix in this case has a single entry which, as proved in [20] is given by

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{5}(s)=\sqrt{\unicode[STIX]{x1D70B}}\frac{\unicode[STIX]{x1D6E4}(s-1/2)}{\unicode[STIX]{x1D6E4}(s)}\left(\frac{5^{s}+5}{5^{s}(5^{s}+1)}\right)\cdot \frac{\unicode[STIX]{x1D701}_{\mathbb{Q}}(2s-1)}{\unicode[STIX]{x1D701}_{\mathbb{Q}}(2s)},\end{eqnarray}$$

hence, one immediately can show that $(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}=4$ .

It is easy to confirm that $\unicode[STIX]{x1D6FE}=\Bigl(\begin{smallmatrix}0 & -1/\sqrt{5}\\ \sqrt{5} & \sqrt{5}\end{smallmatrix})\in \unicode[STIX]{x1D6E4}_{0}(5)^{+}$ . The trace of $\unicode[STIX]{x1D6FE}$ is $\sqrt{5}>2$ , hence $\unicode[STIX]{x1D6FE}$ is hyperbolic. Therefore, $e^{\ell _{M_{5},0}}\leqslant u$ where $u$ is a positive solution of $u^{1/2}+u^{-1/2}=\sqrt{5}$ . Solving, we have that $u=((1+\sqrt{5})/2)^{2}<4$ .

With all this, we have proved that $e^{\ell _{M_{5},0}}<(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ .

The surface $\overline{\unicode[STIX]{x1D6E4}_{0}(5)^{+}}\backslash \mathbb{H}$ has a signature (0;2,2,2;1) meaning that its genus is zero, it has three inequivalent elliptic points of order two and one cusp. The surface $\overline{\unicode[STIX]{x1D6E4}_{0}(6)^{+}}\backslash \mathbb{H}$ has the same signature, as shown in [6, Table C]. The scattering matrix in this case has a single entry which is given by

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{6}(s)=\sqrt{\unicode[STIX]{x1D70B}}\frac{\unicode[STIX]{x1D6E4}(s-1/2)}{\unicode[STIX]{x1D6E4}(s)}\left(\frac{(2^{s}+2)(3^{s}+3)}{6^{s}(2^{s}+1)(3^{s}+1)}\right)\cdot \frac{\unicode[STIX]{x1D701}_{\mathbb{Q}}(2s-1)}{\unicode[STIX]{x1D701}_{\mathbb{Q}}(2s)}.\end{eqnarray}$$

Obviously, $\mathfrak{g}_{1}=\sqrt{6}$ , $\mathfrak{g}_{2}=2\sqrt{3}$ , hence $(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}=2$ .

On the other hand, $\min \{|\text{Tr}A|:A\in {\mathcal{H}}(\unicode[STIX]{x1D6E4}_{0}(6)^{+})\}=\sqrt{6}$ , hence $e^{\ell _{M_{6},0}}\geqslant u$ where $u>1$ is a solution of the equation $u^{1/2}+u^{-1/2}=\sqrt{6}$ . Since $u=((\sqrt{6}+\sqrt{2})/2)^{2}>2$ , we see that $e^{\ell _{M_{6},0}}>(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ . This completes the proof of Proposition 1(ii).

7.3 On existence of surfaces where $e^{\ell _{M,0}}=(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$

We now argue the existence of an abundance of surfaces for which $e^{\ell _{M,0}}<(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ and prove part (iii) of Proposition 1.

Let $M_{\unicode[STIX]{x1D70F}}$ denote a degenerating family of Riemann surfaces, parameterized by the holomorphic parameter $\unicode[STIX]{x1D70F}$ , which approach the Deligne–Mumford boundary of moduli space when $\unicode[STIX]{x1D70F}$ approaches zero. One can select distinguished points of $M_{\unicode[STIX]{x1D70F}}$ which are either removed or whose local coordinates $z$ are replaced by fractional powers $z^{1/n}$ . By doing so, one obtains a degenerating sequence of hyperbolic Riemann surfaces of any signature; we refer the reader to [15] and references therein for further details regarding the construction of the sequence of degenerating hyperbolic Riemann surfaces.

By construction, the length of the smallest geodesic on $M_{\ell }$ approaches zero, so then $\exp (\ell _{M_{\unicode[STIX]{x1D70F}},0})$ approaches one as $\unicode[STIX]{x1D70F}$ approaches zero. In [9], the authors prove that through degeneration, parabolic Eisenstein series on $M_{\unicode[STIX]{x1D70F}}$ converge to parabolic Eisenstein series on the limit surface; see part (ii) of the Main Theorem of [9]. To be precise, one needs that the holomorphic parameter $s$ of the parabolic Eisenstein series lies in the half-plane $\text{Re}(s)>1$ and the spatial parameter $z$ to lie in a bounded region of $M_{\unicode[STIX]{x1D70F}}$ . However, in these ranges, one can compute the scattering matrix by computing the zeroth Fourier coefficient of the parabolic Eisenstein series, and, subsequently, compute the ratio $\mathfrak{g}_{2}/\mathfrak{g}_{1}$ on $M_{\unicode[STIX]{x1D70F}}$ . Since the parabolic Eisenstein series converge through degeneration to the parabolic Eisenstein series on the limit surface, the associated scattering matrix converges to a submatrix $\unicode[STIX]{x1D6F7}$ of the full scattering matrix on the limit surface. Clearly, the determinant of $\unicode[STIX]{x1D6F7}$ can be decomposed into a product of Gamma functions and a Dirichlet series, where the Dirichlet series is such that $\mathfrak{g}_{2}/\mathfrak{g}_{1}>1$ .

Therefore, we conclude that for all $\unicode[STIX]{x1D70F}$ sufficiently close to zero, we have that $e^{\ell _{M_{\unicode[STIX]{x1D70F}},0}}<(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ . In fact, all surfaces near the Deligne–Mumford boundary of any given moduli space satisfy the inequality $e^{\ell _{M_{\unicode[STIX]{x1D70F}},0}}<(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ .

In addition, let us assume that one is considering a moduli space which contains a congruence subgroup so then there exists a surface where $e^{\ell _{M_{\unicode[STIX]{x1D70F}},0}}>(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ . Then by combining the above argument with the computations from § 7.1, there exists surfaces for which $e^{\ell _{M_{\unicode[STIX]{x1D70F}},0}}=(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ . However, we have not been successful in our attempts to explicitly construct such a surface. In a sense, our Main Theorem shows that surfaces for which $e^{\ell _{M_{\unicode[STIX]{x1D70F}},0}}=(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ have a larger number of zeros of $(Z_{M}H_{M})^{\prime }$ than nearby surfaces for which the inequality holds.

8 Higher derivatives

In this section, we outline the proof of the Main Theorem for higher order derivatives of $Z_{M}H_{M}$ . The results are analogous to theorems proved for the zeros of the higher order derivatives of the Riemann zeta function; see [4] and [21].

8.1 Preliminary lemmas on higher derivatives

In order to deduce the vertical and weighted vertical distribution of zeros of the higher order derivatives of $(Z_{M}H_{M})$ we prove some preliminary lemmas, analogous to lemmas in § 4.

Lemma 18. Let $f_{M}(s)$ be defined by (18) and $\widetilde{Z}_{M}(s)$ defined by (19). Let us define, inductively, the functions $\widetilde{Z}_{M,j}(s)$ as $\widetilde{Z}_{M,0}(s):=Z_{M}(s)$ , $\widetilde{Z}_{M,1}(s):=\widetilde{Z}_{M}(s)$ and, for $j\geqslant 2$ ,

(56) $$\begin{eqnarray}\displaystyle \widetilde{Z}_{M,j}(1-s)=\frac{1}{f_{M}(s)}\!\left(\!(j-1)\frac{f_{M}^{\prime }}{f_{M}}(s)+\frac{\unicode[STIX]{x1D702}_{M}^{\prime }}{\unicode[STIX]{x1D702}_{M}}(s)-\frac{K_{M}^{\prime }}{K_{M}}(s)-\!\overset{j-1}{\underset{i=0}{\sum }}\frac{\widetilde{Z}_{M,i}^{\prime }}{\widetilde{Z}_{M,i}}(1-s)\!\right)\!. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Then for every positive integer $k$ the $k$ th derivative of the function $Z_{M}H_{M}$ can be represented as

(57) $$\begin{eqnarray}(Z_{M}H_{M})^{(k)}(s)=\left(f_{M}(s)\right)^{k}\unicode[STIX]{x1D702}_{M}(s)K_{M}^{-1}(s)Z_{M}(1-s)\overset{k}{\underset{i=1}{\prod }}\widetilde{Z}_{M,i}(1-s).\end{eqnarray}$$

Proof. The proof is based on a rather obvious induction argument.◻

Lemma 19. For $j\geqslant 1$ , let $Z_{M,j}(s):=\widetilde{Z}_{M,j}(s)-1$ . For small $\unicode[STIX]{x1D6FF}>0$ and $\unicode[STIX]{x1D6FF}_{1}>0$ , let $\unicode[STIX]{x1D70E}_{1}$ be a real number such that $\unicode[STIX]{x1D70E}_{1}\geqslant 1/2+\unicode[STIX]{x1D6FF}_{1}>1/2$ and $(\unicode[STIX]{x1D70E}_{1}\pm iT)$ is away from circles of a fixed, small radius $\unicode[STIX]{x1D6FF}>0$ , centered at integers. Then for $k=0,1$

(58) $$\begin{eqnarray}Z_{M,j}^{(k)}(\unicode[STIX]{x1D70E}_{1}\pm iT)=O\left(\frac{(T\log T)^{2-2\unicode[STIX]{x1D70E}_{1}}\log ^{k}T}{(\unicode[STIX]{x1D70E}_{1}-1/2)T}\right)\quad \text{as }T\rightarrow \infty ,\end{eqnarray}$$

and

(59) $$\begin{eqnarray}\frac{\widetilde{Z}_{M,j}^{\prime }}{\widetilde{Z}_{M,j}}(\unicode[STIX]{x1D70E}_{1}\pm iT)=O\left(\frac{(T\log T)^{2-2\unicode[STIX]{x1D70E}_{1}}\log T}{(\unicode[STIX]{x1D70E}_{1}-1/2)T}\right)\quad \text{as }T\rightarrow \infty .\end{eqnarray}$$

Proof. We prove the statement by induction in $j\geqslant 1$ . When $j=1$ , we use formula (26), which we differentiate, use the bound on the growth of the derivative of the digamma function (see [1, formula 6.4.12.]) and the bound (20) with $k=0$ or $k=1$ . These computations, which are elementary, allow one to prove (58) for $\unicode[STIX]{x1D70E}_{1}\geqslant 1/2+\unicode[STIX]{x1D6FF}_{1}>1/2$ in the case when $j=1$ . In addition,

$$\begin{eqnarray}\displaystyle \frac{\widetilde{Z}_{M,1}^{\prime }}{\widetilde{Z}_{M,1}}(\unicode[STIX]{x1D70E}_{1}\pm iT) & = & \displaystyle \frac{Z_{M,1}^{\prime }(\unicode[STIX]{x1D70E}_{1}\pm iT)}{1+Z_{M,1}(\unicode[STIX]{x1D70E}_{1}\pm iT)}\nonumber\\ \displaystyle & = & \displaystyle O\left(\frac{(T\log T)^{2-2\unicode[STIX]{x1D70E}_{1}}\log T}{(\unicode[STIX]{x1D70E}_{1}-1/2)T}\right)\quad \text{ as }T\rightarrow \infty .\nonumber\end{eqnarray}$$

With all this, we have proved (59) for $j=1$ .

Assume now that (58) and (59) hold true for all $1\leqslant m\leqslant j$ . Then, by (56) we get

$$\begin{eqnarray}1+Z_{M,j+1}(s)=\widetilde{Z}_{M,j+1}(s)=1+Z_{M,j}(s)+\frac{1}{f_{M}(s)}\left(\frac{f_{M}^{\prime }}{f_{M}}(s)-\frac{\widetilde{Z}_{M,k,j}^{\prime }}{\widetilde{Z}_{M,j}}(s)\right).\end{eqnarray}$$

Therefore, by the inductive assumption on $\widetilde{Z}_{M,j}^{\prime }/\widetilde{Z}_{M,j}$ and $Z_{M,j}$ , we have for $k=0,1$ ,

$$\begin{eqnarray}Z_{M,j+1}^{(k)}(\unicode[STIX]{x1D70E}_{1}\pm iT)=O\left(\frac{(T\log T)^{2-2\unicode[STIX]{x1D70E}_{1}}\log ^{k}T}{(\unicode[STIX]{x1D70E}_{1}-1/2)T}\right)\quad \text{ as }T\rightarrow \infty .\end{eqnarray}$$

In other words, (58) holds true with $m=j+1$ . In addition,

$$\begin{eqnarray}\displaystyle \frac{\widetilde{Z}_{M,j+1}^{\prime }}{\widetilde{Z}_{M,j+1}}(\unicode[STIX]{x1D70E}_{1}\pm iT) & = & \displaystyle \frac{Z_{M,j+1}^{\prime }(\unicode[STIX]{x1D70E}_{1}\pm iT)}{1+Z_{M,j+1}(\unicode[STIX]{x1D70E}_{1}\pm iT)}\nonumber\\ \displaystyle & = & \displaystyle O\left(\frac{(T\log T)^{2-2\unicode[STIX]{x1D70E}_{1}}\log T}{(\unicode[STIX]{x1D70E}_{1}-1/2)T}\right)\quad \text{ as }T\rightarrow \infty .\nonumber\end{eqnarray}$$

The proof is complete.◻

For any integer $k\geqslant 2$ , let us define $a_{M,k}:=(-1)^{k-1}a_{M}\log ^{k-1}A_{M}$ , where we set $a_{M,1}:=a_{M}$ . The analogue of the function $X_{M}(s)$ , defined by (12), is

(60) $$\begin{eqnarray}X_{M,k}(s):=\frac{A_{M}^{s}}{a_{M,k}}(Z_{M}H_{M})^{(k)}(s),\end{eqnarray}$$

where, of course, $X_{M,1}(s)=X_{M}(s)$ .

Lemma 20. For any integer $k\geqslant 1$ , there exists constants $\unicode[STIX]{x1D70E}_{k}>1$ and $0<c_{\unicode[STIX]{x1D6E4},k}<1$ such that for all $\unicode[STIX]{x1D70E}=\operatorname{Re}(s)\geqslant \unicode[STIX]{x1D70E}_{k}$ ,

$$\begin{eqnarray}X_{M,k}(s)=1+O(c_{\unicode[STIX]{x1D6E4},k}^{\unicode[STIX]{x1D70E}})\neq 0\quad \text{ as }\unicode[STIX]{x1D70E}\rightarrow +\infty .\end{eqnarray}$$

Proof. For $k=1$ , the statement is Lemma 10. Furthermore, from the proof of Lemma 10 and the definition of constants $A_{M}$ and $a_{M,1}$ , we see that

(61) $$\begin{eqnarray}(Z_{M}H_{M})^{\prime }(s)=Z_{M}(s)H_{M}(s){\mathcal{D}}_{1}(s),\end{eqnarray}$$

where ${\mathcal{D}}_{1}(s)$ is a Dirichlet series, converging absolutely for $\text{Re}(s)>\unicode[STIX]{x1D70E}_{1}$ , for sufficiently large $\unicode[STIX]{x1D70E}_{1}$ , with the leading term equal to $a_{M,1}\cdot A_{M}^{-s}$ as $\operatorname{Re}(s)\rightarrow +\infty$ .

Let us define, for $k\geqslant 1$ and $\operatorname{Re}(s)\gg 0$

$$\begin{eqnarray}(Z_{M}H_{M})^{(k)}(s)=Z_{M}(s)H_{M}(s){\mathcal{D}}_{k}(s).\end{eqnarray}$$

We claim that ${\mathcal{D}}_{k}(s)$ is a Dirichlet series with the leading term equal to $a_{M,k}\cdot A_{M}^{-s}$ as $\operatorname{Re}(s)\rightarrow +\infty$ . The statement is obviously true for $k=1$ . A simple inductive argument shows that the statement is true for all $k\geqslant 1$ . Therefore, for $\operatorname{Re}(s)=\unicode[STIX]{x1D70E}\gg 0$ , we may write

$$\begin{eqnarray}(Z_{M}H_{M})^{(k)}(s)=Z_{M}(s)H_{M}(s)\frac{a_{M,k}}{A_{M}^{s}}\left(1+O(A_{\unicode[STIX]{x1D6E4},k}^{-\unicode[STIX]{x1D70E}})\right)\quad \text{ as }\operatorname{Re}(s)\rightarrow \infty .\end{eqnarray}$$

Equation (32) implies that there exists $\unicode[STIX]{x1D70E}_{k}\geqslant 1$ and a constant $C_{\unicode[STIX]{x1D6E4},k}>1$ such that for $\operatorname{Re}s>\unicode[STIX]{x1D70E}_{k}$ , we have

$$\begin{eqnarray}(Z_{M}H_{M})^{(k)}(s)=\frac{a_{M,k}}{A_{M}^{s}}\left[1+O\left(\frac{1}{C_{\unicode[STIX]{x1D6E4},k}^{\operatorname{Re}(s)}}\right)\right]\quad \text{ as }\operatorname{Re}(s)\rightarrow \infty .\end{eqnarray}$$

Setting $c_{\unicode[STIX]{x1D6E4},k}=1/C_{\unicode[STIX]{x1D6E4},k}$ completes the proof.◻

Lemma 21. For arbitrary $\unicode[STIX]{x1D716}>0$ , $t\geqslant 1$ and $\unicode[STIX]{x1D70E}_{2}\geqslant 1$ such that $-\unicode[STIX]{x1D70E}_{2}$ is not a pole of $(Z_{M}H_{M})$ we have, for any positive integer $k$

$$\begin{eqnarray}(Z_{M}H_{M})^{(k)}(\unicode[STIX]{x1D70E}+it)=\left\{\begin{array}{@{}ll@{}}O(\exp \unicode[STIX]{x1D716}t) & \text{ for }\frac{1}{2}\leqslant \unicode[STIX]{x1D70E}\leqslant \unicode[STIX]{x1D70E}_{0},\\ O\left(\exp (1/2-\unicode[STIX]{x1D70E}+\unicode[STIX]{x1D716})t\right) & \text{ for }-\unicode[STIX]{x1D70E}_{2}\leqslant \unicode[STIX]{x1D70E}<1/2,\end{array}\right.\end{eqnarray}$$

as $t\rightarrow \infty$ .

Proof. When $k=1$ , the statement is proved in Lemma 14. Assume that the statement of Lemma holds for an integer $k\geqslant 1$ . Then for $1/2\leqslant \operatorname{Re}(s)=\unicode[STIX]{x1D70E}\leqslant \unicode[STIX]{x1D70E}_{0}$ the Cauchy integral formula yields

$$\begin{eqnarray}(Z_{M}H_{M})^{(k+1)}(s)=\frac{1}{2\unicode[STIX]{x1D70B}i}\int _{C}\frac{(Z_{M}H_{M})^{(k)}(z)}{(z-s)^{2}}dz\end{eqnarray}$$

where $C$ is a circle of a small, fixed radius $r<\unicode[STIX]{x1D716}$ , centered at $s$ . Using the inductive assumption on $(Z_{M}H_{M})^{(k)}(z)$ , we then get the bounds

$$\begin{eqnarray}(Z_{M}H_{M})^{(k+1)}(\unicode[STIX]{x1D70E}+it)=O\left(\exp ((r+\unicode[STIX]{x1D716})t)/r\right)=O\left(\exp (2\unicode[STIX]{x1D716}t)\right),\end{eqnarray}$$

for $1/2\leqslant \unicode[STIX]{x1D70E}\leqslant \unicode[STIX]{x1D70E}_{0}$ and $t\geqslant 1$ . This proves the first part of Lemma for $(Z_{M}H_{M})^{(k+1)}(z)$ , hence, the first part of the Lemma holds true for all $k\geqslant 1$ .

In the case when $\unicode[STIX]{x1D70E}<1/2$ , we employ the functional equation (57) for $(Z_{M}H_{M})^{(k)}$ to deduce that

$$\begin{eqnarray}\displaystyle |(Z_{M}H_{M})^{(k+1)}(-\unicode[STIX]{x1D70E}_{2}+it)| & = & \displaystyle |(Z_{M}H_{M})^{(k)}(-\unicode[STIX]{x1D70E}_{2}+it)|\nonumber\\ \displaystyle & & \displaystyle \cdot \,\Bigg|\Bigg[k\frac{f^{\prime }}{f}(-\unicode[STIX]{x1D70E}_{2}+it)+\frac{\unicode[STIX]{x1D702}_{M}^{\prime }}{\unicode[STIX]{x1D702}_{M}}(-\unicode[STIX]{x1D70E}_{2}+it)\nonumber\\ \displaystyle & & \displaystyle -\,\frac{K_{M}^{\prime }}{K_{M}}(-\unicode[STIX]{x1D70E}_{2}+it)-\overset{k}{\underset{i=0}{\sum }}\frac{\widetilde{Z}_{M,i}^{\prime }}{\widetilde{Z}_{M,i}}(1+\unicode[STIX]{x1D70E}_{2}-it)\Bigg]\Bigg|.\nonumber\end{eqnarray}$$

Since $\unicode[STIX]{x1D70E}_{2}\geqslant 1$ , we have $Z_{M}^{\prime }/Z_{M}(1+\unicode[STIX]{x1D70E}_{2}-it)=O(1)$ as $t\rightarrow +\infty$ . Furthermore, formula (59) and the same computations as in the proof of Lemma 14 imply that

$$\begin{eqnarray}\displaystyle & & \displaystyle k\frac{f^{\prime }}{f}(-\unicode[STIX]{x1D70E}_{2}+it)+\frac{\unicode[STIX]{x1D702}_{M}^{\prime }}{\unicode[STIX]{x1D702}_{M}}(-\unicode[STIX]{x1D70E}_{2}+it)-\frac{K_{M}^{\prime }}{K_{M}}(-\unicode[STIX]{x1D70E}_{2}+it)\nonumber\\ \displaystyle & & \displaystyle \qquad -\overset{k}{\underset{i=0}{\sum }}\frac{\widetilde{Z}_{M,i}^{\prime }}{\widetilde{Z}_{M,i}}(1+\unicode[STIX]{x1D70E}_{2}-it)=O(t)\quad \text{ as }t\rightarrow \infty ,\nonumber\end{eqnarray}$$

since the leading term in the above expression is $\text{vol}(M)(1/2+\unicode[STIX]{x1D70E}_{2}-it)$ $\tan (\unicode[STIX]{x1D70B}(1/2+\unicode[STIX]{x1D70E}_{2}-it))$ . By the inductive assumption on $(Z_{M}H_{M})^{(k)}(-\unicode[STIX]{x1D70E}_{2}+it)$ , we get

$$\begin{eqnarray}\displaystyle |(Z_{M}H_{M})^{(k+1)}(-\unicode[STIX]{x1D70E}_{2}+it)| & = & \displaystyle O\left(\exp \left(\left(\frac{1}{2}+\unicode[STIX]{x1D70E}_{2}+\unicode[STIX]{x1D716}\right)\text{vol}(M)t\right)\right),\nonumber\\ \displaystyle & & \displaystyle \text{as }t\rightarrow \infty .\nonumber\end{eqnarray}$$

As in the proof of Lemma 14, one applies the Phragmen–Lindelöf theorem to the function $(Z_{M}H_{M})^{(k+1)}$ in the open sector bounded by the lines $\operatorname{Im}(s)=1$ , $\operatorname{Re}(s)=-\unicode[STIX]{x1D70E}_{2}$ and $\operatorname{Re}(s)=1/2$ . As a result, the proof of the second part of the Lemma is complete for $(Z_{M}H_{M})^{(k+1)}$ .◻

8.2 Distribution of zeros of $(Z_{M}H_{M})^{(k)}$

The following theorem is the analogue of the Main Theorem for zeros of higher derivatives of $(Z_{M}H_{M})$ .

Theorem 22. With the notation as above, the following statements are true for any integer $k\geqslant 2$ .

1. (a) For $\unicode[STIX]{x1D70E}<1/2$ , there exist $t_{0}>0$ such that $(Z_{M}H_{M})^{(k)}(\unicode[STIX]{x1D70E}+it)\neq 0$ for all $|t|>t_{0}$ .

2. (b)

   (62) $$\begin{eqnarray}N_{\text{ver}}(T;(Z_{M}H_{M})^{(k)})=N_{\text{ver}}(T;(Z_{M}H_{M})^{\prime })+o(T)\quad \text{ as }T\rightarrow \infty .\end{eqnarray}$$

3. (c)

   (63) $$\begin{eqnarray}\displaystyle \hspace{-10.00002pt}N_{\text{w}}(T;(Z_{M}H_{M})^{(k)}) & = & \displaystyle N_{\text{w}}(T;(Z_{M}H_{M})^{\prime })+\frac{(k-1)T}{2\unicode[STIX]{x1D70B}}[\log (T\cdot \text{vol}(M))-1]\nonumber\\ \displaystyle & & \displaystyle -\,\frac{T}{2\unicode[STIX]{x1D70B}}\log ((k-1)\log A_{M})+o(T)\quad \text{as }T\rightarrow \infty .\end{eqnarray}$$

Proof. We first outline the proof of part (a). For $k\geqslant 2$ , $\unicode[STIX]{x1D70E}<1/2$ and $s=\unicode[STIX]{x1D70E}\pm iT$ equation (57) yields

(64) $$\begin{eqnarray}\displaystyle \frac{(Z_{M}H_{M})^{(k)}}{(Z_{M}H_{M})^{(k-1)}}(s) & = & \displaystyle \log \left((Z_{M}H_{M})^{(k-1)}(s)\right)^{\prime }\nonumber\\ \displaystyle & = & \displaystyle (k-1)\frac{f^{\prime }}{f}(s)+\frac{\unicode[STIX]{x1D702}_{M}^{\prime }}{\unicode[STIX]{x1D702}_{M}}(s)-\frac{K_{M}^{\prime }}{K_{M}}(s)-\frac{Z_{M}^{\prime }}{Z_{M}}(1-s)\nonumber\\ \displaystyle & & \displaystyle -\,\overset{k-1}{\underset{i=1}{\sum }}\frac{\widetilde{Z}_{M,i}^{\prime }}{\widetilde{Z}_{M,i}}(1-s).\end{eqnarray}$$

We now apply (59) with $\unicode[STIX]{x1D70E}_{1}=1-\unicode[STIX]{x1D70E}>1/2$ and (20) to deduce that

$$\begin{eqnarray}\frac{Z_{M}^{\prime }}{Z_{M}}(1-s)+\overset{k-1}{\underset{i=1}{\sum }}\frac{\widetilde{Z}_{M,i}^{\prime }}{\widetilde{Z}_{M,i}}(1-s)=O\left(\frac{(T\log T)^{2\unicode[STIX]{x1D70E}}\log T}{(1/2-\unicode[STIX]{x1D70E})}\right)\quad \text{as }T\rightarrow \infty .\end{eqnarray}$$

Since $\operatorname{Re}(\unicode[STIX]{x1D702}_{M}^{\prime }/\unicode[STIX]{x1D702}_{M}(\unicode[STIX]{x1D70E}\pm iT))=-\text{vol}(M)t+O(\log t)$ and $K_{M}^{\prime }/K_{M}(\unicode[STIX]{x1D70E}\pm it)=O(\log t)$ as $t\rightarrow +\infty$ , we immediately deduce from (64) that

$$\begin{eqnarray}\displaystyle \text{Re}\left(-\frac{(Z_{M}H_{M})^{(k)}}{(Z_{M}H_{M})^{(k-1)}}(\unicode[STIX]{x1D70E}\pm it)\right) & = & \displaystyle \text{vol}(M)t\nonumber\\ \displaystyle & & \displaystyle +\,O\left(\max \left\{\log t,\frac{(t\log t)^{2\unicode[STIX]{x1D70E}}\log t}{(1/2-\unicode[STIX]{x1D70E})}\right\}\right)\nonumber\\ \displaystyle & & \displaystyle \quad \text{as }t\rightarrow +\infty ,\nonumber\end{eqnarray}$$

for any $\unicode[STIX]{x1D70E}<1/2$ . This proves part (a).

The proof of parts (b) and (c) closely follows lines of the proof of parts (b) and (c) of the Main Theorem. We fix a large positive number $T$ and choose number $T^{\prime }$ to be a bounded distance from $T$ such that $T^{\prime }$ is distinct from the imaginary part of any zero of $Z_{M}H_{M}$ . We fix a number $a\in (0,1/2)$ and use part (a) of the Theorem to choose $t_{0}>0$ to be the number such that $(Z_{M}H_{M})^{(k)}(\unicode[STIX]{x1D70E}+it)\neq 0$ for all $\unicode[STIX]{x1D70E}\leqslant a$ and $|t|>t_{0}$ . Let $\unicode[STIX]{x1D70E}_{0}$ be a constant such that $\unicode[STIX]{x1D70E}_{0}\geqslant \max \{\unicode[STIX]{x1D70E}_{0}^{\prime },\unicode[STIX]{x1D70E}_{k}\}$ , where $\unicode[STIX]{x1D70E}_{0}^{\prime }$ is defined in Lemma 3 and $\unicode[STIX]{x1D70E}_{k}$ is defined in Lemma 20.

We apply Littlewood’s theorem to the function $X_{M,k}(s)$ , defined by (60) which is holomorphic in the rectangle $R(a,T^{\prime })$ with vertices $a+it_{0}$ , $\unicode[STIX]{x1D70E}_{0}+it_{0}$ , $\unicode[STIX]{x1D70E}_{0}+iT^{\prime }$ , $a+iT^{\prime }$ . The resulting formula is

(65) $$\begin{eqnarray}\displaystyle 2\unicode[STIX]{x1D70B}\underset{t_{0}<\unicode[STIX]{x1D6FE}^{(k)}<T^{\prime },\unicode[STIX]{x1D6FD}^{(k)}>a}{\underset{\unicode[STIX]{x1D70C}^{(k)}=\unicode[STIX]{x1D6FD}^{(k)}+i\unicode[STIX]{x1D6FE}^{(k)}}{\sum }}\left(\unicode[STIX]{x1D6FD}^{(k)}-a\right) & = & \displaystyle \int _{t_{0}}^{T^{\prime }}\log \left|X_{M,k}(a+it)\right|dt\nonumber\\ \displaystyle & & \displaystyle -\,\int _{t_{0}}^{T^{\prime }}\log \left|X_{M,k}(\unicode[STIX]{x1D70E}_{0}+it)\right|dt\nonumber\\ \displaystyle & & \displaystyle -\,\int _{a}^{\unicode[STIX]{x1D70E}_{0}}\arg X_{M,k}(\unicode[STIX]{x1D70E}+it_{0})\,d\unicode[STIX]{x1D70E}\nonumber\\ \displaystyle & & \displaystyle +\,\int _{a}^{\unicode[STIX]{x1D70E}_{0}}\arg X_{M,k}(\unicode[STIX]{x1D70E}+iT^{\prime })\,d\unicode[STIX]{x1D70E}\nonumber\\ \displaystyle & = & \displaystyle I_{1,k}+I_{2,k}+I_{3,k}+I_{4,k},\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}^{(k)}$ denotes the zero of $(Z_{M}H_{M})^{(k)}$ . By the choice of $t_{0}$ , the sum on the left-hand side of (65) is actually taken over all zeros $\unicode[STIX]{x1D70C}^{(k)}$ of $(Z_{M}H_{M})^{(k)}$ with imaginary part in the interval $(t_{0},T^{\prime })$ .

Trivially, $I_{3,k}=O(1)$ as $T\rightarrow +\infty$ . The application of Lemma 20 immediately yields that $I_{2,k}=O(1)$ as $T\rightarrow +\infty$ , once we apply the same method as in evaluation of $I_{2}$ .

One can follow the steps of the proof that $|\!\arg X_{M}(\unicode[STIX]{x1D70E}+iT^{\prime })|=o(T)$ as $T\rightarrow +\infty$ in the present setting. One uses function $X_{M,k}$ instead of $X_{M}$ and Lemma 21 instead of Lemma 14. From this, we deduce that $|\!\arg X_{M,k}(\unicode[STIX]{x1D70E}+iT^{\prime })|=o(T)$ as $T\rightarrow +\infty$ . Therefore, it is left to evaluate $I_{1,k}$ .

From definition of $X_{M,k}$ , using the functional equation (57) for $(Z_{M}H_{M})^{(k)}$ , we get for $k\geqslant 2$ , the expression

(66) $$\begin{eqnarray}\displaystyle I_{1,k} & = & \displaystyle \int _{t_{0}}^{T^{\prime }}\log \left|A_{M}^{(a+it)}a_{M,k}^{-1}\right|dt+k\int _{t_{0}}^{T^{\prime }}\log |f_{M}(a+it)|dt\nonumber\\ \displaystyle & & \displaystyle +\,\!\int _{t_{0}}^{T^{\prime }}\log |\unicode[STIX]{x1D702}_{M}(a+it)|dt\nonumber\\ \displaystyle & & \displaystyle +\,\int _{t_{0}}^{T^{\prime }}\log |K_{M}^{-1}(a+it)|dt+\int _{t_{0}}^{T^{\prime }}\log |Z_{M}(1-a-it)|dt\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{i=1}^{k-1}\int _{t_{0}}^{T^{\prime }}\log |1+Z_{M,i}(1-a-it)|dt.\end{eqnarray}$$

By employing equation (58) with $k=0$ , we get

$$\begin{eqnarray}\int _{t_{0}}^{T^{\prime }}\log |1+Z_{M,i}(1-a-it)|dt=O((T\log T)^{2a})\quad \text{as }T\rightarrow \infty\end{eqnarray}$$

for all $i=1,\ldots ,k-1$ . Substituting this equation, together with (42), (44), (45) and (47) into (66), we immediately deduce that

$$\begin{eqnarray}\displaystyle I_{1,k} & = & \displaystyle \left(\frac{1}{2}-a\right)\frac{\text{vol}(M)}{2}T^{2}+\left(\frac{n_{1}}{2}+k\right)T\log T\nonumber\\ \displaystyle & & \displaystyle +\,C_{M,a,k}T+O((T\log T)^{2a})\quad \text{as }T\rightarrow \infty ,\nonumber\end{eqnarray}$$

where

$$\begin{eqnarray}\displaystyle C_{M,a,k} & = & \displaystyle 2\left(a-\frac{1}{2}\right)n_{1}\log 2+a\log A_{M}-\log |a_{M,k}|\nonumber\\ \displaystyle & & \displaystyle +\,k(\log (\text{vol}(M))-1)+2a\log \mathfrak{g}_{1}-\log |d(1)|-\frac{n_{1}}{2}(\log \unicode[STIX]{x1D70B}+1).\nonumber\end{eqnarray}$$

Combining this equation with the bounds on $I_{2,k}$ , $I_{3,k}$ and $I_{4,k}$ and (65), we get

(67) $$\begin{eqnarray}\displaystyle 2\unicode[STIX]{x1D70B}\underset{t_{0}<\unicode[STIX]{x1D6FE}^{(k)}<T^{\prime }}{\underset{\unicode[STIX]{x1D70C}^{(k)}=\unicode[STIX]{x1D6FD}^{(k)}+i\unicode[STIX]{x1D6FE}^{(k)}}{\sum }}\left(\unicode[STIX]{x1D6FD}^{(k)}-a\right) & = & \displaystyle \left(\frac{1}{2}-a\right)\frac{\text{vol}(M)}{2}T^{2}+\left(\frac{n_{1}}{2}+k\right)T\log T\nonumber\\ \displaystyle & & \displaystyle +\,C_{M,a,k}T+o(T)\quad \text{as }T\rightarrow \infty .\end{eqnarray}$$

Replacing $a$ by $a/2$ in (67) and subtracting proves part (b). Part (c) is proved by employing an analogue of equation (51), with $\unicode[STIX]{x1D6FD}^{\prime }$ and $\unicode[STIX]{x1D70C}^{\prime }$ replaced by $\unicode[STIX]{x1D6FD}^{(k)}$ and $\unicode[STIX]{x1D70C}^{(k)}$ .◻

Remark 23. The statement of Theorem 22 is true in the case of co-compact Riemann surfaces $\unicode[STIX]{x1D6E4}\setminus \mathbb{H}$ when taking $H_{M}=1$ and $A_{M}=\exp (\ell _{M,0})$ in (62) and (63).

In the case when $\unicode[STIX]{x1D6E4}\setminus \mathbb{H}$ is compact the statement (b) of Theorem 22 was announced by Luo in [22], with the weaker error term $O(T)$ . As one can see, we put considerable effort into the analysis yielding the error term $o(T)$ , and the structure of the constant $C_{M,a,k}$ is, in our opinion, fascinating.

Remark 24. From the formula (63) for the weighted vertical distribution of zeros of $(Z_{M}H_{M})^{(k)}$ , we see that the differentiation of $(Z_{M}H_{M})^{(k)}$ increases the sum $N_{\text{w}}(T;(Z_{M}H_{M})^{(k)})$ by the quantity $[(1/(2\unicode[STIX]{x1D70B}))\cdot T\log T+O(T)]$ as $T\rightarrow \infty$ . Hence, after each differentiation, zeros of $(Z_{M}H_{M})^{\prime }$ move further to the right of $1/2$ . Since every zero of $(Z_{M}H_{M})^{\prime }$ on the line $\operatorname{Re}(s)=1/2$ (up to finitely many of them) is a multiple zero of $Z_{M}$ , this result fully supports the “bounded multiplicities conjecture”. To recall, the “bounded multiplicities conjecture” asserts that the order of every multiple zero of $Z_{M}$ is uniformly bounded, or, equivalently, that the dimension of every eigenspace associated to the discrete eigenvalue of the Laplacian on $M$ is uniformly bounded, with a bound depending solely upon $M$ .

9 Concluding remarks

9.1 Revisiting Weyl’s law

Weyl’s law for an arbitrary finite volume hyperbolic Riemann surface $M$ is the following asymptotic formula, which we quote from [14, p. 466]:

(68) $$\begin{eqnarray}\displaystyle N_{M,\text{dis}}(T)+N_{M,\text{con}}(T) & = & \displaystyle \frac{\text{vol}(M)}{4\unicode[STIX]{x1D70B}}T^{2}-\frac{n_{1}}{\unicode[STIX]{x1D70B}}T\log T+\frac{n_{1}T}{\unicode[STIX]{x1D70B}}(1-\log 2)\nonumber\\ \displaystyle & & \displaystyle +\,O\,(T/\log T)\quad \text{as }T\rightarrow \infty ,\end{eqnarray}$$

where

$$\begin{eqnarray}N_{M,\text{dis}}(T)=\#\{s=1/2+it|Z_{M}(s)=0\text{ and }0\leqslant t\leqslant T\}\end{eqnarray}$$

and

$$\begin{eqnarray}N_{M,\text{con}}(T)=\frac{1}{4\unicode[STIX]{x1D70B}}\int _{-T}^{T}\frac{-\unicode[STIX]{x1D719}_{M}^{\prime }}{\unicode[STIX]{x1D719}_{M}}(1/2+it)\,dt.\end{eqnarray}$$

The term $N_{M,\text{dis}}(T)$ counts the number of zeros of the Selberg zeta function $Z_{M}(s)$ on the critical line $\text{Re}(s)=1/2$ , whereas the term $N_{M,\text{con}}(T)$ is related to the number of zeros of $Z_{M}(s)$ off the critical line but within the critical strip. In the following proposition, we relate the counting function $N_{\text{ver}}(T;\unicode[STIX]{x1D719}_{M})$ with the function $N_{M,\text{con}}(T)$ , showing that the constant $\mathfrak{g}_{1}$ appears in the resulting asymptotic formula.

Proposition 25. There exists a sequence $\{T_{n}\}$ of positive numbers tending toward infinity such that, with the notation as above, we have the asymptotic formula

$$\begin{eqnarray}N_{\text{ver}}(T_{n};\unicode[STIX]{x1D719}_{M})=N_{M,\text{con}}(T_{n})-\frac{\log \mathfrak{g}_{1}}{\unicode[STIX]{x1D70B}}T_{n}+O\left(\log T_{n}\right)\quad \text{as }n\rightarrow \infty .\end{eqnarray}$$

Proof. Let $R(T)$ denote the rectangle with vertexes $1/2-iT$ , $\unicode[STIX]{x1D70E}_{0}^{\prime }-iT$ , $\unicode[STIX]{x1D70E}_{0}^{\prime }+iT$ , $1/2+iT$ , where $\unicode[STIX]{x1D70E}_{0}^{\prime }>\unicode[STIX]{x1D70E}_{0}$ , where $\unicode[STIX]{x1D70E}_{0}$ is defined in § 1.3. Therefore, the series (8) converges uniformly and absolutely for Re $s\geqslant \unicode[STIX]{x1D70E}_{0}^{\prime }$ , and all zeros of $\unicode[STIX]{x1D719}_{M}$ with real part greater than 1/2 lie inside $R(T)$ . Recall that the zeros of $\unicode[STIX]{x1D719}_{M}$ appear in pairs of the form $\unicode[STIX]{x1D70C}$ and $\overline{\unicode[STIX]{x1D70C}}$ . As a result, the proposition follow by studying the expression

$$\begin{eqnarray}\displaystyle 2N_{\text{ver}}(T;\unicode[STIX]{x1D719}_{M}) & = & \displaystyle \frac{1}{2\unicode[STIX]{x1D70B}i}\int _{R(T)}\frac{\unicode[STIX]{x1D719}_{M}^{\prime }}{\unicode[STIX]{x1D719}_{M}}(s)\,ds=\frac{-1}{2\unicode[STIX]{x1D70B}}\int _{-T}^{T}\frac{\unicode[STIX]{x1D719}_{M}^{\prime }}{\unicode[STIX]{x1D719}_{M}}\left(\frac{1}{2}+it\right)dt\nonumber\\ \displaystyle & & \displaystyle +\,\frac{1}{2\unicode[STIX]{x1D70B}}\int _{-T}^{T}\frac{\unicode[STIX]{x1D719}_{M}^{\prime }}{\unicode[STIX]{x1D719}_{M}}(\unicode[STIX]{x1D70E}_{0}^{\prime }+it)\,dt+I_{1}(T)+I_{2}(T)\nonumber\end{eqnarray}$$

where $I_{1}$ and $I_{2}$ denote the integrals along the horizontal lines which bound $R(T)$ . In [18, Theorem 7.1] it is proved that $\unicode[STIX]{x1D719}_{M}$ is of regularized product type with order $M=0$ . As a result, from [19, Chapter 1], we have the existence of a sequence of real numbers $\{T_{n}\}$ tending to infinity such that $I_{1}(T_{n})=O(\log T_{n})$ and $I_{2}(T_{n})=O(\log T_{n})$ when $n\rightarrow \infty$ , so then

$$\begin{eqnarray}\displaystyle 2N_{\text{ver}}(T_{n};\unicode[STIX]{x1D719}_{M}) & = & \displaystyle -\frac{1}{2\unicode[STIX]{x1D70B}}\int _{-T_{n}}^{T_{n}}\frac{\unicode[STIX]{x1D719}_{M}^{\prime }}{\unicode[STIX]{x1D719}_{M}}\left(\frac{1}{2}+it\right)dt\nonumber\\ \displaystyle & & \displaystyle +\,\frac{1}{2\unicode[STIX]{x1D70B}}\int _{-T_{n}}^{T_{n}}\frac{\unicode[STIX]{x1D719}_{M}^{\prime }}{\unicode[STIX]{x1D719}_{M}}(\unicode[STIX]{x1D70E}_{0}^{\prime }+it)\,dt+O(\log T_{n})\quad \text{when }n\rightarrow \infty .\nonumber\end{eqnarray}$$

Using the notation as above, we now write

$$\begin{eqnarray}\displaystyle \int _{-T_{n}}^{T_{n}}\frac{\unicode[STIX]{x1D719}_{M}^{\prime }}{\unicode[STIX]{x1D719}_{M}}(\unicode[STIX]{x1D70E}_{0}^{\prime }+it)\,dt & = & \displaystyle \mathop{\sum }_{i=1}^{\infty }\frac{b\left(q_{i}\right)}{q_{i}^{\unicode[STIX]{x1D70E}_{0}^{\prime }}}\int _{-T_{n}}^{T_{n}}\frac{dt}{q_{i}^{it}}-4T\log \mathfrak{g}_{1}\nonumber\\ \displaystyle & & \displaystyle +\,n_{1}\int _{-T_{n}}^{T_{n}}\left(\frac{\unicode[STIX]{x1D6E4}^{\prime }}{\unicode[STIX]{x1D6E4}}\left(\unicode[STIX]{x1D70E}_{0}^{\prime }+it-\frac{1}{2}\right)-\frac{\unicode[STIX]{x1D6E4}^{\prime }}{\unicode[STIX]{x1D6E4}}(\unicode[STIX]{x1D70E}_{0}^{\prime }+it)\right)dt.\nonumber\end{eqnarray}$$

Interchanging the sum and the integral above is justified by the fact that the series defining $H_{M}^{\prime }/H_{M}(s)$ converges absolutely and uniformly for $\operatorname{Re}(s)>\unicode[STIX]{x1D70E}_{0}$ . Furthermore, we also have that

$$\begin{eqnarray}\mathop{\sum }_{i=1}^{\infty }\frac{b\left(q_{i}\right)}{q_{i}^{\unicode[STIX]{x1D70E}_{0}^{\prime }}}\int _{-T_{n}}^{T_{n}}\frac{dt}{q_{i}^{it}}=O\left(1\right)\quad \text{as }n\rightarrow \infty ,\end{eqnarray}$$

Using the series representation of the digamma function we get that

$$\begin{eqnarray}\int _{-T_{n}}^{T_{n}}\left(\frac{\unicode[STIX]{x1D6E4}^{\prime }}{\unicode[STIX]{x1D6E4}}\left(\unicode[STIX]{x1D70E}_{0}^{\prime }+it-\frac{1}{2}\right)-\frac{\unicode[STIX]{x1D6E4}^{\prime }}{\unicode[STIX]{x1D6E4}}\left(\unicode[STIX]{x1D70E}_{0}^{\prime }+it\right)\right)dt=O(1)\quad \text{ as }n\rightarrow \infty .\end{eqnarray}$$

With all this, the proof of the Proposition is complete.◻

Remark 26. The above proposition shows that the term $-(\log \mathfrak{g}_{1}/\unicode[STIX]{x1D70B})T$ measures the discrepancy between the number of zeros of $\unicode[STIX]{x1D719}_{M}$ with real part greater that $1/2$ , meaning $N_{\text{ver}}(T;\unicode[STIX]{x1D719}_{M})$ , and the quantity $N_{M,\text{con}}(T)$ , appearing in the classical version of the Weyl’s law.

Furthermore, one can restate Proposition 25 as the relation representing Weyl’s law

(69) $$\begin{eqnarray}\displaystyle N_{\text{ver}}(T_{n};Z_{M}H_{M}) & = & \displaystyle \frac{\text{vol}(M)}{4\unicode[STIX]{x1D70B}}T_{n}^{2}-\frac{n_{1}}{\unicode[STIX]{x1D70B}}T_{n}\log T_{n}+\frac{T_{n}}{\unicode[STIX]{x1D70B}}\left(n_{1}(1-\log 2)-\log \mathfrak{g}_{1}\right)\nonumber\\ \displaystyle & & \displaystyle +\,O\left(T_{n}/\log T_{n}\right),\end{eqnarray}$$

as $n\rightarrow \infty$ .

A direct consequence of the relation (69), Main Theorem and Theorem 22 is the following reformulation of the Weyl’s law:

Corollary 27. There exist a sequence $\{T_{n}\}$ of positive real numbers tending to infinity such that, for every positive integer $k$

$$\begin{eqnarray}\displaystyle N_{\text{ver}}(T_{n};Z_{M}H_{M}) & = & \displaystyle N_{\text{ver}}(T_{n};(Z_{M}H_{M})^{(k)})-\frac{n_{1}}{\unicode[STIX]{x1D70B}}T_{n}\log T_{n}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{T_{n}}{2\unicode[STIX]{x1D70B}}(2n_{1}+\log A_{M})+o(T_{n})\quad \text{as }n\rightarrow \infty .\nonumber\end{eqnarray}$$

Remark 28. An interpretation of the constant $\log \mathfrak{g}_{1}$ , similar to the one derived in Proposition 25 is obtained in [7, formula (3.4.15)], where it is shown, in our notation, that

$$\begin{eqnarray}\log \mathfrak{g}_{1}=\lim _{x\rightarrow \infty }\lim _{y\rightarrow \infty }\!\left[2x\!\int _{0}^{\infty }\!\frac{N_{M,\text{con}}(t)-N_{\text{ver}}(t;\unicode[STIX]{x1D719}_{M})}{t}\!\left(\!\frac{1}{t^{2}+x^{2}}-\frac{1}{t^{2}+y^{2}}\!\right)\!dt\right]\!.\end{eqnarray}$$

A geometric interpretation of the constant $\mathfrak{g}_{1}$ , in the case when the surface has one cusp $\mathfrak{a}$ is derived in [17]. In that case, $\mathfrak{g}_{1}^{-1}$ is the radius of the largest isometric circle arising in the construction of the standard polygon for the group $\unicode[STIX]{x1D70E}_{\mathfrak{a}}^{-1}\unicode[STIX]{x1D6E4}\unicode[STIX]{x1D70E}_{\mathfrak{a}}$ , where $\unicode[STIX]{x1D70E}_{\mathfrak{a}}$ denotes the scaling matrix of the cusp $\mathfrak{a}$ .

As stated in the introduction, we do not know of a spectral or geometric interpretation of the constant $\mathfrak{g}_{2}$ , besides the trivial one which realizes $\mathfrak{g}_{2}$ as the second largest denominator of the Dirichlet series portion of the scattering determinant. Therefore, we view Corollary 27 as giving rise to a new spectral invariant.

9.2 A comparison of counting functions

In this section, we will prove Theorem 2. In effect, it is necessary to recall results from [14], translate the notation in [14] to the notation in the present paper, then combine the result with (2) and parts (c) of the Main Theorem and Theorem 22.

From [14, Theorem 2.22] we have the asymptotic relation

(70) $$\begin{eqnarray}\displaystyle N_{\text{w}}(T;H_{M}) & = & \displaystyle \frac{n_{1}}{2}\cdot \frac{T\log T}{2\unicode[STIX]{x1D70B}}+\frac{T}{2}\left(-\frac{n_{1}}{2\unicode[STIX]{x1D70B}}-\frac{1}{\unicode[STIX]{x1D70B}}\log \left|b_{2}\right|\right)+O(\log T)\nonumber\\ \displaystyle & & \displaystyle \qquad \text{as }T\rightarrow \infty .\end{eqnarray}$$

Note that in [14], the author counts the zeros of $H_{M}$ in both the upper and lower half-planes, whereas the counting function $N_{\text{w}}(T;H_{M})$ only considers those zeros in the upper half-plane. Recall that the zeros and poles of $H$ appear symmetrically about the real axis. As a result, the relation (70) differs from [14, Theorem 2.22] by a factor of two. Comparing [14, eq. (2.15) on p. 445] with our notation we deduce that $b_{2}=\unicode[STIX]{x1D70B}^{n_{1}/2}\mathfrak{g}_{1}^{-1}d(1)$ , hence, we are able to rewrite [14, Theorem 2.22] as

(71) $$\begin{eqnarray}\displaystyle N_{\text{w}}(T;H_{M}) & = & \displaystyle \frac{n_{1}}{2}\cdot \frac{T\log T}{2\unicode[STIX]{x1D70B}}-\frac{T}{2\unicode[STIX]{x1D70B}}\left(\frac{n_{1}}{2}+\frac{n_{1}}{2}\log \unicode[STIX]{x1D70B}+\log \left|d(1)\right|-\log \mathfrak{g}_{1}\right)\nonumber\\ \displaystyle & & \displaystyle +\,O(\log T)\quad \text{as }T\rightarrow \infty .\end{eqnarray}$$

Comparing (71) with part (c) of Main Theorem we deduce that

$$\begin{eqnarray}N_{\text{w}}(T;(Z_{M}H_{M})^{\prime })-N_{\text{w}}(T;H_{M})\displaystyle =\frac{T\log T}{2\unicode[STIX]{x1D70B}}+\frac{T}{2\unicode[STIX]{x1D70B}}C+o(T)\quad \text{as }T\rightarrow \infty\end{eqnarray}$$

with $C=\frac{1}{2}\log A_{M}-\log |a_{M}|+\log \text{vol}(M)-1.$

Assume that $e^{\ell _{M,0}}<(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ , so then

$$\begin{eqnarray}C=\log \left(\frac{2\text{vol}(M)\sinh (\ell _{M,0}/2)}{e\cdot m_{M,0}\ell _{M,0}}\right).\end{eqnarray}$$

Let $\widetilde{M}$ be any co-compact hyperbolic Riemann surface such that $\text{vol}(\widetilde{M})=\text{vol}(M)$ . Assume that $M$ and $\widetilde{M}$ have systoles of equal length, and the same number of inconjugate classes of systoles. Then, using (2), we arrive at the conclusion that

(72) $$\begin{eqnarray}N_{\text{w}}(T;(Z_{M}H_{M})^{\prime })-N_{\text{w}}(T;H_{M})=N_{\text{w}}(T;Z_{\widetilde{M}}^{\prime })+o(T)\quad \text{as }T\rightarrow \infty .\end{eqnarray}$$

Furthermore, when $e^{\ell _{M,0}}<(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ , comparing (72) with part (c) of Theorem 22, for $k\geqslant 2$ we arrive at

$$\begin{eqnarray}\displaystyle N_{\text{w}}(T;(Z_{M}H_{M})^{(k)})-N_{\text{w}}(T;H_{M}) & = & \displaystyle N_{\text{w}}(T;Z_{\widetilde{M}}^{\prime })\nonumber\\ \displaystyle & & \displaystyle +\,\frac{(k-1)T}{2\unicode[STIX]{x1D70B}}[\log (T\text{vol}(M))-1]\nonumber\\ \displaystyle & & \displaystyle -\,\frac{T}{2\unicode[STIX]{x1D70B}}\log ((k-1)\ell _{M,0})+o(T)\nonumber\\ \displaystyle & & \displaystyle \quad \text{as }T\rightarrow \infty .\nonumber\end{eqnarray}$$

Then, from part (c) of Theorem 22 applied to the zeta function $Z_{\widetilde{M}}$ we deduce

$$\begin{eqnarray}N_{\text{w}}(T;(Z_{M}H_{M})^{(k)})-N_{\text{w}}(T;H_{M})\displaystyle =N_{\text{w}}(T;(Z_{\widetilde{M}})^{(k)})+o(T)\quad \text{as }T\rightarrow \infty .\end{eqnarray}$$

This proves Theorem 2.

We find the comparison of counting functions, as summarized in (13) very interesting, especially since the coefficients in the asymptotic expansions in (70) and part (c) of the Main Theorem are somewhat involved and dissimilar from other known asymptotic expansions.

9.3 Concluding remarks

In [5] the authors defined $213$ genus zero subgroups of which $171$ are associated to “Moonshine”. It would be interesting to compute the invariant $A_{M}$ for each of these groups to see if further information regarding the groups, possibly related to “moonshine”, is uncovered.

Is it possible to explicitly determine an example of a surface where $e^{\ell _{M,0}}=(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ ? More generally, one could study the set of such surfaces, as a subset of moduli space. Is the set of surfaces where $e^{\ell _{M,0}}>(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ a connected subset of moduli space, or are there several components? Is there another characterization of surfaces where $e^{\ell _{M,0}}=(\mathfrak{g}_{2}/\mathfrak{g}_{1})^{2}$ ? Many other basic questions can be easily posed, and we find these problems very interesting.

In [2], the authors determined the asymptotic behavior of Selberg’s zeta function through degeneration up to the critical line. It would be interesting to study the asymptotic behavior of the zeros of the derivative of Selberg’s zeta function through degeneration, either in moduli space or through elliptic degeneration.

To come full circle, we return to the setting of the Riemann zeta function and speculate if one can attempt to mimic results which follow from the Levinson–Montgomery article [21]. Specifically, we recall, that Levinson used results from the distribution of zeros of $\unicode[STIX]{x1D701}_{\mathbb{Q}}^{\prime }$ to prove that more than $1/3$ of the zeros of the Riemann zeta function lie on the critical line. Can one follow a similar investigation in the setting of the Selberg zeta function associated to a noncompact, finite volume surface? To do so, we note that a starting point would be to establish an analogue of the approximate functional equation for the Selberg zeta function. Results in this direction would be very significant, and we plan to undertake the project in the near future.