Zeros of Rankin–Selberg L-functions in families

Abstract

Let $\mathfrak {F}_n$ be the set of all cuspidal automorphic representations $\pi$ of $\mathrm {GL}_n$ with unitary central character over a number field $F$. We prove the first unconditional zero density estimate for the set $\mathcal {S}=\{L(s,\pi \times \pi ')\colon \pi \in \mathfrak {F}_n\}$ of Rankin–Selberg $L$-functions, where $\pi '\in \mathfrak {F}_{n'}$ is fixed. We use this density estimate to establish: (i) a hybrid-aspect subconvexity bound at $s=\frac {1}{2}$ for almost all $L(s,\pi \times \pi ')\in \mathcal {S}$; (ii) a strong on-average form of effective multiplicity one for almost all $\pi \in \mathfrak {F}_n$; and (iii) a positive level of distribution for $L(s,\pi \times \widetilde {\pi })$, in the sense of Bombieri–Vinogradov, for each $\pi \in \mathfrak {F}_n$.

1. Introduction and statement of the main result

Let $\mathbb {A}_F$ be the ring of adèles over a number field $F$ with absolute norm $\mathrm {N}=\mathrm {N}_{F/\mathbb {Q}}$ and absolute discriminant $D_F$. Let $\mathfrak {F}_{n}$ be the set of cuspidal automorphic representations $\pi =\bigotimes _{v} \pi _{v}$ of $\mathrm {GL}_{n}(\mathbb {A}_F)$, where the (restricted) tensor product runs over all places of $F$ and $\pi$ is normalized so that its central character is trivial on the diagonally embedded copy of the positive reals. Let $\mathfrak {q}_{\pi }$ be the arithmetic conductor of $\pi$, $C(\pi )\geq 1$ the analytic conductor of $\pi$ (see (3.4)), and $\mathfrak {F}_n(Q)=\{\pi \in \mathfrak {F}_n\colon C(\pi )\leq Q\}$. The analytic conductor $C(\pi )$ is a useful measure for the arithmetic and spectral complexity of $\pi$. Our normalization for the central characters ensures that $|\mathfrak {F}_n(Q)|$ is finite.

Given $\pi \in \mathfrak {F}_n$ and $\pi '\in \mathfrak {F}_{n'}$, let $L(s,\pi \times \pi ')$ be the associated Rankin–Selberg $L$-function, and let $\widetilde {\pi }\in \mathfrak {F}_n$ and $\widetilde {\pi }'\in \mathfrak {F}_{n'}$ be the contragredient representations. When $\pi '\in \{\widetilde {\pi },\widetilde {\pi }'\}$, work of Brumley [Hum19, Appendix] and the authors [HT22] shows that there exists an effectively computable constant $c_{1}=c_{1}(n,n')>0$ such that $L(s,\pi \times \pi ')$ has a ‘standard’ zero-free region of the shape

(1.1)\begin{align} \text{Re}(s)\geq 1-\frac{c_{1}}{\log(C(\pi)C(\pi')(|\text{Im}(s)|+3)^{[F:\mathbb{Q}]})} \end{align}

apart from at most one real simple zero. This is comparable to the classical zero-free region for Dirichlet $L$-functions. Brumley (see [Bru06a] and [Lap13, Appendix]) established a much narrower zero-free region for all choices of $\pi$ and $\pi '$. The generalized Riemann hypothesis (GRH) asserts that $L(s,\pi \times \pi ')\neq 0$ for $\text {Re}(s)>\frac {1}{2}$. Zeros near the line $\text {Re}(s)=1$ are typically most damaging in applications, but even a zero-free region of the shape $\text {Re}(s)\geq 1-\delta$ for some constant $\delta =\delta (n,n',[F:\mathbb {Q}])>0$ would be sufficient to obtain many spectacular arithmetic consequences.

Since such strong zero-free regions for Rankin–Selberg $L$-functions remain out of reach, it is useful to show that zeros near the line $\text {Re}(s)=1$ must be ‘sparse’. A suitable quantitative formulation can serve as a proxy for a zero-free region of the shape $\text {Re}(s)\geq 1-\delta$. Famous consequences of this philosophy include Hoheisel's proof [Hoh30] that $p_{n+1}-p_n\ll p_n^{1-1/33\,000}$ (where $p_n$ is the $n$th prime) and Linnik's proof [Lin44] that if $\gcd (a,q)=1$, then there exists an absolute, effectively computable constant $B>0$ and a prime $p\leq q^B$ such that $p\equiv a\,(\mathrm {mod}\,\,q)$.

To quantify our notion of ‘sparse’, we define for $\sigma \geq 0$ and $T\geq 1$ the quantity

\[ N_{\pi\times\pi'}(\sigma,T)=|\{\rho=\beta+i\gamma\colon L(\rho,\pi\times\pi')=0,~\beta\geq\sigma,~|\gamma|\leq T\}|. \]

Note that $N_{\pi \times \pi '}(\frac {1}{2},T)$ is roughly $T\log (C(\pi )C(\pi ')T)$ via the argument principle and the functional equation, and GRH can be restated as $N_{\pi \times \pi '}(\sigma,T)=0$ for all $\sigma >\frac {1}{2}$. The zero density estimate

(1.2)\begin{equation} N_{\pi\times\pi'}(\sigma,T)\ll_{n,n',[F:\mathbb{Q}]}(C(\pi)C(\pi')T^{[F:\mathbb{Q}]})^{10^7 (n'n)^4(1-\sigma)}, \end{equation}

follows from work of Soundararajan and Thorner [ST19, Corollary 2.6]. Therefore, while an arbitrary Rankin–Selberg $L$-function $L(s,\pi \times \pi ')$ is not yet known to have the standard zero-free region (1.1), the bound (1.2) ensures that the number of zeros in the region (1.1) is $O_{n,n',[F:\mathbb {Q}]}(1)$.

Let $\mathcal {S}\subseteq \mathfrak {F}_n$, and let $\mathcal {S}(Q)=\{\pi \in \mathcal {S}\colon C(\pi )\leq Q\}$. In this article, we seek a strong averaged form of (1.2), namely

(1.3)\begin{equation} \sum_{\pi\in\mathcal{S}(Q)}N_{\pi\times\pi'}(\sigma,T)\ll_{n,n',[F:\mathbb{Q}],\varepsilon}(Q|\mathcal{S}(Q)|C(\pi')T^{[F:\mathbb{Q}]})^{A(1-\sigma)+\varepsilon}, \end{equation}

where $A=A(n,n',[F:\mathbb {Q}])>0$ is a constant and $\varepsilon >0$. The bound (1.3) follows from the works of Brumley, Thorner, and Zaman under at least one of the following hypotheses:¹

• • $\pi '\in \mathfrak {F}_1$ is trivial [TZ21, Theorem 1.2];

• • $\max \{n,n'\}\leq 4$ (see [BTZ22, Theorem 1.3]); or

• • $\pi '$ and each $\pi \in \mathcal {S}(Q)$ satisfy certain unproven partial progress towards the generalized Ramanujan conjecture (GRC) [BTZ22, Hypothesis 1.1 and Theorem 1.3].²

Here, we prove the first completely unconditional zero density estimate of the form (1.3).

Theorem 1.1 Let $n,n'\geq 1$ and $\varepsilon >0$. Let $\mathcal {S}\subseteq \mathfrak {F}_n$ and $\mathcal {S}(Q)=\{\pi \in \mathcal {S}\colon C(\pi )\leq Q\}$. If $0\leq \sigma \leq 1$, $\pi '\in \mathfrak {F}_{n'}$, and $Q,T\geq 1$, then

\[ \sum_{\pi\in\mathcal{S}(Q)}N_{\pi\times\pi'}(\sigma,T)\ll_{n,n',[F:\mathbb{Q}],\varepsilon} \big(|\mathcal{S}(Q)|^{4} \big(C(\pi') Q T^{[F:\mathbb{Q}]}\big)^{6.15\max\{n^2,n'n\}}\big)^{1-\sigma+\varepsilon}. \]

Theorem 1.1 is non-trivial when

(1.4)\begin{equation} \delta_{\mathcal{S}}=\liminf_{Q\to\infty}\frac{\log|\mathcal{S}(Q)|}{\log Q}>0. \end{equation}

This is important because in applications, it is usually convenient to bound $Q$ by a power of $|\mathcal {S}(Q)|$ or vice versa. When $\mathcal {S}=\mathfrak {F}_n$, we have the bounds

(1.5)\begin{equation} Q^{n+1}\ll_{n,F}|\mathfrak{F}_n(Q)|\ll_{\varepsilon}D_F^{-n^2}Q^{2n+\varepsilon}. \end{equation}

The upper bound in (1.5) was proved by Brumley et al. [BTZ22, Theorem A.1]. The lower bound in (1.5) follows from work of Brumley and Milićević [BM18, Theorem 1.1], who computed a constant $c_{n,F}>0$ such that if $\mathfrak {F}_n^*(Q)$ is the subset of $\pi \in \mathfrak {F}_n(Q)$ that are spherical at the archimedean places of $F$, then $|\mathfrak {F}_n^*(Q)|\sim c_{n,F}Q^{n+1}$. (The lower bound in (1.5) reflects the conjectured order of growth; see [BM18].) Together, Theorem 1.1 and (1.5) imply that

(1.6)\begin{equation} \sum_{\pi\in\mathfrak{F}_n(Q)}N_{\pi\times\pi'}(\sigma,T)\ll_{n,n',F,\varepsilon}(|\mathfrak{F}_n(Q)|C(\pi')^{n} T^{n[F:\mathbb{Q}]})^{7.1\max\{n,n'\}(1-\sigma)+\varepsilon}. \end{equation}

Furthermore, if $n'=n\geq 3$ and $\pi '\in \mathfrak {F}_{n'}$, then Theorem 1.1 and (1.5) together imply that

(1.7)\begin{equation} \sum_{\pi\in\mathfrak{F}_n(Q)}N_{\pi\times\pi'}(\sigma,T)\ll_{\varepsilon}(C(\pi') Q T^{[F:\mathbb{Q}]})^{9n^2(1-\sigma)+\varepsilon}. \end{equation}

2. Applications

We now describe some applications of Theorem 1.1. In what follows, we write $f\ll _{\nu }g$, $f=O_{\nu }(g)$, and $g\gg _{\nu }f$ to denote that there exists a constant $c>0$ such that $|f|\leq c|g|$ in the stated range. The implied constant $c$, which is effectively computable unless otherwise stated, will depend at most on $\nu$, $n$, $n'$, and $[F:\mathbb {Q}]$. The expression $f\asymp _{\nu } g$ means that $f\ll _{\nu }g$ and $g\ll _{\nu }f$. We use $\varepsilon >0$ to denote an arbitrarily small quantity that depends at most on $n$, $n'$, and $[F:\mathbb {Q}]$.

2.1 Bounds for Rankin–Selberg $L$-functions

It is a classical problem for Dirichlet $L$-functions to find strong bounds on the critical line $\text {Re}(s)=\frac {1}{2}$. The Phragmén–Lindelöf convexity principle shows that if $q_{\chi}$ is the conductor of a primitive Dirichlet character $\chi$, then $L(\tfrac {1}{2},\chi ) \ll q^{1/4}$; improving this bound by replacing $1/4$ with a smaller exponent is known as a subconvex bound. The multiplicative version of the classical large sieve inequality combined with an approximate functional equation shows that for almost all $\chi$, we have the bound $L(\frac {1}{2},\chi )\ll _{\varepsilon }q_{\chi }^{\varepsilon }$ for all $\varepsilon >0$, consistent with the generalized Lindelöf hypothesis (GLH).

For $\pi \in \mathfrak {F}_2$, GLH predicts that $L(\tfrac {1}{2},\pi )\ll _{\varepsilon }C(\pi )^{\varepsilon }$. Michel and Venkatesh [MV10] proved that there exists a fixed positive $\delta >0$ such that $L(\frac {1}{2},\pi )\ll _F C(\pi )^{1/4-\delta }$, the culmination of several decades of research. When $F=\mathbb {Q}$, a sharp mean value estimate for Hecke eigenvalues proved by Deshouillers and Iwaniec [DI82], in conjunction with the approximate functional equation, implies the bound $L(\frac {1}{2},\pi )\ll _{\varepsilon } (qT)^{\varepsilon }$ for almost all $\pi \in \mathfrak {F}_2$ of arithmetic conductor $q$, trivial central character, and archimedean complexity (Laplace eigenvalue or weight squared) lying in the dyadic interval $[T,2T]$. Note that in this case, $C(\pi )\asymp qT$.

For $\pi \in \mathfrak {F}_n$ with $n\geq 3$, the best uniform result towards the bound $L(\frac {1}{2},\pi )\ll _{\varepsilon }C(\pi )^{\varepsilon }$ predicted by GLH is that of Soundararajan and Thorner [ST19, Corollary 2.7], namely

(2.1)\begin{equation} L(\tfrac{1}{2},\pi)\ll C(\pi)^{{1}/{4}}(\log C(\pi))^{-1/(10^{17}n^3)}. \end{equation}

We mention three results that improve upon (2.1) in an average sense, each having complementary strengths. Jana [Jan21, Theorem 6] extended the GLH-on-average bound of Deshouillers and Iwaniec to the family of cuspidal automorphic representations of $\mathrm {GL}_n(\mathbb {A}_{\mathbb {Q}})$ of arithmetic conductor 1 and growing analytic conductor. Blomer [Blo23, Corollary 5] proved the corresponding result for the family of cuspidal automorphic representations of $\mathrm {GL}_n(\mathbb {A}_{\mathbb {Q}})$ of a large given prime arithmetic conductor $q$, trivial central character, and whose archimedean components are principal series representations confined to a compact subset of the unitary dual. Thorner and Zaman [TZ21, Theorem 1.3] proved that there exists a constant $c_{2} = c_{2}(n,[F:\mathbb {Q}])>0$ such that if $\varepsilon >0$, then

(2.2)\begin{equation} |\{\pi\in\mathfrak{F}_n(Q)\colon |L(\tfrac{1}{2},\pi)|\geq c_{2} C(\pi)^{{1}/{4}-{\varepsilon}/{10^{16}n^3}}\}|\ll_{F}|\mathfrak{F}_n(Q)|^{\varepsilon}. \end{equation}

Unlike the preceding results, (2.2) is uniform in both the arithmetic conductor and spectral aspects and holds over number fields other than $\mathbb {Q}$, but the savings over (2.1) is not comparable to GLH on average.

Given $\pi \in \mathfrak {F}_n$ and $\pi '\in \mathfrak {F}_{n'}$, Soundararajan and Thorner [ST19, Corollary 2.7] proved when $F=\mathbb {Q}$ that if $C(\pi \times \pi ')$ is the analytic conductor of $L(s,\pi \times \pi ')$, then

(2.3)\begin{equation} L(\tfrac{1}{2},\pi\times\pi')\ll |L(\tfrac{3}{2},\pi\times\pi')|^2 C(\pi\times\pi')^{{1}/{4}} (\log C(\pi\times\pi'))^{-1/(10^{17}(n'n)^3)}. \end{equation}

As of now, the best general upper bound for $|L(\tfrac {3}{2},\pi \times \pi ')|^2$ is larger than any fixed power of $\log C(\pi \times \pi ')$ (see [Li10, Theorem 2]). The factor of $|L(\tfrac {3}{2},\pi \times \pi ')|^2$ can be removed under certain partial progress toward GRC. The bound $L(\frac {1}{2},\pi \times \pi ')\ll _{\varepsilon }C(\pi \times \pi ')^{\varepsilon }$ is predicted by GLH.

In order to improve (2.3) on average with uniformity in $\pi$ and $\pi '$, one might first try to mimic the approach that worked well for Dirichlet $L$-functions and $\mathrm {GL}_2$ $L$-functions using trace formulae, approximate functional equations, the spectral large sieve, Voronoĭ summation, etc. While such methods have seen great success for $\mathrm {GL}_n\times \mathrm {GL}_{n'}$ with $n,n' \in \{1,2\}$, suitably uniform and flexible versions of these tools do not appear to be available yet in the general setting. The special case where $|n-n'|\leq 1$ exhibits some nice structural properties, lending itself to approaches via period integrals that completely avoids the aforementioned tools. To describe work in this direction, let $F=\mathbb {Q}$, $\mathcal {F}_n\subseteq \mathfrak {F}_n$ be the subset of cuspidal automorphic representations of $\mathrm {GL}_n(\mathbb {A}_{\mathbb {Q}})$ of arithmetic conductor 1, and $\mathcal {F}_n(Q)=\{\pi \in \mathcal {F}_n\colon C(\pi )\leq Q\}$. It follows from work of Jana [Jan22, Corollary 2.2] that if $\varepsilon >0$, $\pi '\in \mathcal {F}_n$, and the spectral parameters of $\pi '$ have real part at least $- {1}/({n^2+1})$ (which is far stronger than the best known unconditional lower bound $-\frac {1}{2}+ {1}/({n^2+1})$ due to Luo, Rudnick, and Sarnak [LRS99]), then

(2.4)\begin{equation} \sum_{\pi\in\mathcal{F}_n(Q)}\bigg|L\bigg(\frac{1}{2},\pi\times\pi'\bigg)\bigg|^2\ll_{\pi',\varepsilon}|\mathcal{F}_n(Q)|^{1+\varepsilon}. \end{equation}

Therefore, for fixed $\pi '\in \mathcal {F}_n$, GLH for $L(\frac {1}{2},\pi \times \pi ')$ holds on average over the $\pi \in \mathcal {F}_n$. Using Chebyshev's inequality, we conclude that for all $\delta >0$, there exists a constant $c_{\pi ',\delta }>0$ such that

(2.5)\begin{equation} |\{\pi\in\mathcal{F}_n(Q)\colon |L(\tfrac{1}{2},\pi\times \pi')|\geq c_{\pi',\delta} C(\pi\times \pi')^{2\delta}\}|\ll_{\pi',\delta}|\mathcal{F}_n(Q)|^{1-\delta}. \end{equation}

See also the work of Blomer [Blo12, Theorem 2], which proves a variant of (2.4) for families of ‘spectrally close’ Hecke–Maaß newforms on $\mathrm {SL}_n(\mathbb {Z})$.

Along the same lines as (2.2), we use (1.6) to prove the following result.

Theorem 2.1 Let $n,n'\geq 1$ and $Q\geq 1$. If $\varepsilon >0$ and $\pi '\in \mathfrak {F}_{n'}$, then

\[ |\{\pi\in\mathfrak{F}_n(Q)\colon |L(\tfrac{1}{2},\pi\times\pi')|\geq C(\pi\times\pi')^{{1}/{4}-{\varepsilon}/{(10^{10}\max\{n,n'\})}}\}|\ll_{\varepsilon}(C(\pi')^{n} |\mathfrak{F}_n(Q)|)^{\varepsilon}. \]

Remark 2.2 Given a subset $\mathcal {S}\subseteq \mathfrak {F}_n$, a similar result can be proved for $\pi \in \mathcal {S}(Q)$ using Theorem 1.1. Such a result would depend effectively on $\delta _{\mathcal {S}}$ in (1.4).

In contrast with Jana's work in (2.5), Theorem 2.1 provides a smaller power-saving improvement over (2.3), but the improvement is uniform in the arithmetic conductor and spectral aspects as well as in $\pi '$. The exceptional set in Theorem 2.1 is a much smaller than in (2.5). Theorem 2.1 removes the requirements that $n=n'$, that $q_{\pi }=q_{\pi '}=1$, and that the spectral parameters of $\pi '$ have real part at least $- {1}/({n^2+1})$. Finally, Theorem 2.1 is proved over any number field, while [Jan22] is only proved over $\mathbb {Q}$.

2.2 Effective multiplicity one

Let $\pi =\bigotimes _v \pi _v$ and $\pi '=\bigotimes _v \pi _v'$ be cuspidal automorphic representations in $\mathfrak {F}_n(Q)$. Under the assumption of GRH for $L(s,\pi \times \widetilde {\pi })$ and $L(s,\pi \times \pi ')$ and that $\pi _{\mathfrak {p}}$ and $\pi _{\mathfrak {p}}'$ are tempered for all prime ideals $\mathfrak {p} \mid \mathfrak {q}_{\pi }\mathfrak {q}_{\pi '}$, it is known that if $Y=(\log Q)^2$ and $\pi _{\mathfrak {p}}\cong \pi _{\mathfrak {p}}'$ for all $\mathfrak {p}\nmid \mathfrak {q}_{\pi }\mathfrak {q}_{\pi '}$ with $\mathrm {N}\mathfrak {p}\ll Y$, then $\pi =\pi '$ (see [IK04, Proposition 5.22]). Brumley [Bru06a], improving on work of Moreno [Mor85], proved that there exists a constant $B_n>0$ such that this result holds unconditionally with $Y=Q^{B_n}$. This result makes effective the multiplicity one theorems of Jacquet and Shalika [JS81, Theorem 4.8] and Piatetski-Shapiro [Pia79]. Any fixed $B_n>2n$ suffices [LW09].

When $n=2$, we have an average result that nearly achieves what GRH predicts. Specifically, let $\mathfrak {F}_2^{\flat }$ be the subset of $\pi \in \mathfrak {F}_2$ with squarefree conductor and trivial central character, and let $\pi '\in \mathfrak {F}_2^{\flat }$. For all $\varepsilon >0$, there exists an effectively computable constant $N_{\varepsilon }>0$, depending at most on $\varepsilon$ and $[F:\mathbb {Q}]$, such that

(2.6)\begin{equation} |\{\pi\in\mathfrak{F}_2^{\flat}(Q)\colon \text{$\pi_{\mathfrak{p}}\cong\pi_{\mathfrak{p}}'$ for all $\mathfrak{p}\nmid\mathfrak{q}_{\pi}$ with $\mathrm{N}\mathfrak{p}\leq (\log Q)^{N_{\varepsilon}}$}\}|\ll_{\varepsilon} Q^{\varepsilon}. \end{equation}

In particular, the implied constant does not depend on $\pi '$. This was proved by Duke and Kowalski [DK00, Theorem 3] when $F=\mathbb {Q}$ in a stronger form under the assumption of GRC. See Brumley's Ph.D. thesis [Bru04, Corollary 5.2.2] for a proof that does not use GRC.

If $\pi \in \mathfrak {F}_2$ and $\widetilde {\pi }\in \mathfrak {F}_2$ is the contragredient, then $L(s,\pi \times \widetilde {\pi })=\zeta _F(s)L(s,\pi,\mathrm {Ad})$, where $\zeta _F(s)$ is the Dedekind zeta function of $F$ and $\mathrm {Ad}$ is the adjoint square lift from a representation of $\mathrm {GL}_2(\mathbb {A}_F)$ to a representation of $\mathrm {GL}_3(\mathbb {A}_F)$. The arguments in [Bru04, DK00] rely on two key results as follows.

1. (1) Gelbart and Jacquet [GJ78] proved that if $\pi \in \mathfrak {F}_2$, then $L(s,\pi,\mathrm {Ad})$ is the $L$-function of an automorphic representation of $\mathrm {GL}_3(\mathbb {A}_F)$, denoted $\mathrm {Ad}\,\pi$, complete with a criterion by which one can determine whether $\mathrm {Ad}\,\pi$ is cuspidal (and lies in $\mathfrak {F}_3$).

2. (2) At most $O_{\varepsilon }(Q^{1/2+\varepsilon })$ representations $\pi \in \mathfrak {F}_2(Q)$ have the same adjoint lift. In addition, Ramakrishnan (see [DK00, Appendix] and [Ram00]) proved that $\mathrm {Ad}\colon \mathfrak {F}_2^{\flat }\to \mathfrak {F}_3$ is injective.

In attempting to generalize the strategy of Duke and Kowalski to $\mathrm {GL}_n$ for $n\geq 3$, one encounters some deep open problems. If $\pi \in \mathfrak {F}_n$, then for $\text {Re}(s)$ sufficiently large, $L(s,\pi \times \widetilde {\pi })$ factors as $\zeta _F(s)L(s,\pi,\mathrm {Ad})$, where $\mathrm {Ad}\,$ is the adjoint square lift from $\mathrm {GL}_{n}$ to $\mathrm {GL}_{n^2-1}$. Apart from some special cases, the following obstacles arise.

1. (1) The adjoint lift is not yet known to be automorphic for $n\geq 3$, and if it were, there is no known criterion for cuspidality.

2. (2) Let $\mathcal {H}_n\subseteq \mathfrak {F}_n(Q)$ be the subset of $\pi \in \mathfrak {F}_n(Q)$ such that $\mathrm {Ad}\,\pi \in \mathfrak {F}_{n^2-1}$. It is not known how many $\pi \in \mathcal {H}_n$ have the same adjoint lift.

Despite these setbacks, we can use (1.7) (more specifically, Corollary 7.1) to prove a $\mathrm {GL}_n$ analogue of (2.6) when $\pi '$ depends mildly on $Q$.

Theorem 2.3 Let $n\geq 3$ and $Q\geq 1$. There exists an absolute, effectively computable constant $c_{3}>0$ such that if $0<\varepsilon <1$ and $\pi '\in \mathfrak {F}_n((\log Q)^{c_{3}/(n^2 [F:\mathbb {Q}]^2)})$, then

\[ |\{\pi\in\mathfrak{F}_n(Q)\colon \pi_{\mathfrak{p}}\cong\pi_{\mathfrak{p}}'~\text{for all}~\mathfrak{p}\nmid\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'}~\text{with}~\mathrm{N}\mathfrak{p}\leq (\log Q)^{41n^2/\varepsilon}\}|\ll_{\varepsilon}Q^{\varepsilon}. \]

Remark 2.4 Given a subset $\mathcal {S}\subseteq \mathfrak {F}_n$, a similar result can be proved for $\pi \in \mathcal {S}(Q)$ using Theorem 1.1. Such a result would depend effectively on $\delta _{\mathcal {S}}$ in (1.4).

The proof is very flexible. For example, if $\varepsilon >0$ and $C(\pi ')\ll _{\varepsilon }Q^{c_{3}\varepsilon ^2/(n^2[F:\mathbb {Q}]^2)}$, then the same proof with minor changes in choices of parameters produces the bound

\[ |\{\pi\in\mathfrak{F}_n(Q)\colon \pi_{\mathfrak{p}}\cong\pi_{\mathfrak{p}}'~\text{for all}~\mathfrak{p}\nmid\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'}~\text{with}~\mathrm{N}\mathfrak{p}\leq Q^{\varepsilon}\}|\ll_{\varepsilon}Q^{\varepsilon}. \]

While the number of $\mathfrak {p}$ for which one needs to check that $\pi _{\mathfrak {p}}\cong \pi _{\mathfrak {p}}'$ is larger, the range of $C(\pi ')$ is greatly extended, and the threshold $\mathrm {N}\mathfrak {p}\leq Q^{\varepsilon }$ (reminiscent of Vinogradov's conjecture on the size of the least quadratic non-residue) still greatly improves on the unconditional range $\mathrm {N}\mathfrak {p}\ll _{\varepsilon }Q^{2n+\varepsilon }$ from [LW09].

2.3 Automorphic level of distribution

Let $\Lambda (m)$ be the von Mangoldt function, equal to $\log p$ if $m$ is a power of a prime $p$ and zero otherwise. The celebrated Bombieri–Vinogradov theorem states that if $\theta <\frac {1}{2}$ is fixed, then for all $A>0$, we have

(2.7)\begin{equation} \sum_{q\leq x^{\theta}}~\max_{\gcd(a,q)=1}\max_{y\leq x}\Bigg|\sum_{\substack{m\leq y \\ m\equiv a\,(\mathrm{mod}\,\,q)}}\Lambda(m)-\frac{y}{\varphi(q)}\Bigg|\ll_A\frac{x}{(\log x)^A}. \end{equation}

This may be viewed as an average form of GRH for Dirichlet $L$-functions. As part of his proof [Bom65], Bombieri proved a strong form of the zero density estimate in Theorem 1.1 for Dirichlet $L$-functions. We call any $\theta$ for which (2.7) holds a level of distribution for the primes. Elliott and Halberstam conjectured that any fixed $\theta <1$ is a level of distribution for the primes.

Number theorists have proved several interesting extensions and variations of (2.7). For example, Murty and Murty [MM87] proved that primes in the Chebotarev density theorem have a positive level of distribution. To describe a different direction for automorphic representations over $\mathbb {Q}$, we let $n\geq 2$ and consider $\pi \in \mathfrak {F}_{n}$ with conductor $q_{\pi }$. Let $\Lambda (m)$ be the von Mangoldt function, and define the numbers $a_{\pi }(m)$ by

\[ -\frac{L'}{L}(s,\pi)=\sum_{p}\sum_{k=1}^{\infty}\frac{\sum_{j=1}^{n}\alpha_{j,\pi}(p)^k\log p}{p^{ks}}=\sum_{n=1}^{\infty}\frac{a_{\pi}(m)\Lambda(m)}{m^s},\quad \text{Re}(s)>1. \]

Note that $a_{\pi }(p)=\lambda _{\pi }(p)$. For fixed $\theta < {1}/({n^2-2})$, Wong [Won20, Theorem 9] proved that if $\pi$ satisfies GRC and $L(s,\pi \times (\widetilde {\pi }\otimes \chi ))$ has no Landau–Siegel zero for all Dirichlet characters $\chi$, then for any $A>0$,

(2.8)\begin{equation} \sum_{q\leq x^{\theta}}~\max_{\gcd(a,q)=1}\max_{y\leq x}\Bigg|\sum_{\substack{m\leq y \\ \gcd(m,q_{\pi})=1 \\ m\equiv a\,(\mathrm{mod}\,\,q)}}|a_{\pi}(m)|^2\Lambda(m)-\frac{y}{\varphi(q)}\Bigg|\ll_{A,\pi}\frac{x}{(\log x)^A}. \end{equation}

This conditionally endows $L(s,\pi\times\widetilde{\pi})$ with a positive level of distribution $\theta$. The hypotheses for (2.8) hold for $\pi$ attached to non-CM holomorphic cuspidal newforms on congruence subgroups of $\mathrm {SL}_2(\mathbb {Z})$.

Let $\pi \in \mathfrak {F}_n$. Using (1.7), we unconditionally endow $L(s,\pi\times\widetilde{\pi})$ with a notion of positive level of distribution. In particular, we avoid recourse to unproven progress toward GRC or the absence of Landau–Siegel zeros.

Theorem 2.5 Let $F=\mathbb {Q}$ and $\pi \in \mathfrak {F}_{n}$. Fix $\theta <1/(9n^3)$. If $A>0$, then

\[ \sum_{\substack{q\leq x^{\theta} \\ \gcd(q,q_{\pi}\!)=1}}\max_{\gcd(a,q)=1}\max_{y\leq x} \Bigg|\sum_{\substack{m\leq y \\ m\equiv a\,(\mathrm{mod}\,\,q)}}|a_{\pi}(m)|^2\Lambda(m)-\frac{y}{\varphi(q)}\Bigg| \ll_{A,\pi}\frac{x}{(\log x)^A}. \]

The implied constants are ineffective.

Remark 2.6 One can prove an analogue of Theorem 2.5 with $F\neq \mathbb {Q}$, replacing residue classes modulo $q$ with ray classes modulo $\mathfrak {q}$. We restrict to $F=\mathbb {Q}$ for notational simplicity.

Overview of the paper

In § 3, we recall basic properties of standard $L$-functions and Rankin–Selberg $L$-functions that we use in our proofs. In § 4, we prove a large sieve inequality for the Dirichlet coefficients of $L(s,\pi \times \pi ')^{-1}$ and a corollary on mean values of Dirichlet polynomials, which we use in our proof of Theorem 1.1 in § 5. We then prove Theorem 2.1 in § 6, Theorem 2.3 in § 7, and Theorem 2.5 in § 8.

3. Properties of $L$-functions

We recall some standard facts about $L$-functions arising from automorphic representations and their Rankin–Selberg convolutions. See [Bru06a, GJ72, JPS83, MW89, ST19].

3.1 Standard $L$-functions

Given $\pi \in \mathfrak {F}_n$, let $\widetilde {\pi }\in \mathfrak {F}_n$ be the contragredient representation and $\mathfrak {q}_{\pi }$ be the conductor of $\pi$. We express $\pi$ as a restricted tensor product $\bigotimes _v \pi _v$ of smooth admissible representations of $\mathrm {GL}_n(F_v)$, where $v$ varies over places of $F$. When $v$ is a non-archimedean place corresponding with a prime ideal $\mathfrak {p}$, then the local $L$-function $L(s,\pi _{\mathfrak {p}})$ is defined in terms of the Satake parameters $A_{\pi }(\mathfrak {p})=\{\alpha _{1,\pi }(\mathfrak {p}),\ldots,\alpha _{n,\pi }(\mathfrak {p})\}$ by

(3.1)\begin{equation} L(s,\pi_{\mathfrak{p}})=\prod_{j=1}^{n}(1-\alpha_{j,\pi}(\mathfrak{p})\mathrm{N}\mathfrak{p}^{-s})^{-1}=\sum_{k=0}^{\infty}\frac{\lambda_{\pi}(\mathfrak{p}^k)}{\mathrm{N}\mathfrak{p}^{ks}}. \end{equation}

We have $\alpha _{j,\pi }(\mathfrak {p})\neq 0$ for all $j$ whenever $\mathfrak {p}\nmid \mathfrak {q}_{\pi }$, and when $\mathfrak {p} \mid \mathfrak {q}_{\pi }$, it might be the case that there exist $j$ such that $\alpha _{j,\pi }(\mathfrak {p})=0$. The standard $L$-function $L(s,\pi )$ associated to $\pi$ is of the form

\[ L(s,\pi)=\prod_{\mathfrak{p}} L(s,\pi_{\mathfrak{p}})=\sum_{\mathfrak{n}}\frac{\lambda_{\pi}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^s}. \]

The Euler product and Dirichlet series converge absolutely when $\text {Re}(s)>1$.

At each archimedean place $v$ of $F$, there are $n$ Langlands parameters $\mu _{j,\pi }(v)\in \mathbb {C}$ such that

\[ L(s,\pi_{\infty}) = \prod_{v|\infty}\prod_{j=1}^{n}\Gamma_{ v}(s+\mu_{j,\pi}(v)),\quad \Gamma_{v}(s):= \begin{cases} \pi^{-s/2}\Gamma(s/2) & \mbox{if}\ F_{ v}=\mathbb{R},\\ 2(2\pi)^{-s}\Gamma(s) & \mbox{if}\ F_{ v}=\mathbb{C}. \end{cases} \]

By combining the work in [BB11, BB13, LRS99, MS04], we know that there exists

(3.2)\begin{equation} 0\leq\theta_n\leq\begin{cases} 0 & \mbox{if}\ n=1,\\ 7/64 & \mbox{if}\ n=2,\\ 5/14 & \mbox{if}\ n=3,\\ 9/22 & \mbox{if}\ n=4,\\ 1/2-1/(n^2+1) & \mbox{if}\ n\geq 5,\end{cases} \end{equation}

such that

(3.3)\begin{equation} |\alpha_{j,\pi}(\mathfrak{p})|\leq \mathrm{N}\mathfrak{p}^{\theta_n}\quad\text{and}\quad\text{Re}(\mu_{j,\pi}(v))\geq -\theta_n. \end{equation}

GRC asserts that in (3.2), one may take $\theta _n=0$. We have $\mathfrak {q}_{\pi }=\mathfrak {q}_{\widetilde {\pi }}$, and for each $\mathfrak {p}$ and each $v$, we have the equalities of sets $\{\alpha _{j,\widetilde {\pi }}(\mathfrak {p})\}=\{\overline { \alpha _{j,\pi }(\mathfrak {p})}\}$ and $\{\mu _{j,\widetilde {\pi }}(v)\}=\{\overline {\mu _{j,\pi }(v)}\}$.

Let $r_{\pi }$ be the order of the pole of $L(s,\pi )$ at $s=1$. The completed $L$-function

\[ \Lambda(s,\pi) = (s(s-1))^{r_{\pi}}(D_F^n \mathrm{N}\mathfrak{q}_{\pi})^{s/2}L(s,\pi)L(s,\pi_{\infty}) \]

is entire of order 1, and there exists a complex number $W(\pi )$ of modulus 1 such that for all $s\in \mathbb {C}$, we have the functional equation $\Lambda (s,\pi )=W(\pi )\Lambda (1-s,\widetilde {\pi })$. Let $d(v)=1$ if $F_{ v}=\mathbb {R}$ and $d(v)=2$ if $F_{ v}=\mathbb {C}$. The analytic conductor of $\pi$ (see [IS00]) is given by

(3.4)\begin{equation} C(\pi,t):= D_F^n \mathrm{N}\mathfrak{q}_{\pi}\prod_{v|\infty}\prod_{j=1}^n(3+|it+\mu_{j,\pi}(v)|^{d(v)}),\quad C(\pi):= C(\pi,0). \end{equation}

Since $\Lambda (s,\pi )$ is entire of order 1, there exist complex numbers $a_{\pi }$ and $b_{\pi }$ such that

\[ \Lambda(s,\pi)=e^{a_{\pi}+b_{\pi}s}\prod_{\Lambda(\rho,\pi)=0}\bigg(1-\frac{s}{\rho}\bigg)e^{s/\rho}. \]

The zeros $\rho$ in the above Hadamard product are the non-trivial zeros of $L(s,\pi )$, and the zeros of $L(s,\pi )$ that arise as poles of $s^{r_{\pi }}L(s,\pi _{\infty })$ are the trivial zeros.

3.2 Rankin–Selberg $L$-functions

Let $\pi \in \mathfrak {F}_n$ and $\pi '\in \mathfrak {F}_{n'}$. At each prime ideal $\mathfrak {p}$, Jacquet et al. [JPS83] associate to $\pi _{\mathfrak {p}}$ and $\pi _{\mathfrak {p}}'$ a local Rankin–Selberg $L$-function

(3.5)\begin{equation} L(s,\pi_{\mathfrak{p}}\times\pi_{\mathfrak{p}}')=\prod_{j=1}^{n}\prod_{j'=1}^{n'}(1-\alpha_{j,j',\pi \times\pi'}(\mathfrak{p}) \mathrm{N}\mathfrak{p}^{-s})^{-1}=\sum_{k=0}^{\infty}\frac{\lambda_{\pi\times\pi'}(\mathfrak{p}^k)}{\mathrm{N}\mathfrak{p}^{ks}} \end{equation}

and a local conductor $\mathfrak {q}_{\pi _{\mathfrak {p}}\times \pi _{\mathfrak {p}}'}$. If $\mathfrak {p}\nmid \mathfrak {q}_{\pi }\mathfrak {q}_{\pi '}$, then we have the equality of sets

(3.6)\begin{equation} \{\alpha_{j,j',\pi\times\pi'}(\mathfrak{p})\}=\{\alpha_{j,\pi}(\mathfrak{p})\alpha_{j',\pi'}(\mathfrak{p})\}. \end{equation}

The Rankin–Selberg $L$-function $L(s,\pi \times \pi ')$ associated to $\pi$ and $\pi '$ and its arithmetic conductor are

\[ L(s,\pi\times\pi')=\prod_{\mathfrak{p}}L(s,\pi_{\mathfrak{p}}\times\pi_{\mathfrak{p}}')=\sum_{\mathfrak{n}}\frac{\lambda_{\pi\times\pi'}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^s},\quad \mathfrak{q}_{\pi\times\pi'}=\prod_{\mathfrak{p}}\mathfrak{q}_{\pi_{\mathfrak{p}}\times\pi_{\mathfrak{p}}'}. \]

At an archimedean place $v$ of $F$, Jacquet, Piatetski-Shapiro, and Shalika associate $n'n$ complex Langlands parameters $\mu _{j,j',\pi \times \pi '}(v)$ to $\pi _v$ and $\pi _v'$, from which one defines

\[ L(s,\pi_{\infty}\times\pi_{\infty}') = \prod_{v|\infty}\prod_{j=1}^{n}\prod_{j'=1}^{n'}\Gamma_{ v}(s+\mu_{j,j',\pi\times\pi'}(v)). \]

Using the explicit descriptions of $\alpha _{j,j',\pi \times \pi '}(\mathfrak {p})$ and $\mu _{j,j',\pi \times \pi '}(v)$ in [Hum19, ST19], one sees that

(3.7)\begin{equation} |\alpha_{j,j',\pi\times\pi'}(\mathfrak{p})|\leq\mathrm{N}\mathfrak{p}^{\theta_n + \theta_{n'}},\quad \text{Re}(\mu_{j,j',\pi\times\pi'}(v))\geq -\theta_n - \theta_{n'}. \end{equation}

Let $r_{\pi \times \pi '} = -\mathrm {ord}_{s=1}L(s,\pi \times \pi ')$. By our normalization for the central characters of $\pi$ and $\pi '$, we have that $r_{\pi \times \pi '}=0$ if and only if $\pi \neq \widetilde {\pi }'$, and $r_{\pi \times \widetilde {\pi }}=1$ otherwise. The function

(3.8)\begin{equation} \Lambda(s,\pi\times\pi')=(s(s-1))^{r_{\pi\times\pi'}}(D_F^{n'n}\mathrm{N}\mathfrak{q}_{\pi\times\pi'})^{s/2}L(s,\pi\times\pi')L(s,\pi_{\infty}\times\pi_{\infty}') \end{equation}

is entire of order 1, and there exists a complex number $W(\pi \times \pi ')$ of modulus 1 such that $\Lambda (s,\pi \times \pi ')$ satisfies the functional equation $\Lambda (s,\pi \times \pi ')=W(\pi \times \pi ')\Lambda (1-s,\widetilde {\pi }\times \widetilde {\pi }')$. As with $L(s,\pi )$, the analytic conductor of $L(s,\pi \times \pi ')$ is given by

(3.9)\begin{equation} C(\pi\times\pi',t):= D_F^{n'n}\mathrm{N}\mathfrak{q}_{\pi\times\pi'}\prod_{v|\infty}\prod_{j=1}^n \prod_{j'=1}^{n'}(3+|it+\mu_{j,j',\pi\times\pi'}(v)|^{d(v)}),\quad C(\pi\times\pi'):= C(0,\pi\times\pi'). \end{equation}

The combined work of Bushnell and Henniart [BH97] and Brumley [Hum19, Appendix] yields

(3.10)\begin{equation} C(\pi\times\pi',t)\ll C(\pi\times\pi')(3+|t|)^{[F:\mathbb{Q}] n'n},\quad C(\pi\times\pi')\ll C(\pi)^{n'}C(\pi')^{n}. \end{equation}

Since $\Lambda (s,\pi \times \pi ')$ is entire of order 1, there exist complex numbers $a_{\pi \times \pi '}$ and $b_{\pi \times \pi '}$ such that the Hadamard factorization

(3.11)\begin{equation} \Lambda(s,\pi\times\pi')=e^{a_{\pi\times\pi'}+b_{\pi\times\pi'}s}\prod_{\Lambda(\rho,\pi\times\pi')=0} \bigg(1-\frac{s}{\rho}\bigg)e^{s/\rho} \end{equation}

holds. The zeros $\rho$ in (3.11) are the non-trivial zeros of $L(s,\pi \times \pi ')$, and the zeros of $L(s,\pi \times \pi ')$ that arise as poles of $s^{r_{\pi \times \pi '}}L(s,\pi _{\infty }\times \pi _{\infty }')$ are the trivial zeros.

It follows from work of Li [Li10, Theorem 2] (with minor adjustments when $F\neq \mathbb {Q}$) that there exists an absolute and effectively computable constant $c_{4}>0$, which we assume to be sufficiently large for future convenience, such that

(3.12)\begin{equation} \lim_{\sigma_0\to\sigma}(\sigma_0-1)^{r_{\pi\times\pi'}}L(\sigma_0,\pi\times\pi')\ll \exp\bigg(c_{4}n'n[F:\mathbb{Q}]\frac{\log C(\pi\times\pi')}{\log\log C(\pi\times\pi')}\bigg),\quad \sigma\in[1,3]. \end{equation}

We can change $\pi '$ to $\pi '\otimes |\det |^{it}$; at the archimedean places, this has the effect of adding $it$ to each $\mu _{j,j',\pi \times \pi '}(v)$. We then apply functional equation, the Phragmén–Lindelöf convexity principle, and (3.10) to obtain for all $\sigma \geq 0$

(3.13)\begin{align} &\lim_{\sigma_0\to\sigma}\bigg(\frac{\sigma_0+it-1}{\sigma_0+it+1}\bigg)^{r_{\pi\times\pi'}}L(\sigma_0+it,\pi\times\pi')\nonumber\\ &\quad\ll_{\varepsilon} C(\pi\times\pi',t)^{{\max\{1-\sigma,0\}}/{2}+{\varepsilon}/{n'n[F:\mathbb{Q}]}}\nonumber\\ &\quad\ll_{\varepsilon} (C(\pi)^{n'} C(\pi')^n(3+|t|)^{n'n[F:\mathbb{Q}]})^{{\max\{1-\sigma,0\}}/{2}+{\varepsilon}/{n'n[F:\mathbb{Q}]}}. \end{align}

Lemma 3.1 If $\pi \in \mathfrak {F}_n$, $X\geq 3$, and $\varepsilon >0$, then $\sum _{\mathrm {N}\mathfrak {n}\leq X}\lambda _{\pi \times \widetilde {\pi }}(\mathfrak {n})/\mathrm {N}\mathfrak {n}\ll _{\varepsilon } C(\pi )^{\varepsilon }\log X$.

Proof. Since $\lambda _{\pi \times \widetilde {\pi }}(\mathfrak {n})\geq 0$ for all $\mathfrak {n}$ by [HR95, Lemma a], we observe by (3.12) that

\[ \sum_{\mathrm{N}\mathfrak{n}\leq X}\frac{\lambda_{\pi\times\widetilde{\pi}}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}}\leq e \sum_{\mathfrak{n}}\frac{\lambda_{\pi\times\widetilde{\pi}}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{1+{1}/{\!\log X}}}\ll (\log X)\mathop{\mathrm{Res}}\limits_{s=1}L(s,\pi\times\widetilde{\pi}). \]

The desired bounded now follows from (3.13) with $\sigma =1$, $t=0$, and $\pi '=\widetilde {\pi }$.

Lemma 3.2 Let $J\geq 1$ be an integer. For all $j\in \{1,\ldots,J\}$, let $t_j\in \mathbb {R}$; $n_j,n_j'$ be positive integers; and $\pi _j\in \mathfrak {F}_{n_j}$ and $\pi _j'\in \mathfrak {F}_{n_j'}$. Let

\[ D(s) = \prod_{j=1}^J L(s+it_j,\pi_j\times\widetilde{\pi}_j'),\quad \Delta(s) = \prod_{j=1}^J \Lambda(s+it_j,\pi_j\times\widetilde{\pi}_j'),\quad Q=\prod_{j=1}^J C(\pi_j)^{n_j'}C(\pi_j')^{n_j}. \]

Let $R=-\mathrm {ord}_{s=1}D(s)$. If $1<\sigma <2$ and the $\mathfrak {n}$th Dirichlet coefficient of $- ({D'}/{D})(s)$ is non-negative when $\gcd (\mathfrak {n},\prod _{j=1}^J \mathfrak {q}_{\pi _j}\mathfrak {q}_{\pi _j'})=\mathcal {O}_F$, then

\[ \sum_{\Delta(\rho)=0}\text{Re}\bigg(\frac{1}{\sigma-\rho}\bigg) <\frac{R}{\sigma-1} + \sum_{\substack{1\leq j\leq J \\ \pi_j=\pi_j',~t_j\neq 0}}\frac{\sigma-1}{(\sigma-1)^2+t_j^2}+O(\log Q). \]

Proof. Let $1<\sigma <2$ and

\[ D_{\infty}(s) = \prod_{j=1}^{J}L(s,(\pi_j)_{\infty}\times(\widetilde{\pi}_j')_{\infty}),\quad \mathfrak{q}_{D}=\prod_{j=1}^{J}\mathfrak{q}_{\pi_j\times\widetilde{\pi}_j'}. \]

By comparing the logarithmic derivative of $\Delta (s)$ with the logarithmic derivative of its Hadamard factorization

\[ \Delta(s)=e^{a_{D}+b_{D}s}\prod_{\Delta(\rho)=0}\bigg(1-\frac{s}{\rho}\bigg)e^{s/\rho}, \]

we find that

\begin{align*} &\sum_{\Delta(\rho)=0}\bigg(\frac{1}{\sigma-\rho}+\frac{1}{\rho}\bigg)+ b_{D}\\ &\quad =\frac{D'}{D}(\sigma)+\frac{\log\mathrm{N}\mathfrak{q}_{D}}{2}+\frac{R}{\sigma-1}+\sum_{\substack{1\leq j\leq J\\ \pi_j = \pi_j'}}\bigg(\frac{1}{\sigma+it_j-1}+\frac{1}{\sigma+it_j}\bigg) +\frac{D_{\infty}'}{D_{\infty}}(\sigma). \end{align*}

Since $\text {Re}(b_{D})=-\sum _{\Delta (\rho )=0}\text {Re}(\rho ^{-1})$ (see [IK04, Proposition 5.7(3)]), we take real parts and obtain

\[ \sum_{\Delta(\rho)=0}\text{Re}\bigg(\frac{1}{\sigma-\rho}\bigg)=\text{Re} \bigg(\frac{D'}{D}(s)+\frac{\log\mathrm{N}\mathfrak{q}_{D}}{2}+\frac{R}{\sigma-1}+\sum_{\substack{1\leq j\leq J \\ \pi_j=\pi_j' \\ t_j\neq 0}}\frac{\sigma-1}{(\sigma-1)^2+t_j^2}+\frac{D_{\infty}'}{D_{\infty}}(s)\bigg)+O(1). \]

By (3.3), (3.7), and (3.10), the bound $\text {Re}( ({D'}/{D})(\sigma ))\ll \log Q$ (respectively, $ ({D_{\infty }'}/{D_{\infty }})(\sigma )\ll \log Q$) follows from our hypothesis on the Dirichlet coefficients of $- ({D'}/{D})(s)$ (respectively, Stirling's formula).

3.3 Rankin–Selberg combinatorics

A partition $\mu =(\mu _i)_{i=1}^{\infty }$ is a sequence of non-increasing nonnegative integers $\mu _1\geq \mu _2\geq \cdots$ with only finitely many non-zero entries. For a partition $\mu$, let $\ell (\mu )$ be the number of non-zero $\mu _i$, and let $|\mu |=\sum _{i=1}^{\infty } \mu _i$. For a set $\{\alpha _{1},\ldots,\alpha _n \}$ of real numbers and a partition $\mu$ with $\ell (\mu )\leq n$, let $s_{\mu }(\{\alpha _1,\ldots,\alpha _n\})$ be the Schur polynomial $\det [(\alpha _{i}^{\lambda (j)+n-j})_{ij}] / \det [(\alpha _{i}^{n-j})_{ij}]$ associated to $\mu$. If $|\mu |=0$, then $s_{\mu }(\{\alpha _1,\ldots,\alpha _n\})$ is identically one. By convention, if $\ell (\mu )>n$, then $s_{\mu }(\{\alpha _1,\ldots,\alpha _n\})$ is identically zero.

Let $\pi \in \mathfrak {F}_n$ and $\pi '\in \mathfrak {F}_{n'}$. By (3.5), (3.6), and Cauchy's identity [Bum13, Theorem 38.1], we have

\[ \sum_{k=0}^{\infty}\frac{\lambda_{\pi\times\pi'}(\mathfrak{p}^k)}{\mathrm{N}\mathfrak{p}^{ks}}=L(s,\pi_{\mathfrak{p}}\times\pi_{\mathfrak{p}}')=\sum_{\mu}\frac{s_{\mu}(A_{\pi}(\mathfrak{p}))s_{\mu}(A_{\pi'}(\mathfrak{p}))}{\mathrm{N}\mathfrak{p}^{s|\mu|}},\quad \mathfrak{p}\nmid\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'}, \]

where the sum ranges over all partitions. This yields

\[ \lambda_{\pi\times\pi'}(\mathfrak{p}^k) = \sum_{ |\mu|=k}s_{\mu}(A_{\pi}(\mathfrak{p}))s_{\mu}(A_{\pi'}(\mathfrak{p})),\quad \mathfrak{p}\nmid\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'}. \]

For an integral ideal $\mathfrak {n}$ with factorization $\mathfrak {n}=\prod _{\mathfrak {p}}\mathfrak {p}^{\mathrm {ord}_{\mathfrak {p}}(\mathfrak {n})}$ (with $\mathrm {ord}_{\mathfrak {p}}(\mathfrak {n})=0$ for all but finitely many $\mathfrak {p}$), the multiplicativity of $\lambda _{\pi \times \pi '}(\mathfrak {n})$ tells us that if $\gcd (\mathfrak {n},\mathfrak {q}_{\pi }\mathfrak {q}_{\pi '})=\mathcal {O}_F$, then

(3.14)\begin{equation} \lambda_{\pi\times\pi'}(\mathfrak{n})=\prod_{\mathfrak{p}}\lambda_{\pi\times\pi'}(\mathfrak{p}^{\mathrm{ord}_{\mathfrak{p}}(\mathfrak{n})})=\sum_{(\mu_{\mathfrak{p}}\!)_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]}\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}}(A_{\pi}(\mathfrak{p}))s_{\mu_{\mathfrak{p}}}(A_{\pi'}(\mathfrak{p})), \end{equation}

where $(\mu _{\mathfrak {p}})_{\mathfrak {p}}$ denotes a sequence of partitions indexed by prime ideals and

(3.15)\begin{equation} \underline{\mu}[\mathfrak{n}]:= \{(\mu_{\mathfrak{p}}\!)_{\mathfrak{p}}\colon |\mu_{\mathfrak{p}}|=\mathrm{ord}_{\mathfrak{p}}(\mathfrak{n})\text{ for all $\mathfrak{p}$}\}. \end{equation}

Define the numbers $\mu _{\pi \times \pi '}(\mathfrak {n})$ on unramified prime powers by

\[ \sum_{k=0}^{\infty}\frac{\mu_{\pi\times\pi'}(\mathfrak{p}^k)}{\mathrm{N}\mathfrak{p}^{ks}}=L(s,\pi_{\mathfrak{p}}\times\pi_{\mathfrak{p}}')^{-1}=\prod_{j=1}^{n} \prod_{j'=1}^{n'}(1-\alpha_{j,\pi}(\mathfrak{p})\alpha_{j',\pi'}(\mathfrak{p})\mathrm{N}\mathfrak{p}^{-s}),\quad \mathfrak{p}\nmid\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'}. \]

By multiplicativity, this defines $\mu _{\pi \times \pi '}(\mathfrak {n})$ when $\gcd (\mathfrak {n},\mathfrak {q}_{\pi }\mathfrak {q}_{\pi '})=\mathcal {O}_F$. For a partition $\mu =(\mu _i)_{i=1}^{\infty }$, let $\mu ^*=(\mu _i^*)_{i=1}^{\infty }$ be the dual partition defined by $\mu _i^*=|\{j\colon \mu _j\geq i\}|$. It follows from the dual Cauchy identity [Bum13, Chapter 38] and (3.6) that

\[ \sum_{k=0}^{\infty}\frac{\mu_{\pi\times\pi'}(\mathfrak{p}^k)}{\mathrm{N}\mathfrak{p}^{ks}}=\sum_{\mu}\frac{s_{\mu}(A_{\pi}(\mathfrak{p}))s_{\mu^*}(-A_{\pi'}(\mathfrak{p}))}{\mathrm{N}\mathfrak{p}^{|\mu|s}},\quad \mathfrak{p}\nmid\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'}, \]

where $-A_{\pi '}(\mathfrak {p})=\{-\alpha _{1,\pi '}(\mathfrak {p}),\ldots,-\alpha _{n,\pi '}(\mathfrak {p})\}$. Hence, we have

(3.16)\begin{equation} \mu_{\pi\times\pi'}(\mathfrak{n})=\sum_{(\mu_{\mathfrak{p}}\!)_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]}\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}}(A_{\pi}(\mathfrak{p}))s_{\mu^*_{\mathfrak{p}}}(-A_{\pi}(\mathfrak{p})),\quad \gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'})=\mathcal{O}_F . \end{equation}

Lemma 3.3 If $\gcd (\mathfrak {n},\mathfrak {q}_{\pi }\mathfrak {q}_{\pi '})=\mathcal {O}_F$, then we have $|\mu _{\pi \times \pi '}(\mathfrak {n})|\leq \frac {1}{2}(\lambda _{\pi \times \widetilde {\pi }}(\mathfrak {n})+\lambda _{\pi '\times \widetilde {\pi }'}(\mathfrak {n}))$.

Proof. We apply the inequality of arithmetic and geometric means to (3.16):

\[ |\mu_{\pi\times\pi'}(\mathfrak{n})|\leq \frac{1}{2}\bigg(\sum_{(\mu_{\mathfrak{p}}\!)_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]}\bigg|\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}}(A_{\pi}(\mathfrak{p}))\bigg|^2+\sum_{(\mu_{\mathfrak{p}}\!)_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]}\bigg|\prod_{\mathfrak{p}}s_{\mu^*_{\mathfrak{p}}}(-A_{\pi'}(\mathfrak{p}))\bigg|^2\bigg). \]

The first sum equals $\lambda _{\pi \times \widetilde {\pi }}(\mathfrak {n})$ by (3.14). For the second sum, note that since $|\mu |=|\mu ^*|$, we have $(\mu _{\mathfrak {p}})_{\mathfrak {p}}\in \underline {\mu }[\mathfrak {n}]$ if and only if $(\mu _{\mathfrak {p}}^*)_{\mathfrak {p}}\in \underline {\mu }[\mathfrak {n}]$. Hence, by rearranging, we see that

(3.17)\begin{equation} \sum_{(\mu_{\mathfrak{p}}\!)_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]}\bigg|\prod_{\mathfrak{p}}s_{\mu^*_{\mathfrak{p}}}(-A_{\pi'}(\mathfrak{p}))\bigg|^2= \sum_{(\mu_{\mathfrak{p}}\!)_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]}\bigg|\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}}(-A_{\pi'}(\mathfrak{p}))\bigg|^2= \lambda_{\pi\times\widetilde{\pi}'}(\mathfrak{n}), \end{equation}

where the last equality holds because $\alpha _{j,\pi '}(\mathfrak {p})\overline {\alpha _{j',\pi '}(\mathfrak {p})}=(-\alpha _{j,\pi '}(\mathfrak {p}))\overline {(-\alpha _{j',\pi '}(\mathfrak {p}))}$.

4. A new mean value estimate

Our proof of Theorem 1.1 uses the following new mean value estimate for the Dirichlet coefficients of $L(s,\pi \times \pi ')$ and $L(s,\pi \times \pi ')^{-1}$. Let $\mathcal {S}\subseteq \mathfrak {F}_n$, and let $\mathcal {S}(Q)=\{\pi \in \mathcal {S}\colon C(\pi )\leq Q\}$.

Theorem 4.1 Let $b$ be a complex-valued function supported on the integral ideals of $\mathcal {O}_F$. Let $n,n'\geq 1$, and let $\pi '\in \mathfrak {F}_{n'}$. Let $Q,T\geq 1$, $\varepsilon >0$, and $x\geq 1$. Both

\[ \sum_{\pi\in\mathcal{S}(Q)}\bigg|\sum_{\substack{\mathrm{N}\mathfrak{n}\in(x,xe^{1/T}] \\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'})=\mathcal{O}_F }}\mu_{\pi\times\pi'}(\mathfrak{n})b(\mathfrak{n})\bigg|^2\quad\text{and}\quad \sum_{\pi\in\mathcal{S}(Q)}\bigg|\sum_{\substack{\mathrm{N}\mathfrak{n}\in(x,xe^{1/T}] \\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'})=\mathcal{O}_F }}\lambda_{\pi\times\pi'}(\mathfrak{n})b(\mathfrak{n})\bigg|^2 \]

are

\[ \ll_{\varepsilon} Q^{\varepsilon}\bigg(\frac{x}{T}+Q^{4n^2\theta_n+n}T^{({1}/{2})n^2 [F:\mathbb{Q}]+\varepsilon}|\mathcal{S}(Q)|\bigg)\sum_{\substack{\mathrm{N}\mathfrak{n}\in(x,xe^{1/T}] \\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi'})=\mathcal{O}_F}}\lambda_{\pi'\times\widetilde{\pi}'}(\mathfrak{n})|b(\mathfrak{n})|^2, \]

where $\theta _n\in [0,\frac {1}{2}- {1}/({n^2+1})]$ is the best known exponent towards GRC for $\pi \in \mathcal {S}$.

Remark 4.2 If $\pi '= \mathbb{1}$ and $\mathcal {S}=\mathfrak {F}_n$, then Theorem 4.1 recovers [TZ21, Theorem 1.1]:³

\begin{align*} &\sum_{\pi\in\mathfrak{F}_n(Q)}\bigg|\sum_{\substack{\mathrm{N}\mathfrak{n}\in(x,xe^{1/T}]\\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi}\!)=\mathcal{O}_F }}\lambda_{\pi}(\mathfrak{n})b(\mathfrak{n})\bigg|^2\\ &\quad\ll_{\varepsilon} Q^{\varepsilon}\bigg(\frac{x}{T}+Q^{4n^2\theta_n+n}T^{({1}/{2})n^2 [F:\mathbb{Q}]+\varepsilon}|\mathfrak{F}_n(Q)|\bigg)\sum_{\substack{\mathrm{N}\mathfrak{n}\in(x,xe^{1/T}] \\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi'})=\mathcal{O}_F}}|b(\mathfrak{n})|^2. \end{align*}

One could try to prove Theorem 4.1 starting with this, replacing $b(\mathfrak {n})$ with $\lambda _{\pi '}(\mathfrak {n})\mu _F(\mathfrak {n})b(\mathfrak {n})$ (where $\mu _F(\mathfrak {n})$ is the $\mathfrak {n}$th Dirichlet coefficient of $\zeta _F(s)^{-1}$), and try to recover a version of Theorem 4.1 with $\mu _{\pi \times \pi '}(\mathfrak {n})$ replaced by $\lambda _{\pi }(\mathfrak {n})\lambda _{\pi '}(\mathfrak {n})\mu _F(\mathfrak {n})$. Note that $\mu _{\pi \times \pi '}(\mathfrak {n})=\lambda _{\pi }(\mathfrak {n})\lambda _{\pi '}(\mathfrak {n})\mu _F(\mathfrak {n})$ when $\mathfrak {n}$ is squarefree and coprime to $\mathfrak {q}_{\pi }\mathfrak {q}_{\pi '}$. Otherwise, equality is not guaranteed. If one wants to approximate $L(s,\pi \times \pi ')$ with $\sum _{\mathfrak{n}\ {\rm squarefree}}\lambda _{\pi }(\mathfrak {n})\lambda _{\pi '}(\mathfrak {n})\mathrm {N}\mathfrak {n}^{-s}$ and extend into the critical strip when $\pi,\pi '\in \mathfrak {F}_n$ and $n\geq 5$, then one must have progress towards GRC well beyond what is known unconditionally [Bru06b, DK00]. Such progress would then be a hypothesis for Theorem 1.1.

Theorem 4.1 provides the first non-trivial unconditional mean value estimates of large sieve type for the Dirichlet coefficients $\lambda _{\pi \times \pi '}(\mathfrak {n})$ or $\mu _{\pi \times \pi '}(\mathfrak {n})$ for arbitrary $n$ and $n'$. Theorem 4.1 follows from a more general result, Proposition 4.3, for sequences of products of Schur polynomials evaluated on the set $A_{\pi }(\mathfrak {p})$ of Satake parameters of $\pi$ at $\mathfrak {p}$. We begin with a mean value estimate for the Satake parameters of $\pi$ as $\pi \in \mathcal {S}$ varies. Let

(4.1)\begin{equation} a: \bigcup_{x<\mathrm{N}\mathfrak{n}\leq xe^{1/T}}\underline{\mu}[\mathfrak{n}]\to\mathbb{C},\quad \alpha:\mathcal{S}(Q)\to\mathbb{C} \end{equation}

be functions that are not identically zero. Their $\ell ^2$ norms $\|a\|_2$ and $\|\alpha \|_2$ are defined by

\[ \|a\|_2 = \bigg(\sum_{x<\mathrm{N}\mathfrak{n}\leq xe^{1/T}}~\sum_{(\mu_{\mathfrak{p}}\!)_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]}|a((\mu_{\mathfrak{p}}\!)_{\mathfrak{p}})|^2\bigg)^{1/2},\quad \|\alpha\|_2=\bigg(\sum_{\pi\in\mathcal{S}(Q)}|\alpha(\pi)|^2\bigg)^{1/2}. \]

For convenience, we define $\mathbf {1}_{(\mathfrak {n},\mathfrak {q})}$ to equal one when $\gcd (\mathfrak {n},\mathfrak {q})=\mathcal {O}_F$ and zero otherwise.

Proposition 4.3 Let $x\geq 1$ and $Q,T\geq 1$. Define

\[ C(Q,T,x):= \sup_{\|a\|_2\neq 0}\frac{1}{\|a\|_2^2}\sum_{\pi\in\mathcal{S}(Q)}\bigg|\sum_{\substack{x<\mathrm{N}\mathfrak{n}\leq xe^{1/T} \\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi}\!)=\mathcal{O}_F }}\sum_{(\mu_{\mathfrak{p}}\!)_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]}\bigg[\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}}(A_{\pi}(\mathfrak{p}))\bigg] a((\mu_{\mathfrak{p}}\!)_{\mathfrak{p}})\bigg|^2. \]

We have the bound $C(Q,T,x)\ll _{\varepsilon } Q^{\varepsilon }(T^{-1}x+Q^{4n^2\theta _n+n} T^{ {n^2[F:\mathbb {Q}]}/{2}+\varepsilon }|\mathcal {S}(Q)|)$.

Proof. We observe that

(4.2)\begin{equation} C(Q,T,x)=\sup_{\|a\|_2=1}\sum_{\pi\in\mathcal{S}(Q)}\bigg|\sum_{\substack{x<\mathrm{N}\mathfrak{n}\leq xe^{1/T}}}~\sum_{(\mu_{\mathfrak{p}}\!)_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]}\bigg[\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}}(A_{\pi}(\mathfrak{p}))\bigg] \mathbf{1}_{(\mathfrak{n},\mathfrak{q}_{\pi}\!)}a((\mu_{\mathfrak{p}}\!)_{\mathfrak{p}})\bigg|^2. \end{equation}

By the duality principle for bilinear forms, (4.2) is bounded by the supremum over the functions $\alpha \colon \mathcal {S}(Q)\to \mathbb {C}$ such that $\|\alpha \|_2=1$ of

(4.3)\begin{equation} \sum_{x<\mathrm{N}\mathfrak{n}\leq xe^{1/T}}~\sum_{(\mu_{\mathfrak{p}}\!)_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]} \bigg|\sum_{\pi\in\mathcal{S}(Q)}\bigg[\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}}(A_{\pi}(\mathfrak{p}))\bigg] \mathbf{1}_{(\mathfrak{n},\mathfrak{q}_{\pi}\!)}\alpha(\pi)\bigg|^2. \end{equation}

Let $\phi$ be a fixed smooth test function, supported in a compact subset of $[-2,2]$, such that $\phi (t)=1$ for $t\in [0,1]$ and $\phi (t)\in [0,1)$ otherwise. Then (4.3) is at most

(4.4)\begin{equation} \sum_{\mathfrak{n}}\sum_{(\mu_{\mathfrak{p}}\!)_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]}\bigg| \sum_{\pi\in\mathcal{S}(Q)}\bigg[\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}}(A_{\pi}(\mathfrak{p}))\bigg] \mathbf{1}_{(\mathfrak{n},\mathfrak{q}_{\pi}\!)}\alpha(\pi)\bigg|^2\phi\bigg(T\log\frac{\mathrm{N}\mathfrak{n}}{x}\bigg). \end{equation}

We expand the square, interchange the order of summation, and apply (3.14) to find that (4.4) equals

(4.5)\begin{align} &\sum_{\pi,\pi'\in\mathcal{S}(Q)}\alpha(\pi)\overline{\alpha(\pi')}\sum_{\mathfrak{n}}\sum_{(\mu_{\mathfrak{p}}\!)_{\mathfrak{p}}\in \underline{\mu}[\mathfrak{n}]}\bigg[\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}}(A_{\pi}(\mathfrak{p}))\bigg] \overline{\bigg[\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}}(A_{\pi'}(\mathfrak{p}))\bigg]} \mathbf{1}_{(\mathfrak{n},\mathfrak{q}_{\pi}\!)}\mathbf{1}_{(\mathfrak{n},\mathfrak{q}_{\pi'})}\phi\bigg(T\log\frac{\mathrm{N}\mathfrak{n}}{x}\bigg)\nonumber\\ &\quad =\sum_{\pi,\pi'\in\mathcal{S}(Q)}\alpha(\pi)\overline{\alpha(\pi')}\sum_{\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'})=\mathcal{O}_F }\lambda_{\pi\times\widetilde{\pi}'}(\mathfrak{n})\phi\bigg(T\log\frac{\mathrm{N}\mathfrak{n}}{x}\bigg). \end{align}

Let $\kappa _{\pi \times \pi '}={\mathrm {Res}}_{s=1}L(s,\pi \times \pi ')\prod _{\mathfrak {p} \mid \mathfrak {q}_{\pi }\mathfrak {q}_{\pi '}}L(s,\pi _{\mathfrak {p}}\times \pi _{\mathfrak {p}}')^{-1}$. Note that $\kappa _{\pi \times \pi '}\geq 0$, with equality if and only if $\pi '\neq \widetilde {\pi }$. Since $|\alpha _{j,j',\pi \times \pi '}(\mathfrak {p})|\leq \mathrm {N}\mathfrak {p}$, we have the bound

\[ \prod_{\mathfrak{p} \mid \mathfrak{q}_{\pi}\mathfrak{q}_{\pi'}}|L(1,\pi_{\mathfrak{p}}\times\pi_{\mathfrak{p}}')|^{-1}=\prod_{\mathfrak{p} \mid \mathfrak{q}_{\pi}\mathfrak{q}_{\pi'}}\prod_{j=1}^n \prod_{j'=1}^{n'}\bigg|1-\frac{\alpha_{j,j',\pi\times\pi'}(\mathfrak{p})}{\mathrm{N}\mathfrak{p}}\bigg|\leq \prod_{\mathfrak{p} \mid\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'}}2^{n'n}. \]

Since $|\{\mathfrak {p}\colon \mathfrak {p} \mid \mathfrak {n}\}|\ll (\log \mathrm {N}\mathfrak {n})/\!\log \log \mathrm {N}\mathfrak {n}$ (see [Wei83, Lemma 1.13b]), it follows from (3.12) that

(4.6)\begin{equation} \kappa_{\pi\times\widetilde{\pi}}\ll_{\varepsilon} C(\pi)^{\varepsilon}. \end{equation}

Let $\widehat {\phi }(s)=\int _{\mathbb {R}}\phi (y)e^{sy}\,{{d}y}$. It follows from a standard contour integral calculation using (3.7) and (3.13) that (4.5) equals

(4.7) \begin{align} &\sum_{\pi,\pi'\in\mathcal{S}(Q)}\alpha(\pi)\overline{\alpha(\pi')} \bigg(\frac{1}{2\pi iT}\int_{3-i\infty}^{3+i\infty}\frac{L(s,\pi\times\widetilde{\pi}')}{\prod_{\mathfrak{p} \mid \mathfrak{q}_{\pi}\mathfrak{q}_{\pi'}}L(s,\pi_{\mathfrak{p}}\times\widetilde{\pi}_{\mathfrak{p}}')}x^s\widehat{\phi}(s/T)\,ds\bigg)\nonumber\\ &\quad=\sum_{\pi,\pi'\in\mathcal{S}(Q)}\alpha(\pi)\overline{\alpha(\pi')} \bigg(\kappa_{\pi\times\widetilde{\pi}'}x\frac{\widehat{\phi}({1}/{T})}{T}+\frac{1}{2\pi iT}\int_{{1}/{4\log(ex)}-i\infty}^{{1}/{4\log(ex)}+i\infty} \frac{L(s,\pi\times\widetilde{\pi}')x^s\widehat{\phi}(s/T)}{\prod_{\mathfrak{p} \mid \mathfrak{q}_{\pi}\mathfrak{q}_{\pi'}}L(s,\pi_{\mathfrak{p}}\times\widetilde{\pi}_{\mathfrak{p}}')}\,ds\bigg)\nonumber\\ &\quad=\sum_{\pi,\pi'\in\mathcal{S}(Q)}\alpha(\pi)\overline{\alpha(\pi')}\bigg(\kappa_{\pi\times\widetilde{\pi}'}x \frac{\widehat{\phi}({1}/{T})}{T}+O_{\phi,\varepsilon}(Q^{4n^2\theta_n+n+\varepsilon}T^{({1}/{2})n^2 [F:\mathbb{Q}]+\varepsilon})\bigg). \end{align}

Recall that $\kappa _{\pi \times \widetilde {\pi }'}>0$ when $\pi =\pi '$, and $\kappa _{\pi \times \widetilde {\pi }'}=0$ otherwise. Since $\|\alpha \|_2=1$ and $\phi$ is fixed, it follows from the inequality of arithmetic and geometric means that (4.7) equals

\begin{align*} &\frac{\widehat{\phi}({1}/{T})}{T}x\sum_{\pi\in\mathcal{S}(Q)}|\alpha(\pi)|^2\kappa_{ \pi\times\widetilde{\pi}'}+O_{\phi}(Q^{4n^2\theta_n+n+\varepsilon}T^{({1}/{2})n^2 [F:\mathbb{Q}]+\varepsilon}|\mathcal{S}(Q)|)\\ &\quad \ll_{\phi,\varepsilon}\frac{x}{T}\max_{\pi\in\mathcal{S}(Q)}\kappa_{\pi\times\widetilde{\pi}'}+Q^{4n^2\theta_n+n+\varepsilon}T^{({1}/{2}) n^2 [F:\mathbb{Q}]+\varepsilon}|\mathcal{S}(Q)|. \end{align*}

We estimate the maximum using (4.6), and the desired result follows.

We use Proposition 4.3 to prove Theorem 4.1.

Proof of Theorem 4.1 For the sum involving $\mu _{\pi \times \pi '}(\mathfrak {n})$, we apply Proposition 4.3 with

\[ a((\mu_{\mathfrak{p}}\!)_{\mathfrak{p}}) = b\bigg(\prod_{\mathfrak{p}}\mathrm{N}\mathfrak{p}^{|\mu_{\mathfrak{p}}|}\bigg)\mathbf{1}_{(\prod_{\mathfrak{p}}\mathrm{N}\mathfrak{p}^{|\mu_{\mathfrak{p}}|},\mathfrak{q}_{\pi'})}\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}^*}(-A_{\pi'}(\mathfrak{p})). \]

If $(\mu _{\mathfrak {p}})_{\mathfrak {p}}\in \underline {\mu }[\mathfrak {n}]$, then by (3.15), we have that $|\mu _{\mathfrak {p}}|=\mathrm {ord}_{\mathfrak {p}}(\mathfrak {n})$ and

\[ a((\mu_{\mathfrak{p}}\!)_{\mathfrak{p}})=b(\mathfrak{n})\mathbf{1}_{(\mathfrak{n},\mathfrak{q}_{\pi'})}\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}^*}(-A_{\pi'}(\mathfrak{p})). \]

By (3.16), the left-hand side of Proposition 4.3 becomes

\begin{align*} &\sum_{\pi\in\mathcal{S}(Q)}\bigg|\sum_{\substack{x<\mathrm{N}\mathfrak{n}\leq xe^{1/T} \\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi}\!)= \mathcal{O}_F }}~\sum_{(\mu_{\mathfrak{p}}\!)_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]} \bigg[\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}}(A_{\pi}(\mathfrak{p}))\bigg] a((\mu_{\mathfrak{p}}\!)_{\mathfrak{p}})\bigg|^2\\ &\quad=\sum_{\pi\in\mathcal{S}(Q)}\bigg|\sum_{\substack{x<\mathrm{N}\mathfrak{n}\leq xe^{1/T} \\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi}\!)= \mathcal{O}_F }}b(\mathfrak{n})\mathbf{1}_{(\mathfrak{n},\mathfrak{q}_{\pi'})} \sum_{(\mu_{\mathfrak{p}}\!)_{\mathfrak{p}}\in \underline{\mu}[\mathfrak{n}]} \prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}}(A_{\pi}(\mathfrak{p}))s_{\mu_{\mathfrak{p}}^*}(-A_{\pi'}(\mathfrak{p})\bigg|^2\\ &\quad=\sum_{\pi\in\mathcal{S}(Q)}\bigg|\sum_{\substack{x<\mathrm{N}\mathfrak{n}\leq xe^{1/T} \\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'})= \mathcal{O}_F }}\mu_{\pi\times\pi'}(\mathfrak{n})b(\mathfrak{n})\bigg|^2. \end{align*}

Similarly, the right-hand side of Proposition 4.3 becomes

\[ Q^{\varepsilon}\bigg(\frac{x}{T}+Q^{4n^2\theta_n+n}T^{({1}/{2})n^2 [F:\mathbb{Q}]+\varepsilon}|\mathcal{S}(Q)|\bigg)\sum_{\substack{\mathrm{N}\mathfrak{n}\in(x,xe^{1/T}] \\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi'})=\mathcal{O}_F }}|b(\mathfrak{n})|^2\sum_{(\mu_{\mathfrak{p}}\!)_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]}\bigg|\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}^*}(-A_{\pi'}(\mathfrak{p}))\bigg|^2. \]

The desired result now follows from (3.17). Apart from the new choice of

\[ a((\mu_{\mathfrak{p}}\!)_{\mathfrak{p}}) = b\bigg(\prod_{\mathfrak{p}}\mathrm{N}\mathfrak{p}^{|\mu_{\mathfrak{p}}|}\bigg)\mathbf{1}_{(\prod_{\mathfrak{p}}\mathrm{N}\mathfrak{p}^{|\mu_{\mathfrak{p}}|},\mathfrak{q}_{\pi'})}\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}}(A_{\pi'}(\mathfrak{p})), \]

the result for the sum involving $\lambda _{\pi \times \pi '}(\mathfrak {n})$ is handled in the same manner.

Corollary 4.4 Let $\pi '\in \mathfrak {F}_{n'}$, $Q,T\geq 1$, and $\varepsilon >0$. If $Y\geq e$ and

\[ X\geq Q^{4n^2\theta_n+n+\varepsilon}T^{({1}/{2})n^2 [F:\mathbb{Q}]+1+\varepsilon}|\mathcal{S}(Q)|,\quad \log Y\asymp_{\varepsilon} \log X, \]

then

\begin{align*} \sum_{\pi\in\mathcal{S}(Q)}\int_{-T}^{T}\bigg|\sum_{\substack{\mathrm{N}\mathfrak{n} > X \\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'})= \mathcal{O}_F }}\frac{\mu_{\pi\times\pi'}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{1+{1}/{\!\log Y}+iv}}\bigg|^2\,dv &\ll_{\varepsilon} C(\pi')^{\varepsilon}Q^{\varepsilon}\log X,\\ \sum_{\pi\in\mathcal{S}(Q)}\int_{-T}^{T}\bigg|\sum_{\substack{\mathrm{N}\mathfrak{n}\leq X \\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'})= \mathcal{O}_F }}\frac{\mu_{\pi\times\pi'}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{{1}/{2}+iv}}\bigg|^2 \,dv& \ll_{\varepsilon} C(\pi')^{\varepsilon}Q^{\varepsilon}X\log X. \end{align*}

Proof. We prove the first result; the second result is proved completely analogously. A formal generalization of [Gal70, Theorem 1] to number fields tells us that if $c(\mathfrak {n})$ is a complex-valued function supported on the integral ideals of $\mathcal {O}_F$ with $\sum _{\mathfrak {n}}|c(\mathfrak {n})|<\infty$, then

\[ \int_{-T}^{T}\bigg|\sum_{\mathfrak{n}}\frac{c(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{it}}\bigg|^2 \,dt\ll T^2 \int_0^{\infty}\bigg|\sum_{\mathrm{N}\mathfrak{n}\in(x,xe^{1/T}]}c(\mathfrak{n})\bigg|^2\frac{{d}x}{x}. \]

We choose $b(\mathfrak {n})=\mathrm {N}\mathfrak {n}^{-1- {1}/{\!\log Y}}$ if $\mathrm {N}\mathfrak {n}>X$ and $b(\mathfrak {n})=0$ otherwise, which leads to

\begin{align*} &\sum_{\pi\in\mathcal{S}(Q)}\int_{-T}^{T}\bigg|\sum_{\substack{\mathrm{N}\mathfrak{n} > X \\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'})=\mathcal{O}_F }}\frac{\mu_{\pi\times\pi'}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{1+{1}/{\!\log Y}+it}}\bigg|^2\,dt\\ &\quad \ll T^2 \int_0^{\infty}\sum_{\pi\in\mathcal{S}(Q)}\bigg|\sum_{\mathrm{N}\mathfrak{n}\in(x,xe^{1/T}]}\mu_{\pi\times\pi'}(\mathfrak{n})b(\mathfrak{n})\bigg|^2 \frac{{d}x}{x}. \end{align*}

Theorem 4.1 and the fact that $\lambda _{\pi '\times \widetilde {\pi }'}(\mathfrak {n})\geq 0$ for all $\mathfrak {n}$ imply that the above display is

\begin{align*} &\ll_{\varepsilon} Q^{\varepsilon}\sum_{\substack{\mathrm{N}\mathfrak{n} > X \\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi'})=\mathcal{O}_F }}\frac{\lambda_{\pi'\times\widetilde{\pi}'}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{1+{2}/{\!\log Y}}}\bigg(1+\frac{Q^{4n^2\theta_n+n}T^{({1}/{2})n^2 [F:\mathbb{Q}]+1+\varepsilon}|\mathcal{S}(Q)|}{\mathrm{N}\mathfrak{n}}\bigg)\\ &\quad \ll_{\varepsilon} Q^{\varepsilon}\sum_{\mathrm{N}\mathfrak{n} > X}\frac{\lambda_{\pi'\times\widetilde{\pi}'}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{1+{2}/{\!\log Y}}}. \end{align*}

The desired result now follows from Lemma 3.1 and our choices of $X$ and $Y$.

5. Proof of Theorem 1.1

Let $n,n'\geq 1$, $\varepsilon >0$, $T\geq 3$, $Q\geq 3$, $\pi \in \mathcal {S}(Q)$, and $\pi '\in \mathfrak {F}_{n'}$. Define

(5.1) \begin{align} & X:= C(\pi')^{\varepsilon}Q^{4n^2\theta_n+n+\varepsilon}T^{({1}/{2})n^2 [F:\mathbb{Q}]+1+\varepsilon}|\mathcal{S}(Q)|,\nonumber\\ & Y:= \big((C(\pi')^{{n}/{2}}Q^{{n'}/{2}} (C(\pi')Q)^{2n'n\max\{0,\theta_n+\theta_{n'}-{1}/{2}\}} |\mathcal{S}(Q)| T^{{n'n[F:\mathbb{Q}]}/{2}+1})^{1+{\varepsilon}/{4n'n[F:\mathbb{Q}]}}X\big)^{{1}/({3-2\sigma})},\nonumber\\ & L^{\mathrm{ur}}(s,\pi\times\pi'):= L(s,\pi\times\pi')\prod_{\mathfrak{p} \mid \mathfrak{q}_{\pi}\mathfrak{q}_{\pi'}}L(s,\pi_{\mathfrak{p}}\times\pi_{\mathfrak{p}}')^{-1},\nonumber\\ & M_X(s,\pi\times\pi'):= \sum_{\substack{\mathrm{N}\mathfrak{n}\leq X \nonumber\\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'})=\mathcal{O}_F }} \frac{\mu_{\pi\times\pi'}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^s},\nonumber\\ & LM_X(s,\pi\times\pi'):= L^{\mathrm{ur}}(s,\pi\times\pi') M_X(s,\pi\times\pi'). \end{align}

Note that $\log (C(\pi )C(\pi ')T)\ll \log X\asymp _{\varepsilon } \log Y$. We assume that $T\geq 2(\log Y)^2$ since the proof does not change appreciably otherwise. In this section, $\varepsilon$ might vary from line to line, and terms of size $(C(\pi ')QT^{[F:\mathbb {Q}]}|\mathcal {S}(Q)|)^{\varepsilon }$ and polynomials in $\log (C(\pi ')QT^{[F:\mathbb {Q}]}|\mathcal {S}(Q)|)$ might be bounded by $X^{\varepsilon }$ without mention.

If $t\in [-T,T]$, then

\[ |\{\rho=\beta+i\gamma\colon \beta\geq 0,~|\gamma-t|\leq 1\}|\ll \log C(\pi\times\pi',t)\ll \log(C(\pi)C(\pi')T) \]

by [IK04, Proposition 5.7]. We observe that the rectangle $[\sigma,1]\times [-T,T]$ is covered by $O(T)$ boxes of the form $[\sigma,1]\times [y,y+2(\log Y)^2]$, each containing $O((\log (C(\pi )C(\pi ')T))^3)$ zeros. If we write $n_{\pi \times \pi '}$ for the number of such boxes containing at least one zero $\rho$ of $L(s,\pi \times \pi ')$, then

\[ N_{\pi\times\pi'}(\sigma,T)\ll (\log(C(\pi')Q T))^3 n_{\pi\times\pi'}. \]

If $\pi '=\widetilde {\pi }$, then it suffices to assume that $|\rho -1|>1$ since there are $O(\log (C(\pi ')QT))$ zeros $\rho$ such that $|\rho -1|\leq 1$.

Let $\rho =\beta +i\gamma$ be a zero of $L(s,\pi \times \pi ')$, in which case $LM_X(\rho,\pi \times \pi ')=0$. We compute

(5.2) \begin{align} e^{-1/Y}&=\frac{1}{2\pi i}\int_{1-\beta+{1}/{\!\log Y}-i\infty}^{1-\beta+{1}/{\!\log Y}+i\infty}(1-LM_X(\rho+w,\pi\times\pi')\Gamma(w)Y^{w}\,dw\nonumber\\ &\quad +\frac{1}{2\pi i}\int_{{1}/{2}-\beta-i\infty}^{{1}/{2}-\beta+i\infty}LM_X(\rho+w,\pi\times\pi')\Gamma(w)Y^{w}\,dw\nonumber\\ &\quad +\kappa_{\pi\times\pi'}\Gamma(1-\rho)Y^{1-\rho}\sum_{\substack{\mathrm{N}\mathfrak{n}\leq X \nonumber\\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'})=\mathcal{O}_F}}\frac{\mu_{\pi\times\pi'}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}}. \end{align}

It follows from (3.13) and Lemmas 3.1 and 3.3 that if $\text {Re}(w) = 1-\beta + {1}/{\!\log Y}$, then

(5.3) \begin{align} &|1-LM_X(\rho+w,\pi\times\pi')|\nonumber\\ &\quad= |L^{\mathrm{ur}}(\rho+w,\pi\times\pi')|\cdot|L^{\mathrm{ur}}(\rho+w,\pi\times\pi')^{-1}-M_X(\rho+w,\pi\times\pi')|\nonumber\\ &\quad\ll_{\varepsilon} C(\pi\times\pi',|\gamma+\text{Im}(w)|+1)^{\varepsilon}\sum_{\substack{\mathrm{N}\mathfrak{n}>X \nonumber\\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'})=\mathcal{O}_F }}\frac{|\mu_{\pi\times\pi'}(\mathfrak{n})|}{\mathrm{N}\mathfrak{n}^{1+{1}/{\!\log Y}}}\nonumber\\ &\quad\ll_{\varepsilon} C(\pi\times\pi',|\gamma+\text{Im}(w)|+1)^{\varepsilon}\sum_{\substack{\mathrm{N}\mathfrak{n}>X \\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'})=\mathcal{O}_F }}\frac{\lambda_{\pi\times\widetilde{\pi}}(\mathfrak{n})+\lambda_{\pi'\times\widetilde{\pi}'}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{1+{1}/{\!\log Y}}}\nonumber\\ &\quad\ll_{\varepsilon} C(\pi)^{\varepsilon}C(\pi')^{\varepsilon}(|\gamma+\text{Im}(w)|+1)^{\varepsilon}\log X. \end{align}

Therefore, we deduce from Stirling's formula that

\begin{align*} &\bigg|\frac{1}{2\pi i}\int_{1-\beta+{1}/{\!\log Y}-i\infty}^{1-\beta+{1}/{\!\log Y}+i\infty} (1-LM_X(\rho+w,\pi\times\pi'))\Gamma(w)Y^{w}\,dw\\ &\quad -\frac{1}{2\pi i}\int_{1-\beta+{1}/{\!\log Y}-i(\log Y)^2}^{1-\beta+{1}/{\!\log Y}+ i(\log Y)^2}(1-LM_X(\rho+w,\pi\times\pi'))\Gamma(w)Y^{w}\,dw\bigg|\ll \frac{1}{Y}. \end{align*}

The other terms in (5.2) are handled similarly. Since $e^{-1/Y}=1+O(1/Y)$, it follows that

(5.4)\begin{align} 1&\ll \frac{1}{2\pi i}\int_{1-\beta+{1}/{\!\log Y}-i(\log Y)^2}^{1-\beta+{1}/{\!\log Y}+i(\log Y)^2} (1-LM_X(\rho+w,\pi\times\pi'))\Gamma(w)Y^{w}\,dw\nonumber\\ &\quad +\frac{1}{2\pi i}\int_{{1}/{2}-\beta-i(\log Y)^2}^{{1}/{2}-\beta+i(\log Y)^2} LM_X(\rho+w,\pi\times\pi')\Gamma(w)Y^{w}\,dw+\kappa_{\pi\times\pi'}Y^{1-\sigma}\frac{\log X}{\max\{1,|\gamma|^3\}}. \end{align}

Write the first integral in (5.4) as $I_1$ and the second as $I_2$. A simple optimization calculation shows that if $c\geq 1$ and $c^{-1}\leq |I_1|+|I_2|$, then $c^{-1}\leq 2c(|I_1|^2+|I_2|)$. We estimate $|I_1|^2$ using the Cauchy–Schwarz inequality, thus obtaining the bound

\begin{align*} 1&\ll (\log Y)^2 Y^{2(1-\sigma)}\int_{\gamma-(\log Y)^2}^{\gamma+(\log Y)^2}\bigg|1-LM_X\bigg(1+\frac{1}{\log Y}+iv, \pi\times\pi'\bigg)\bigg|^2\,dv\\ &\quad\,\, +Y^{{1}/{2}-\sigma}\int_{\gamma-(\log Y)^2}^{\gamma+(\log Y)^2}\bigg|LM_X \bigg(\frac{1}{2}+iv,\pi\times\pi'\bigg)\bigg|\,dv+\kappa_{\pi\times\pi'}Y^{1-\sigma} \frac{\log X}{\max\{1,|\gamma|^3\}}. \end{align*}

Since $T\geq 2(\log Y)^2$ by hypothesis, we conclude that

(5.5) \begin{align} n_{\pi\times\pi'}&\ll (\log Y)^2 Y^{2(1-\sigma)}\int_{-T}^{T}\bigg|1-L\bigg(1+\frac{1}{\log Y}+iv\bigg)M_X \bigg(1+\frac{1}{\log Y}+iv\bigg)\bigg|^2 \,dv\nonumber\\ &\quad\,\, +Y^{{1}/{2}-\sigma}\int_{-T}^{T}\bigg|L\bigg(\frac{1}{2}+iv\bigg)M_X\bigg(\frac{1}{2}+iv\bigg)\bigg|\, dv+\kappa_{\pi\times\pi'}Y^{1-\sigma}\log X+1. \end{align}

Note that $\kappa _{\pi \times \pi '}=0$ unless $\pi '=\widetilde {\pi }$, which occurs for at most one $\pi \in \mathcal {S}(Q)$. When $\kappa _{\pi \times \pi '}\neq 0$, it satisfies $\kappa _{\pi \times \pi '}=\kappa _{\pi \times \widetilde {\pi }}\ll _{\varepsilon } C(\pi )^{\varepsilon }$ per (4.6). Therefore, summing (5.5) over $\pi \in \mathcal {S}(Q)$, we find that

(5.6)\begin{align} \sum_{\pi\in\mathcal{S}(Q)}N_{\pi\times\pi'}(\sigma,T)&\ll (\log C(\pi)C(\pi')T)^3\sum_{\pi\in\mathcal{S}(Q)}n_{\pi\times\pi'}\nonumber\\ &\ll_{\varepsilon} X^{\varepsilon} \bigg(Y^{2(1-\sigma)}\int_{-T}^{T}\sum_{\pi\in\mathcal{S}(Q)} \bigg|1-LM_X\bigg(1+\frac{1}{\log Y}+iv,\pi\times\pi'\bigg)\bigg|^2\,dv\nonumber\\ &\quad\ \ +Y^{{1}/{2}-\sigma}\int_{-T}^{T}\sum_{\pi\in\mathcal{S}(Q)} \bigg|LM_X\bigg(\frac{1}{2}+iv,\pi\times\pi'\bigg)\bigg|\,dv+Y^{1-\sigma}\bigg). \end{align}

For the first integral in (5.6), we deduce from (3.13) that

\begin{align*} &\bigg|1-LM_X\bigg(1+\frac{1}{\log Y}+it,\pi\times\pi'\bigg)\bigg|^2\\ &\quad=\bigg|L^{\mathrm{ur}}\bigg(1+\frac{1}{\log Y}+it,\pi\times\pi'\bigg)\bigg|^2 \bigg|\frac{1}{L^{\mathrm{ur}}(1+\tfrac{1}{\log Y}+it,\pi\times\pi')}-M_X \bigg(1+\frac{1}{\log Y}+it,\pi\times\pi'\bigg)\bigg|^2\\ &\quad \ll_{\varepsilon} X^{\varepsilon}\bigg|\sum_{\substack{\mathrm{N}\mathfrak{n}>X \\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'})= \mathcal{O}_F}}\frac{\mu_{\pi\times\pi'}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{1+{1}/{\!\log Y}}}\bigg|^2. \end{align*}

For the second integral in (5.6), it follows from the Cauchy–Schwarz inequality that

\begin{align*} \int_{-T}^{T}\sum_{\pi\in\mathcal{S}(Q)}\bigg|LM_X\bigg(\frac{1}{2}+iv,\pi\times\pi'\bigg)\bigg|\,dv &\leq \bigg(\int_{-T}^{T}\sum_{\pi\in\mathcal{S}(Q)} \bigg|L^{\mathrm{ur}}\bigg(\frac{1}{2}+iv,\pi\times\pi'\bigg)\bigg|^2 \,dv\bigg)^{{1}/{2}}\\ &\quad \times \bigg(\int_{-T}^{T}\sum_{\pi\in\mathcal{S}(Q)}\bigg|\sum_{\substack{\mathrm{N}\mathfrak{n}\leq X\\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'})=\mathcal{O}_F}}\frac{\mu_{\pi\times\pi'}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{{1}/{2}+iv}}\bigg|^2 \,dv \bigg)^{{1}/{2}}. \end{align*}

We bound the second moment of $L^{\mathrm {ur}}(\frac {1}{2}+iv,\pi \times \pi ')$ trivially as

\begin{align*} &\int_{-T}^{T}\sum_{\pi\in\mathcal{S}(Q)}\bigg|L^{\mathrm{ur}}\bigg(\frac{1}{2}+iv,\pi\times\pi'\bigg)\bigg|^2\,dv \\ &\quad \ll_{\varepsilon} (QC(\pi'))^{2n'n\max\{0,\theta_n+\theta_{n'}-{1}/{2}\}+\varepsilon} Q^{{n'}/{2}}C(\pi')^{{n}/{2}}T^{{n'n[F:\mathbb{Q}]}/{2}+1+\varepsilon}|\mathcal{S}(Q)| \ll_{\varepsilon}\frac{Y^{3-2\sigma}}{X} \end{align*}

using (3.7) (to bound the ramified Euler factors) and (3.13). In summary, we find that

\begin{align*} \sum_{\pi\in\mathcal{S}(Q)}N_{\pi\times\pi'}(\sigma,T)& \ll_{\varepsilon} X^{\varepsilon} \bigg(Y^{2(1-\sigma)}\int_{-T}^{T}\sum_{\pi\in\mathcal{S}(Q)}\bigg |\sum_{\substack{\mathrm{N}\mathfrak{n}>X \\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'})=\mathcal{O}_F}}\frac{\mu_{\pi\times\pi'} (\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{1+{1}/{\!\log Y}+iv}}\bigg|^2\,dv\\ &\quad + Y^{{1}/{2}-\sigma}\bigg(\frac{Y^{3-2\sigma}}{X}\int_{-T}^{T}\sum_{\pi\in\mathcal{S}(Q)}\bigg| \sum_{\substack{\mathrm{N}\mathfrak{n}\leq X\\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'})=\mathcal{O}_F}}\frac{\mu_{\pi\times\pi'}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{{1}/{2}+iv}}\bigg|^2 \,dv\bigg)^{{1}/{2}}+ Y^{1-\sigma}\bigg). \end{align*}

We bound the two $v$-integrals using Corollary 4.4 and (5.1), thus obtaining

\[ \sum_{\pi\in\mathcal{S}(Q)}N_{\pi\times\pi'}(\sigma,T)\ll_{\varepsilon} Y^{2(1-\sigma)}X^{\varepsilon}. \]

Theorem 1.1 now follows from (5.1), the bounds for $\theta _{n}$ and $\theta _{n'}$ in (3.2), and (1.5).

6. Bounds for Rankin–Selberg $L$-functions

In this section, we prove Theorem 2.1. Let $\pi '\in \mathfrak {F}_{n'}$ and $Q\geq 1$. In (1.6), we rescale $\varepsilon$ to $\varepsilon /72$ and define $\alpha = \varepsilon /(7.2\max \{n,n'\})$ so that

(6.1)\begin{equation} |\{\pi\in\mathfrak{F}_n(Q)\colon N_{\pi\times\pi'}(1-\alpha,6)=0\}|\leq \sum_{\pi\in\mathfrak{F}_n(Q)}N_{\pi\times\pi'}(1-\alpha,6)\ll_{F,\varepsilon}(C(\pi')^n |\mathfrak{F}_n(Q)|)^{\varepsilon}. \end{equation}

Let $\pi \in \mathfrak {F}_n(Q)$. Proceeding as in the proof of [ST19, Theorem 1.1], we obtain⁴

\begin{align*} \log\bigg|L\bigg(\frac{1}{2},\pi\times\pi'\bigg)\bigg|&\leq\bigg (\frac{1}{4}-\frac{\alpha}{10^{9}}\bigg)\log C(\pi\times\pi') +2 \log\bigg|L\bigg(\frac{3}{2},\pi\times\pi'\bigg)\bigg|\\ &\quad\ + \frac{\alpha}{10^{7}} N_{\pi\times\pi'}(1-\alpha,6)+O(1). \end{align*}

By (3.12) and (3.13), we find that there exists an effectively computable constant $c_{5}=c_{5}(n,n',[F:\mathbb {Q}],\varepsilon )>0$ such that if $C(\pi \times \pi ')\geq c_{5}$, then

(6.2)\begin{equation} \log\bigg|L\bigg(\frac{1}{2},\pi\times\pi')\bigg|\leq\bigg(\frac{1}{4}-\frac{9\alpha}{10^{10}}\bigg) \log C(\pi\times\pi')+\frac{\alpha}{10^7}N_{\pi\times\pi'}(1-\alpha,6). \end{equation}

Theorem 2.1 now follows from (6.1) and (6.2).

7. Effective multiplicity one on average

In this section, we prove Theorem 2.3. We consider the family of cuspidal automorphic representations $\mathfrak {F}_n(Q)$ over a number field $F$. Recall our convention that implied constants are allowed to depend on $n\geq 3$, $[F:\mathbb {Q}]$, and $\varepsilon$ unless specifically mentioned otherwise.

To prove Theorem 2.3, we use Theorem 1.1 to build large zero-free regions for almost all $L$-functions $L(s,\pi \times \pi ')$ with $\pi \in \mathfrak {F}_n$ varying. If $\pi '\in \{\widetilde {\pi },\widetilde {\pi }'\}$, then $L(s,\pi \times \pi ')$ has the standard zero-free region (1.1) apart from a possible Landau–Siegel zero, which must be both real and simple. In all other cases, only Brumley's much narrower zero-free region is known (see [Bru06a] and [Lap13, Appendix]). We can use Theorem 1.1 to establish a much stronger zero-free region for $L(s,\pi \times \pi ')$ for all $\pi \in \mathfrak {F}_n(Q)$ with very few exceptions. Here is a simple example.

Corollary 7.1 Let $\varepsilon >0$, $n\geq 3$, and $\pi '\in \mathfrak {F}_n(Q)$. For all $\pi \in \mathfrak {F}_n(Q)$ with $O_{\varepsilon }(Q^{\varepsilon })$ exceptions, the $L$-function $L(s,\pi \times \pi ')$ is non-zero in the region

\[ \text{Re}(s)\geq 1-\frac{\varepsilon}{20n^2},\quad |\text{Im}(s)|\leq \log Q. \]

Proof. This follows from (1.7) (once we rescale $\varepsilon$ to $\varepsilon /21$) with $C(\pi ')\leq Q$, $T = \log Q$, and $\sigma = 1-\varepsilon /(20n^2)$.

For $\pi _1,\pi _2\in \mathfrak {F}_n$ define the numbers $\Lambda _{\pi _1\times \pi _2}(\mathfrak {n})$ by the Dirichlet series identity (for $\text {Re}(s)>1$)

(7.1)\begin{equation} \sum_{\mathfrak{n}}\frac{\Lambda_{\pi_1\times\pi_2}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^s}= -\frac{L'}{L}(s,\pi_1\times\pi_2)= \sum_{\mathfrak{p}} \sum_{k = 1}^{\infty} \frac{\sum_{j=1}^{n}\sum_{j'=1}^{n}\alpha_{j,j',\pi_1\times\pi_2}(\mathfrak{p})^k\log\mathrm{N}\mathfrak{p}}{\mathrm{N}\mathfrak{p}^{ks}}. \end{equation}

Lemma 7.2 Let $\pi '\in \mathfrak {F}_n$. There exist absolute and effectively computable constants $c_{6}\in (0,1)$, $c_{7},c_{8},c_{9},c_{10}\geq 1$, and $c_{11}\in (0,1)$ such that the following are true.

1. (1) The $L$-function $L(s,\pi '\times \widetilde {\pi }')$ has at most one zero, say $\beta _1$, in the region

   \[ \text{Re}(s)\geq 1- \frac{c_{6}}{\log(C(\pi')^n(|\text{Im}(s)|+e)^{n^2[F:\mathbb{Q}]})}. \]
   If $\beta _1$ exists, then it must be real and simple and satisfy $\beta _1\leq 1-C(\pi ')^{-c_{7}n}$.

2. (2) If $A\geq c_{8}$, $\log \log C(\pi ')\geq c_{9}n^4[F:\mathbb {Q}]^2$, and $x\geq C(\pi ')^{c_{10}A^2n^3[F:\mathbb {Q}]\log (en[F:\mathbb {Q}])}$, then

   \[ \sum_{{x}/{2}<\mathrm{N}\mathfrak{n}\leq x}\Lambda_{\pi'\times\widetilde{\pi}'}(\mathfrak{n})=\begin{cases} \dfrac{x}{2}(1-\xi^{\beta_1-1})(1+O(e^{-c_{11}A})) & \mbox{if}\ \beta_1\ \mbox{exists,}\\[8pt] \dfrac{x}{2}(1+O(e^{-c_{11}A})) & \mbox{otherwise,} \end{cases} \]
   where $\xi \in [ {x}/{2},x]$ satisfies $x^{\beta _1}-( {x}/{2})^{\beta _1}=\beta _1 ({x}/{2})\xi ^{\beta _1-1}$ and the implied constants are absolute and effectively computable.

Proof. This is [HT22, Theorem 2.1] with $\delta =0$ and $x$ replaced with $x/2$.

Lemma 7.2 informs the following choices of parameters that we will use throughout the rest of this section. Let $0<\varepsilon <1$, let $Q$ be sufficiently large with respect to $\varepsilon$, and let

(7.2)\begin{equation} \begin{aligned} & c_{7}\geq \frac{c_{8}^2c_{10}}{3240},\quad A = \sqrt{\frac{3240 c_{7}}{c_{10}\varepsilon}},\quad B= \frac{41n^2}{\varepsilon},\quad x=\frac{1}{2}(\log Q)^B,\\ & \pi\in\mathfrak{F}_n(Q),\quad \pi'\in \mathfrak{F}_n\big(x^{{1}/{A^2c_{10}n^4[F:\mathbb{Q}]^2}}\big)= \mathfrak{F}_n\big((2^{-{\varepsilon}/{41n^2}}\log Q)^{{41}/{3240c_{7}n^2[F:\mathbb{Q}]^2}}\big). \end{aligned} \end{equation}

Corollary 7.3 Under the notation and hypotheses of (7.2), there holds

\[ \sum_{\substack{{x}/{2}<\mathrm{N}\mathfrak{n}\leq x \\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'})=\mathcal{O}_F }} \Lambda_{\pi'\times\widetilde{\pi}'}(\mathfrak{n})\gg_{\varepsilon} x^{1-{2c_{7}}/{A^2c_{10}n^3[F:\mathbb{Q}]^2}}. \]

Proof. Let $\pi,\pi '\in \mathfrak {F}_n(Q)$. In Lemma 7.2, the lower bound on $C(\pi ')$ only serves to ensure that the implied constants are absolute. This was pertinent in [HT22], but it is not pertinent here. Thus, we may replace the two conditions $\log \log C(\pi ')\geq c_{9}n^4[F:\mathbb {Q}]^2$ and $x\geq C(\pi ')^{A^2 c_{10}n^3[F:\mathbb {Q}]\log (en[F:\mathbb {Q}])}$ with the single condition $C(\pi ')\ll x^{1/(A^2c_{10}n^4[F:\mathbb {Q}]^2)}$. We want to refine Lemma 7.2 so that one only sums over $\mathfrak {n}$ such that $\gcd (\mathfrak {n},\mathfrak {q}_{\pi }\mathfrak {q}_{\pi '})=\mathcal {O}_F$. Using (3.3) and (3.7), we find that the contribution to Lemma 7.2(2) arising from $\mathfrak {n}$ such that $\gcd (\mathfrak {n},\mathfrak {q}_{\pi }\mathfrak {q}_{\pi '})\neq \mathcal {O}_F$ is

\begin{align*} \sum_{\substack{{x}/{2}<\mathrm{N}\mathfrak{n}\leq x \\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'})\neq\mathcal{O}_F}}\Lambda_{\pi'\times\widetilde{\pi}'}(\mathfrak{n})&\ll \sum_{\mathfrak{p} \mid \mathfrak{q}_{\pi}\mathfrak{q}_{\pi'}}\ \sum_{{\log(x/2)}/{\!\log\mathrm{N}\mathfrak{p}}< j\leq {\log x}/{\!\log\mathrm{N}\mathfrak{p}}}\mathrm{N}\mathfrak{p}^{j(1-{2}/({n^2+1}))}\\ &\ll \frac{\log Q}{\log\log Q}x^{1-{2}/({n^2+1})}, \end{align*}

which is $\ll x^{1- {1}/({n^2+1})}$. In the worst case, where $\beta _1$ in Lemma 7.2(1) exists, it follows from Lemma 7.2(2) and the above discussion that if $C(\pi ')\ll _{n,[F:\mathbb {Q}]} x^{1/(A^2c_{10}n^3[F:\mathbb {Q}]\log (en[F:\mathbb {Q}]))}$, then

\[ \sum_{\substack{x<\mathrm{N}\mathfrak{n}\leq 2x \\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'})=\mathcal{O}_F }}\Lambda_{\pi'\times\widetilde{\pi}'}(\mathfrak{n})\gg x(1-\xi^{\beta_1-1})\gg_{n,[F:\mathbb{Q}]} x\min\{1,(1-\beta_1)\log C(\pi')\}\gg \frac{x\log C(\pi')}{C(\pi')^{c_{7}n}}. \]

The desired result now follows.

Lemma 7.4 Let $n\geq 3$. Assume the notation and hypotheses in (7.2). Let $\Phi$ be a fixed smooth function supported on a compact subset of $[\frac {1}{4},2]$ such that $0\leq \Phi (t)\leq 1$ for all $t\in [\frac {1}{4},2]$ and $\Phi (t)=1$ for $t\in (\frac {1}{2},1]$. If $L(s,\pi \times \pi ')$ is entire and does not vanish in the region $\text {Re}(s)\geq 1-\varepsilon /(20n^2)$ and $|\text {Im}(s)|\leq \log Q$, then

\[ \sum_{\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'})=\mathcal{O}_F }\Lambda_{\pi\times\widetilde{\pi}'}(\mathfrak{n})\Phi(\mathrm{N}\mathfrak{n}/x)\ll_{B} x^{1+{2}/{B}-{\varepsilon}/{20n^2}}. \]

Proof. Writing $\widehat {\Phi }(s)=\int _0^{\infty }\Phi (t)t^{s-1}\,dt$ for the Mellin transform of $\Phi$ (which is an entire function of $s$), we observe via Mellin inversion that

\[ \sum_{\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'})=\mathcal{O}_F }\Lambda_{\pi\times\widetilde{\pi}'}(\mathfrak{n})\Phi(\mathrm{N}\mathfrak{n}/x) = \frac{1}{2\pi i}\int_{2-i\infty}^{2+i\infty}-\frac{(L^{\text{ur}})'}{(L^{\text{ur}})}(s,\pi\times\widetilde{\pi}')\widehat{\Phi}(s)x^s \,ds, \]

where $L^{\text {ur}}(s,\pi \times \widetilde {\pi }')=L(s,\pi \times \widetilde {\pi }')\prod _{\mathfrak {p}\mid \mathfrak {q}_{\pi }\mathfrak {q}_{\pi '}}L(s,\pi _{\mathfrak {p}}\times \widetilde {\pi }_{\mathfrak {p}}')^{-1}$. A standard contour integral calculation using the argument principle shows that the above display equals

\[ -\sum_{L^{\text{ur}}(\rho,\pi\times\widetilde{\pi}')=0}\widehat{\Phi}(\rho)x^{\rho}, \]

where $\rho$ ranges over all zeros of $L^{\text {ur}}(s,\pi \times \widetilde {\pi }')$.

Since $\Phi$ is compactly supported and $\widehat {\Phi }$ is entire, it follows that for any $R\geq 2$, we have $|\widehat {\Phi }(s)|\ll _{R}\min \{1,|s|^{-R}e^{\text {Re}(s)}\}$. Note that the reciprocals of the Euler factors of $L(s,\pi \times \widetilde {\pi }')$ at prime ideals $\mathfrak {p} \mid \mathfrak {q}_{\pi }\mathfrak {q}_{\pi '}$ and all of the trivial zeros of $L(s,\pi \times \widetilde {\pi }')$ have real part no larger than $1- {2}/({n^2+1})$ per (3.3) and (3.7). Since for any $t\in \mathbb {R}$ there are $\ll \log Q+\log (|t|+2)$ zeros $\rho =\beta +i\gamma$ of $L^{\text {ur}}(s,\pi \times \widetilde {\pi }')$ that satisfy $0<\beta <1$ and $|\gamma -t|\leq 1$, we find for any $T\geq 1$, $R\geq 2$, and $\sigma _0\geq 1- {2}/({n^2+1})$ that

(7.3)\begin{equation} \sum_{L^{\text{ur}}(\rho,\pi\times\widetilde{\pi}')=0}\widehat{\Phi}(\rho)x^{\rho}=\sum_{\substack{L^{\text{ur}}(\rho,\pi\times\widetilde{\pi}')=0 \\ \beta\geq \sigma_0 \\ |\gamma|\leq T}}\widehat{\Phi}(\rho)x^{\rho}+O(T^{1-R}\log(QT)x+T\log(QT)x^{\sigma_0}). \end{equation}

We choose $T=\log Q=(2x)^{1/B}$ and $\sigma _0=1-\varepsilon /(20n^2)$, in which case our hypotheses imply that the sum over zeros on the right-hand side of (7.3) is empty and

\[ \bigg|\sum_{L^{\text{ur}}(\rho,\pi\times\widetilde{\pi}')=0}\widehat{\Phi}(\rho)x^{\rho}\bigg| \ll_{B} x^{1+({2-R})/{B}}+x^{{2}/{B}+\sigma_0}. \]

The desired result follows from choosing $R=\max \{B(1-\sigma _0),3\}$.

Proof of Theorem 2.3 Suppose to the contrary that $\pi \neq \pi '$ and $\pi _{\mathfrak {p}}\cong \pi _{\mathfrak {p}}'$ for all $\mathfrak {p}\nmid \mathfrak {q}_{\pi }\mathfrak {q}_{\pi '}$ with $\mathrm {N}\mathfrak {p}\leq 2x$. Then $A_{\pi }(\mathfrak {p})=A_{\pi '}(\mathfrak {p})$ for all $\mathfrak {p}\nmid \mathfrak {q}_{\pi }\mathfrak {q}_{\pi '}$ with $\mathrm {N}\mathfrak {p}\leq 2x$. By (3.6) and (7.1), it follows that $\Lambda _{\pi \times \widetilde {\pi }'}(\mathfrak {n})=\Lambda _{\pi '\times \widetilde {\pi }'}(\mathfrak {n})$ for all $\mathfrak {n}$ such that $\gcd (\mathfrak {n},\mathfrak {q}_{\pi }\mathfrak {q}_{\pi '})=\mathcal {O}_F$ and $\mathrm {N}\mathfrak {n}\leq x$. Since $\Lambda _{\pi '\times \widetilde {\pi }'}(\mathfrak {n})\geq 0$ for all such $\mathfrak {n}$, the same holds for $\Lambda _{\pi \times \widetilde {\pi }'}(\mathfrak {n})$. It follows that

\[ \sum_{\substack{{x}/{2}<\mathrm{N}\mathfrak{n}\leq x \\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'})= \mathcal{O}_F }}\Lambda_{\pi'\times\widetilde{\pi}'}(\mathfrak{n})=\sum_{\substack{{x}/{2}<\mathrm{N}\mathfrak{n}\leq x \\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'})=\mathcal{O}_F }}\Lambda_{\pi\times\widetilde{\pi}'}(\mathfrak{n}). \]

By (7.2), we have that

(7.4)\begin{equation} C(\pi')\ll x^{1/(A^2c_{10}n^4[F:\mathbb{Q}]^2)}. \end{equation}

Therefore, if $L(s,\pi \times \widetilde {\pi }')\neq 0$ in the region $\{s\in \mathbb {C}\colon \mathrm {Re}(s)\geq 1-\varepsilon /(20n^2),~|\text {Im}(s)|\leq \log Q\}$, then by Corollary 7.3 and Lemma 7.4, we have that

\[ x^{1-{2c_{7}}/{A^2c_{10}n^3[F:\mathbb{Q}]^2}}\ll \sum_{\substack{{x}/{2}<\mathrm{N}\mathfrak{n}\leq x \\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'})=\mathcal{O}_F }}\Lambda_{\pi'\times\widetilde{\pi}'}(\mathfrak{n})=\sum_{\substack{{x}/{2}<\mathrm{N}\mathfrak{n}\leq x \\ \gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi'})=\mathcal{O}_F }}\Lambda_{\pi\times\widetilde{\pi}'}(\mathfrak{n})\ll_{B}x^{1+{2}/{B}-{\varepsilon}/{20n^2}}. \]

This implies that

\[ 1-\frac{2c_{7}}{A^2c_{10}n^3[F:\mathbb{Q}]^2}\leq 1+\frac{2}{B}-\frac{\varepsilon}{20n^2}, \]

which contradicts our choices of $A$ and $B$ in (7.2). The desired result now follows from Corollary 7.1.

8. Automorphic level of distribution

In this section, we prove Theorem 2.5. In what follows, let $F=\mathbb {Q}$. If $n=2$, then $L(s,\pi \times \widetilde {\pi })$ is the $L$-function of an isobaric automorphic representation of $\mathrm {GL}_4(\mathbb {A}_{\mathbb {Q}})$ whose cuspidal constituents have rank at most 3. Since the $L$-function any Dirichlet character twist of any cuspidal constituents of rank 2 or 3 have no Landau–Siegel zero by [Ban97, HR95], a stronger result than Theorem 2.5 follows from a minor variation of the proof of [JLTW23, Theorem 1.1]. Therefore, we may restrict our consideration to $n\geq 3$.

Let $\pi \in \mathfrak {F}_n$ have arithmetic conductor $q_{\pi }$, and let $\chi$ be a primitive Dirichlet character modulo $q$. We allow $\pi$ to be fixed, so for notational compactness, we introduce $L_{\chi }(s) := L(s,\pi \times (\widetilde {\pi }\otimes \chi ))$ and $L_1(s):= L(s,\pi \times \widetilde {\pi })$. If $q$ and $q_{\pi }$ are coprime, then $L_{\chi }(s)$ is entire if and only if $\chi$ is non-trivial, since $\widetilde {\pi } \otimes \chi \neq \widetilde {\pi }$, which is clear by comparing the arithmetic conductors of $q_{\widetilde {\pi } \otimes \chi }$ and $q_{\widetilde {\pi }}$. If $\chi$ is trivial, then $L_{\chi }(s) = L_1(s)$. We also define $\Lambda _{\chi }(s):= \Lambda (s,\pi \times (\widetilde {\pi }\otimes \chi ))$ and $\Lambda _1(s):= \Lambda (s,\pi \times \widetilde {\pi })$.

8.1 Preliminaries

We define $a_{\pi }(m)$ and $a_{\pi \times \widetilde {\pi }}(m)$ by the Dirichlet series identities

\[ \sum_{n=1}^{\infty}\frac{a_{\pi}(m)\Lambda(m)}{m^s}=-\frac{L'}{L}(s,\pi),\quad \sum_{n=1}^{\infty}\frac{a_{\pi\times\widetilde{\pi}}(m)\Lambda(m)}{m^s}=-\frac{L'}{L}(s,\pi\times\widetilde{\pi}),\quad\text{Re}(s)>1. \]

The local calculations in [LRS95, Lemma 2.1] show that if $\chi \,(\mathrm {mod}\,\,q)$ is a primitive Dirichlet character and $\gcd (q,q_{\pi })=1$, then for all primes $p$, we have that

(8.1)\begin{equation} L(s,\pi_p\times(\widetilde{\pi}\otimes\chi)_p) = \prod_{j=1}^n \prod_{j'=1}^{n}(1-\alpha_{j,j',\pi\times\widetilde{\pi}}(p)\chi(p)p^{-s})^{-1}. \end{equation}

It follows from (3.6) and (8.1) that if $\gcd (m,q_{\pi })=1$, then $a_{\pi \times (\widetilde {\pi }\otimes \chi )}(m) = |a_{\pi }(m)|^2 \chi (m)$. We now provide a convenient expression for $-L_{\chi }'(s)/L_{\chi }(s)$.

Lemma 8.1 Let $\pi \in \mathfrak {F}_n$. Let $\psi \,(\mathrm {mod}\,\,q)$ be a Dirichlet character with $q$ such that $\gcd (q,q_{\pi })=1$ and $\chi$ be the primitive Dirichlet character that induces $\psi$. Let $\delta (\chi )=1$ if $\chi$ is trivial and $\delta (\chi )=0$ otherwise. There exists a function $H_{\pi }(s;\chi,\psi )$ such that in the region $\text {Re}(s)\geq 1- {1}/{n^2}$:

1. (1) $H_{\pi }(s;\chi,\psi )$ is holomorphic;

2. (2) $|H_{\pi }(s;\chi,\psi )|\ll _{\pi } \log (q(3+|\text {Im}(s)|))$; and

3. (3) we have the identity

   \[ \sum_{m=1}^{\infty}\frac{|a_{\pi}(m)|^2\psi(m)\Lambda(m)}{m^s}=\frac{\delta(\chi)}{s-1}-\frac{\Lambda_{\chi}'}{\Lambda_{\chi}}(s) +H_{\pi}(s;\chi,\psi). \]

Proof. Suppose first that $\psi$ is primitive, in which case $\psi =\chi$ and the function $H_{\pi }(s;\chi,\psi )$ is

\begin{align*} & \frac{\log q_{\pi\times(\widetilde{\pi}\otimes\chi)}}{2}+\frac{L'}{L}(s,\pi_{\infty}\times(\widetilde{\pi}\otimes\chi)_{\infty})+\frac{\delta(\chi)}{s}+\sum_{p \mid q_{\pi}q}\frac{L'}{L}(s,\pi_{p}\times(\widetilde{\pi}'\otimes\chi)_p) \\ &\quad -\sum_{p\mid q_{\pi}q}\sum_{\substack{1\leq j\leq n \\ 1\leq j'\leq n}}\frac{\alpha_{j,\pi}(p)\overline{\alpha_{j',\pi'}(p)}\chi(p)\log p}{p^s-\alpha_{j,\pi}(p)\overline{\alpha_{j',\pi}(p)}\chi(p)}. \end{align*}

This is holomorphic and bounded as claimed for $\text {Re}(s)\geq 1- {1}/{n^2}$ by (3.7), (3.10), and Stirling's formula. If $\psi$ is not primitive and $\chi$ is the primitive character that induces $\psi$, then in the same region, we have

\[ \bigg|\sum_{m=1}^{\infty}\frac{|a_{\pi}(m)|^2\psi(m)\Lambda(m)}{m^s}-\sum_{m=1}^{\infty}\frac{|a_{\pi}(m)|^2 \chi(m)\Lambda(m)}{m^s}\bigg|\leq\sum_{p \mid q}\sum_{k\geq 1}\frac{|a_{\pi}(p^k)|^2\log p}{p^{k\,{\text{Re}}(s)}}, \]

which is bounded as desired using (3.7) again.

We use the following zero-free region for $L_{\chi }(s)$ and Siegel-type bound on any real exceptional zeros. We prove this theorem later.

Theorem 8.2 Let $Q\geq 3$ and $\pi \in \mathfrak {F}_n$. There exists an effectively computable constant $c_{12}=c_{12}(\pi )>0$ such that for all primitive Dirichlet characters $\chi \,(\mathrm {mod}\,\,q)$ with $q\leq Q$ and ${\gcd (q,q_{\pi })=1}$ with at most one exception, the $L$-function $L(s,\pi \times (\widetilde {\pi }\otimes \chi ))$ does not vanish in the region

\[ \text{Re}(s)\geq 1-\frac{c_{12}}{\log(Q(|\text{Im}(s)|+3))}. \]

If the exceptional character $\chi _1\,(\mathrm {mod}\,\,q_1)$ exists, then $L(s,\pi \times (\widetilde {\pi }\otimes \chi _1))$ has at most one zero, say $\beta _1$, in this region; $\beta _1$ is both real and simple; and $\chi _1$ must be quadratic. Moreover, for all $\varepsilon >0$, there exists an ineffective constant $c_{\pi }(\varepsilon )>0$ such that $\beta _1\leq 1-c_{\pi }(\varepsilon )q_1^{-\varepsilon }$.

8.2 Proof of Theorem 2.5

We follow Gallagher's proof of the Bombieri–Vinogradov theorem in [Gal68], with $n\geq 2$ and $\pi \in \mathfrak {F}_n$ fixed at the onset. Note that the function

\[ \psi_k(y;q,a):= \frac{1}{k!}\sum_{\substack{m\leq y\\ m\equiv a\,(\mathrm{mod}\,\,q)}}|a_{\pi}(m)|^2\Lambda(m)\Big(\log\frac{y}{m}\Big)^k \]

is monotonically increasing as a function of $y$ for each $k\geq 0$. Thus, if $0<\lambda \leq 1$, then

\[ \frac{1}{\lambda}\int_{e^{-\lambda}y}^{y}\psi_{k-1}(t;q,a)\frac{dt}{t}\leq \psi_{k-1}(y;q,a)\leq \frac{1}{\lambda}\int_{y}^{e^{\lambda}y}\psi_{k-1}(t;q,a)\frac{dt}{t}. \]

The integrals, which equal $\psi _k(y;q,a)-\psi _k(e^{-\lambda }y;q,a)$ and $\psi _k(e^{\lambda }y;q,a)-\psi _k(y;q,a)$, respectively, both have the same asymptotic expansion, namely

\[ (\lambda+O(\lambda^2))\frac{y}{\varphi(q)}+O\bigg(\max_{y\leq ex}|r_k(y;q,a)|\bigg),\quad r_k(y;q,a):= \psi_k(y;q,a)-\frac{y}{\varphi(q)}, \]

where $\varphi$ is Euler's totient function. Thus, we have the bounds

\[ \max_{y\leq x}|r_{k-1}(y;q,a)|\ll \frac{\lambda x}{\varphi(q)}+\frac{1}{\lambda}\max_{y\leq ex}|r_k(y;q,a)| \]

and

\[ \sum_{\substack{q\leq x^{\theta} \\ \gcd(q,q_{\pi}\!)=1}}\max_{\gcd(a,q)=1}\max_{y\leq x}|r_{k-1}(y;q,a)|\ll \lambda x\log x+\frac{1}{\lambda}\sum_{\substack{q\leq x^{\theta} \\ \gcd(q,q_{\pi}\!)=1}}\max_{\gcd(a,q)=1}\max_{y\leq ex}|r_k(y;q,a)|. \]

It follows by induction on $k$ that

\[ \sum_{\substack{q\leq x^{\theta} \\ \gcd(q,q_{\pi}\!)=1}}\max_{\gcd(a,q)=1}\max_{y\leq x}|r_0(y;q,a)|\ll_k \lambda x\log x + \frac{1}{\lambda^{2^k-1}}\sum_{\substack{q\leq x^{\theta} \\ \gcd(q,q_{\pi}\!)=1}}\max_{\gcd(a,q)=1}\max_{y\leq ex}|r_k(y;q,a)|. \]

Proposition 8.3 If $\theta <1/(9n^3)$ is fixed and $k=9n^2+1$, then for any $B>0$, we have

\[ \sum_{\substack{q\leq x^{\theta} \\ \gcd(q,q_{\pi}\!)=1}}\max_{\gcd(a,q)=1}\max_{y\leq ex}|r_k(y;q,a)|\ll_{\pi,B} \frac{x}{(\log x)^B}. \]

Proposition 8.3 implies Theorem 2.5 It follows from Proposition 8.3 that

\[ \sum_{\substack{q\leq x^{\theta} \\ \gcd(q,q_{\pi}\!)=1}}\max_{\gcd(a,q)=1}\max_{y\leq x}|r_0(y;q,a)|\ll_{\pi,B} \lambda x\log x+\frac{1}{\lambda^{2^k-1}}\frac{x}{(\log x)^B}. \]

To finish, we choose $B=2^k(A+1)-1$ and $\lambda = (\log x)^{-(B+1)/2^k}$.

8.3 Proof of Proposition 8.3

Let $\pi \in \mathfrak {F}_n$, $k=9n^2+1$, $Q= x^{\theta }$ for some fixed $0<\theta <1/(9n^3)$, and $q\leq Q$. We have the decomposition

\[ \frac{1}{k!}\sum_{\substack{m\leq y \\ m\equiv a\,(\mathrm{mod}\,\,q)}}|a_{\pi}(m)|^2\Lambda(m)\Big(\log\frac{y}{m}\Big)^k =\frac{1}{k!}\sum_{\psi\,(\mathrm{mod}\,\,q)}\frac{\overline{\psi}(a)}{\varphi(q)}\sum_{\substack{y\leq m}}|a_{\pi}(m)|^2\Lambda(m)\psi(n)\Big(\!\log\frac{y}{m}\Big)^k. \]

By Lemma 8.1 and Mellin inversion, this equals

\[ \frac{1}{\varphi(q)}\sum_{\psi\,(\mathrm{mod}\,\,q)}\sum_{\substack{\text{$\chi$ primitive} \\ \text{$\chi$ induces $\psi$}}}\frac{\overline{\chi}(a)}{2\pi i}\int_{3-i\infty}^{3+i\infty}\bigg({-}\, \frac{\Lambda_{\chi}'}{\Lambda_{\chi}}(s)+H_{\pi}(s;\chi,\psi)\bigg)\frac{y^s}{s^{k+1}}\,ds. \]

Since a primitive character $\chi \,(\mathrm {mod}\,\,q)$ induces characters to moduli that are a multiple of $q$, it follows from the bound $\sum _{f\leq Q,~q|f}\varphi (f)^{-1}\ll ({\log Q})/{\varphi (q)}$ that

(8.2)\begin{align} &\sum_{\substack{q\leq Q \nonumber\\ \gcd(q,q_{\pi}\!)=1}}~\max_{\gcd(a,q)=1}~\max_{y\leq x} \bigg|\frac{1}{k!}\sum_{\substack{m\leq y\\ m\equiv a\,(\mathrm{mod}\,\,q)}}|a_{\pi}(m)|^2\Lambda(m) \bigg(\log\frac{y}{m}\bigg)^k- \frac{y}{\varphi(q)}\bigg|\nonumber\\ &\quad \ll \sum_{\substack{q\leq Q \\ \gcd(q,q_{\pi}\!)=1}}\frac{\log Q}{\varphi(q)}~\sum_{\substack{\chi\,(\mathrm{mod}\,\,q) \\ \text{$\chi$ primitive}}}\max_{y\leq x}\bigg|\frac{1}{2\pi i}\int_{3-i\infty}^{3+i\infty}\bigg({-}\,\frac{\Lambda_{\chi}'}{\Lambda_{\chi}}(s)+H_{\pi}(s;\chi,\chi)\bigg) \frac{y^s}{s^{k+1}}\,ds\bigg|. \end{align}

Observe that by Lemma 8.1 and our range of $\theta$, (8.2) equals

(8.3)\begin{align} & \sum_{\substack{q\leq Q \nonumber\\ \gcd(q,q_{\pi}\!)=1}}\frac{\log Q}{\varphi(q)}~\sum_{\substack{\chi\,(\mathrm{mod}\,\,q) \\ \text{$\chi$ primitive}}}\max_{y\leq x}\bigg|-\sum_{\substack{L_{\chi}(\rho)=0 }}\frac{y^{\rho}}{\rho^{k+1}}+ \frac{1}{2\pi i}\int_{1-{1}/{n^2}-i\infty}^{1-{1}/{n^2}+i\infty}H_{\pi}(s;\chi,\chi)\frac{y^s}{s^{k+1}}\,ds\bigg|\nonumber\\ &\quad \ll_{\pi,B}\log Q \sum_{\substack{q\leq Q \\ \gcd(q,q_{\pi}\!)=1}}\frac{1}{\varphi(q)}~\sum_{\substack{\chi\,(\mathrm{mod}\,\,q) \\ \text{$\chi$ primitive}}}\sum_{\substack{L_{\chi}(\rho)=0 }}\frac{y^{\beta}}{|\rho|^{k+1}}+\frac{x}{(\log x)^B}, \end{align}

where $\rho =\beta +i\gamma$ ranges over the non-trivial zeros of $L_{\chi }(s)$. In light of the bounds $ {1}/{q} \leq {1}/{\varphi (q)}\ll ({\log (eq)})/{q}$, we dyadically decompose $[1,Q]$ into $O(\log Q)$ subintervals and find that (8.3) is

(8.4) \begin{equation} \begin{aligned} & \ll_{\pi,B} x(\log Q)^2\sup_{3\leq R\leq Q}\sum_{\substack{q\leq R \\ \gcd(q,q_{\pi}\!)=1}}\frac{1}{\varphi(q)}~\sum_{\substack{\chi\,(\mathrm{mod}\,\,q) \\ \text{$\chi$ primitive}}}~\sum_{\substack{\rho=\beta+i\gamma }}\frac{x^{\beta-1}}{|\rho|^k}+\frac{x}{(\log x)^B}\\ & \ll_{\pi,B} x(\log Q)^3\sup_{3\leq R\leq x^{\theta}}\frac{1}{R}\sum_{\substack{q\leq R \\ \gcd(q,q_{\pi}\!)=1}}~\sum_{\substack{\chi\,(\mathrm{mod}\,\,q) \\ \text{$\chi$ primitive}}}~\sum_{\substack{\rho=\beta+i\gamma }}\frac{x^{\beta-1}}{|\rho|^k}+\frac{x}{(\log x)^B}\\ & \ll_{\pi,B} x(\log x)^3\sup_{3\leq R\leq x^{\theta}}\frac{1}{R}\bigg(\frac{x^{\beta_1-1}}{\beta_1^k}+\sum_{\substack{q\leq R \\ \gcd(q,q_{\pi}\!)=1}}~\sum_{\substack{\chi\,(\mathrm{mod}\,\,q) \\ \text{$\chi$ primitive}}} \sum_{\substack{\rho=\beta+i\gamma\neq\beta_1 }}\frac{x^{\beta-1}}{|\rho|^k}\bigg)+\frac{x}{(\log x)^B}. \end{aligned} \end{equation}

where $\beta _1$ is the exceptional zero in Theorem 8.2. The term $x^{\beta _1-1}/\beta _1^k$ is omitted if $\beta _1$ does not exist.

If $\beta _1$ exists as in Theorem 8.2 and the supremum is achieved when $R\leq (\log x)^{4B}$, then we apply Theorem 8.2 with $\varepsilon = {1}/{8B}$ and conclude that the contribution from such a zero is absorbed in our error term. If the supremum is achieved when $R>(\log x)^{4B}$, then contribution from $\beta _1$ is trivially absorbed in our error term. Hence, (8.4) is

(8.5)\begin{equation} \ll_{\pi,B} x(\log x)^3\sup_{3\leq R\leq x^{\theta}}\frac{1}{R} \sum_{\substack{q\leq R \\ \gcd(q,q_{\pi}\!)=1}}~\sum_{\substack{\chi\,(\mathrm{mod}\,\,q) \\ \text{$\chi$ primitive}}}\sum_{\substack{\rho=\beta+i\gamma\neq\beta_1 }}\frac{x^{\beta-1}}{|\rho|^k}+\frac{x}{(\log x)^B}. \end{equation}

Now let us consider the zeros $\rho$ with $|\rho |<\frac {1}{4}$. The number of such zeros is $\ll R^2\log R$. From the consideration of the corresponding zeros $1-\rho$ of $L_{\overline {\chi }}(s)$, we deduce that $|\rho |\gg x^{-1/(4k)}$. Thus, the contribution from these zeros is $\ll R x^{ {1}/{4}+ {k}/{4k}}\log R\ll Q x^{ {1}/{2}}\log Q \ll _{B} x(\log x)^{-B}$. Define $T_0=0$ and $T_j=2^{j-1}$ for $j\geq 1$. The above discussion shows that (8.5) is

(8.6) \begin{align} & \ll_{\pi,B} x(\log x)^3\sup_{3\leq R\leq x^{\theta}}\frac{1}{R}\sum_{\substack{q\leq R \\ \gcd(q,q_{\pi}\!)=1}}~\sum_{\substack{\chi\,(\mathrm{mod}\,\,q) \\ \text{$\chi$ primitive}}} \sum_{\substack{\rho=\beta+i\gamma\neq \beta_1 \\ |\rho|>\frac{1}{4}}}\frac{x^{\beta-1}}{|\rho|^k}+\frac{x}{(\log x)^B}\nonumber\\ & \ll_{\pi,B} x(\log x)^3\sup_{3\leq R\leq x^{\theta}}\frac{1}{R}\sum_{j=1}^{\infty}~\sum_{\substack{q\leq R \\ \gcd(q,q_{\pi}\!)=1}}~\sum_{\substack{\chi\,(\mathrm{mod}\,\,q) \\ \text{$\chi$ primitive}}}~\sideset{}{'}\sum_{\substack{\rho=\beta+i\gamma\neq\beta_1 \\ |\rho|\geq\frac{1}{4} \\ T_{j-1}\leq|\gamma|\leq T_j}}\frac{x^{\beta-1}}{|\rho|^k}+\frac{x}{(\log x)^B}. \end{align}

If $|\rho |\geq \frac {1}{4}$ and $T_{j-1}\leq |\gamma |\leq T_j$, then $|\rho |\geq \max \{T_{j-1},\frac {1}{4}\}\geq T_j/4$ and $|\rho |\gg |\gamma |+3$. Therefore, if $\delta = \min \{1-9n^3\theta,\tfrac {1}{2}\}$, then

\begin{align*} x^{\beta-1}|\rho|^{-k}&\ll T_j^{-{1}/{2}}(|\gamma|+1)^{-{1}/{2}}x^{-(1-\beta)\delta}(x^{1-\delta}T_j^{k-1})^{-(1-\beta)}\\ &\ll T_j^{-{1}/{2}}(|\gamma|+1)^{-{1}/{2}}x^{-(1-\beta)\delta}(R^{({1-\delta})/{\theta}}T_j^{k-1})^{\beta-1}. \end{align*}

Since $\rho =\beta +i\gamma \neq \beta _1$, it follows from Theorem 8.2 that

\[ (|\gamma|+1)^{-{1}/{2}}x^{-\delta(1-\beta)}\leq e^{-\delta \eta_{\pi}(x,R)},\quad \eta_{\pi}(x,R) := \inf_{t\geq 3}\bigg[c_{\pi}\frac{\log x}{\log(Rt)}+\log t\bigg], \]

so (8.6) is

\begin{align*} &\ll_{\pi,B}x(\log x)^3 \sup_{3\leq R\leq x^{\theta}}\frac{e^{-\delta \eta(x,R)}}{R}\sum_{j=1}^{\infty}T_j^{-{1}/{2}}\\ &\quad\times\sum_{\substack{q\leq R \\ \gcd(q,q_{\pi}\!)=1}}~\sum_{\substack{\chi\,(\mathrm{mod}\,\,q) \\ \text{$\chi$ primitive}}}~\sum_{\substack{\rho=\beta+i\gamma\neq \beta_1\\ |\gamma|\leq T_j}}(R^{({1-\delta})/{\theta}}T_j^{k-1})^{\beta-1}+\frac{x}{(\log x)^B}. \end{align*}

Define

\[ N_{\pi}^*(\sigma,T,R)=\sum_{\substack{q\leq R \\ \gcd(q,q_{\pi}\!)=1}}~\sum_{\substack{\chi\bmod q \\ \text{$\chi$ primitive}}}|\{\rho=\beta+i\gamma\colon \beta\geq\sigma,~|\gamma|\leq T,~L_{\chi}(\rho)=0\}|. \]

By (3.10), there exists an effectively computable constant $c_{13}=c_{13}(n)>0$ such that

\[ \{\widetilde{\pi}\otimes\chi\colon \text{$ \chi\,(\mathrm{mod}\,\,q_{\chi})$ primitive, $\gcd(q_{\chi},q_{\pi})=1$, $q_{\chi}\leq Q$}\}\subseteq\mathfrak{F}_{n}(c_{13}C(\pi)Q^n). \]

Thus, by (1.7), we have $N_{\pi }^*(\sigma,T,R)\ll _{\pi,\varepsilon }(R^n T)^{9n^2(1-\sigma )+\varepsilon }$. Partial summation yields

(8.7)\begin{align} &\sum_{\substack{q\leq R \nonumber\\ \gcd(q,q_{\pi}\!)=1}}~\sum_{\substack{\chi\,(\mathrm{mod}\,\,q) \\ \text{$\chi$ primitive}}} \sum_{\substack{\rho=\beta+i\gamma\neq \beta_1\\ |\gamma|\leq T_j}}(R^{({1-\delta})/{\theta}}T_j^{k-1})^{-(1-\beta)}\nonumber\\ &\quad\ll (R^{({1-\delta})/{\theta}}T_j^{k-1})^{-1}N_{\pi}^*(0,T_j,R)+\log(RT_j)\int_0^1 (R^{({1-\delta})/{\theta}}T_j^{k-1})^{-\sigma}N_{\pi}^*(1-\sigma,T_j,R)\,d\sigma\nonumber\\ &\quad\ll_{\pi,\varepsilon} (R T_j)^{\varepsilon}\bigg((R^{({1-\delta})/{\theta}}T_j^{k-1})^{-1}R T_j+ \int_0^1 (R^{({1-\delta})/{\theta}}T_j^{k-1})^{-\sigma}(R^n T_j)^{9n^2\sigma}d\sigma\bigg). \end{align}

Our choices for $\theta$, $k$, and $\delta$ ensure that (8.7) is $\ll _{\pi,\varepsilon } (R T_j)^{\varepsilon }$. Thus, (8.6) is

(8.8) \begin{align}& \ll_{\pi,B,\varepsilon} x(\log x)^3 \sup_{3\leq R\leq x^{\theta}}e^{-\delta \eta_{\pi}(x,R)}\frac{R^{\varepsilon}}{R}\sum_{j=1}^{\infty}T_j^{-{1}/{2}+\varepsilon}+\frac{x}{(\log x)^B}\nonumber\\ & \ll_{\pi,B,\varepsilon} x(\log x)^3 \sup_{3\leq R\leq x^{\theta}}e^{-\delta \eta_{\pi}(x,R)}\frac{R^{\varepsilon}}{R}+\frac{x}{(\log x)^B}. \end{align}

A small calculation (cf. [TZ19, § 4]) shows that there is a constant $c_{14}=c_{14}(\pi )>0$ such that $e^{-\delta \eta _{\pi }(x,R)}\ll _{\pi,B,\varepsilon } \exp (-c_{14} {\log x}/{\!\log R})+\exp (-c_{14}\sqrt {\log x})$, and Theorem 2.5 follows.

8.4 Proof of Theorem 8.2

Although the proof of Theorem 8.2 contains only standard techniques, such zero-free regions for $L_{\chi }(s)$ are new even in the case when $\chi$ is trivial. The case when $\chi$ is trivial was only recently handled unconditionally in [HT22].

Lemma 8.4 [HT22, Theorem 2.1(1)]

Let $\pi \in \mathfrak {F}_n$. There exists an absolute and effectively computable constant $c_{15}>0$ such that $L_{1}(s)$ has at most one zero, say $\beta _1$, in the region $\text {Re}(s)\geq 1-c_{15}/\!\log (C(\pi )^n(|\text {Im}(s)|+e)^{n^2})$. If $\beta _1$ exists, then it must be real and simple, and there exists an absolute and effectively computable constant $c_{16}$ such that $\beta _1\leq 1-C(\pi )^{-c_{16}n}$.

We prove the corresponding result for $L_{\chi }(s)$. The ideas in [HT22] inform our approach here.

Lemma 8.5 Let $\pi \in \mathfrak {F}_{n}$. Let $\chi \,(\mathrm {mod}\,\,q)$ be a non-trivial primitive Dirichlet character such that $\gcd (q,q_{\pi })=1$. There exists an effectively computable constant $c_{17}=c_{17}(n)>0$ such that $L_{\chi }(s)$ has at most one zero, say $\beta _1$, in the region

(8.9)\begin{equation} \text{Re}(s)\geq 1-\frac{c_{17}}{\log(qC(\pi)(|\text{Im}(s)|+3))}. \end{equation}

If the exceptional zero $\beta _1$ exists, then it is real and simple, and $\chi$ is quadratic.

Proof. Let $\chi \,(\mathrm {mod}\,\,q)$ be a primitive non-trivial Dirichlet character, let $\psi$ be the primitive character that induces $\chi ^2$, and let $\beta +i\gamma$ be a non-trivial zero of $L_{\chi }(s)$. Define $\Pi _{\chi }=\pi \boxplus \pi \otimes \chi |\cdot |^{it}\boxplus \pi \otimes \overline {\chi }|\cdot |^{-it}$, and define

(8.10)\begin{equation} D(s)=L(s,\Pi_{\chi}\times\widetilde{\Pi}_{\chi})=L_{1}(s)^3 L_{\chi}(s+i\gamma)^2 L_{\overline{\chi}}(s-i\gamma)^2 L_{\psi}(s+2i\gamma)L_{\overline{\psi}}(s-2i\gamma). \end{equation}

The factor $L_1(s)^3$ has a pole of order 3 at $s=1$, and the hypothesis that $\gcd (q,q_{\pi })=1$ ensures that $L_{\chi }(s+i\gamma )^2 L_{\overline {\chi }}(s-i\gamma )^2$ is entire. If $\psi$ is complex, then $L_{\psi }(s+2i\gamma )L_{\overline {\psi }}(s-2i\gamma )$ is entire; otherwise, it has poles of order 1 at $s=1\pm 2i\gamma$. The additional poles when $\psi$ is real require some additional casework when $\gamma$ is close to zero. For notational compactness, let $\mathcal {Q}_{\gamma }=qC(\pi )(|\gamma |+3)$. Note that $-(D'/D)(s)$ has non-negative Dirichlet coefficients per [HR95, Lemma a].

The functional equation for $L_{\chi }(s)$ together with the fact that $L(s,\pi \times \widetilde {\pi })$ is a self-dual $L$-function (even if $L(s,\pi )$ itself is not self-dual) implies that if $\rho$ is a zero of $L_{\chi }(s)$, then $\overline {\rho }$ is a zero of $L_{\overline {\chi }}(s)$. Thus, we have that

(8.11)\begin{equation} \mathop{\mathrm{ord}}\limits_{s=\beta}D(s) \geq 4. \end{equation}

Let $\omega$ denote a non-trivial zero of $D(s)$, $\delta (\psi )=1$ if $\psi$ is trivial, and $\delta (\psi )=0$ otherwise. We apply Lemma 3.2 and (3.10) to (8.10), concluding that if $1<\sigma <2$, then

(8.12)\begin{equation} \sum_{\omega}\text{Re}\bigg(\frac{1}{\sigma-\omega}\bigg)<\frac{3}{\sigma-1}+\delta(\psi)\frac{2(\sigma-1)}{(\sigma-1)^2+4\gamma^2}+c_{18}\log\mathcal{Q}_{\gamma}, \end{equation}

where $c_{18}=c_{18}(n)>1$ is a suitable implied constant. Since $\beta <1$ is, by hypothesis, one of the zeros in the sum in (8.12), we have by (8.11) and non-negativity that

(8.13) \begin{align} \frac{4}{\sigma-\beta}< \frac{3}{\sigma-1}+\delta(\psi)\frac{2(\sigma-1)}{4\gamma^2+(\sigma-1)^2}+c_{18}\log\mathcal{Q}_{\gamma}. \end{align}

Case 1: Either $\chi$ is real and $|\gamma |\geq {1}/{7c_{22}\log \mathcal {Q}_0}$ or $\chi$ is complex

If $\sigma = 1+ {1}/{5c_{18}\log \mathcal {Q}_{\gamma }}$, then

\[ \delta(\psi)\frac{2(\sigma-1)}{4\gamma^2+(\sigma-1)^2}\leq \frac{490}{149}c_{18}\log\mathcal{Q}_{\gamma}. \]

Thus, (8.13) becomes

\[ \frac{4}{1+{1}/{5c_{18}\log\mathcal{Q}_{\gamma}}-\beta}\leq \frac{2874}{149}c_{18}\log\mathcal{Q}_{\gamma}. \]

Upon solving for $\beta$, we conclude that $\beta \leq 1- {1}/{136c_{18}\mathcal {Q}_{\gamma }}$.

Case 2: $\chi$ is real and $\gamma =0$

We start at (8.12) with $\delta (\psi )=1$, $\gamma =0$, and $\sigma = 1+ {1}/{3c_{18}\log \mathcal {Q}_0}$. If there are $N$ zeros $\beta$ (with multiplicity) of $D(s)$ such that $\beta \geq 1- {1}/{96c_{18}\log \mathcal {Q}_0}$, then by (8.12),

\[ \frac{32}{11}c_{18}N\log\mathcal{Q}_0=\frac{N}{\sigma-(1-{1}/{96c_{18}\log\mathcal{Q}_0})}\leq \frac{5}{\sigma-1}+c_{18}\log \mathcal{Q}_0=16c_{18}\log\mathcal{Q}_0. \]

It follows that (since $N$ is an integer) $N\leq \lfloor \frac {11}{2}\rfloor =5$. By the bound (8.11), $L_{\chi }(s)$ has at most one real zero $\beta$, necessarily simple, satisfying $\beta \geq 1- {1}/{96c_{18}\log \mathcal {Q}_0}$.

Case 3: $\chi$ is quadratic and $0<|\gamma |<{1}/{7c_{22}\log \mathcal {Q}_0}$

We apply Lemma 3.2 to $L_{1}(s)L_{\chi }(s)$. The only singularity is a simple pole at $s=1$. Both $L_{1}(s)$ and $L_{\chi }(s)$ are self-dual, so if $\beta +i\gamma$ is a non-trivial zero of $L_1(s)L_{\chi }(s)=0$, then so is $\beta -i\gamma$. By (8.1), the $m$th Dirichlet coefficient of $- {L_{1}'}/{L_{1}}(s)- {L_{\chi }'}/{L_{\chi }}(s)$ is $(1+\chi (m))a_{\pi \times \widetilde {\pi }}(m)\Lambda (m)\geq 0$, so Lemma 3.2 yields

(8.14)\begin{equation} 2\frac{\sigma-\beta}{(\sigma-\beta)^2+\gamma^2}\leq\frac{1}{\sigma-1}+c_{18}\log \mathcal{Q}_{\gamma}. \end{equation}

If $\sigma = 1+ {1}/{2c_{18}\log \mathcal {Q}_{\gamma }}$ and $0<|\gamma |< {1}/{7c_{18}\log \mathcal {Q}_{0}}$ in (8.14), then $\beta \leq 1- {1}/{8c_{18}\log \mathcal {Q}_{\gamma }}$.

We now prove a Siegel-type bound for $\beta _1$ in Lemma 8.5 (if it exists) using the ideas of Hoffstein and Lockhart [HL94]. This is new for all $\pi \in \mathfrak {F}_n$ with $n\geq 3$. We begin with an auxiliary calculation. Let $\chi \,(\mathrm {mod}\,\,q)$ and $\chi '\,(\mathrm {mod}\,\,q')$ be distinct non-trivial quadratic Dirichlet characters such that $\gcd (q'q,q_{\pi })=1$, and let $\psi$ be the primitive character that induces $\chi '\chi$ (whose conductor is necessarily coprime to $q_{\pi }$). These coprimality restrictions ensure that $\pi \neq \pi \otimes \chi$, $\pi \neq \pi \otimes \chi '$, $\pi \neq \pi \otimes \psi$, $q_{\pi \otimes \chi }=q_{\pi }q^n$, $q_{\pi \otimes \chi '}=q_{\pi }(q')^n$, and $q_{\pi \otimes \psi }=q_{\psi }q^n$. We also have that $q_{\psi }|q_{\chi }q_{\chi '}$. Let

(8.15)\begin{equation} L(s,\Pi^{\star})=L_{1}(s)L_{\chi}(s)L_{\chi'}(s)L_{\psi}(s). \end{equation}

By the above discussion, $L(s,\Pi ^{\star })$ is holomorphic apart from a simple pole at $s=1$. It follows from (8.1) that if $k\geq 1$, then the $p^k$th Dirichlet coefficient of $\log L(s,\Pi ^{\star })$ equals

\[ k^{-1}a_{\pi\times\widetilde{\pi}}(p^k)(1+\chi(p^k)+\chi'(p^k)+\psi(p^k))\geq 0. \]

The non-negativity of $1+\chi (p^k)+\chi '(p^k)+\psi (p^k)$ follows from the fact that this sum is a Dirichlet coefficient of the Dedekind zeta function of a biquadratic field. Upon exponentiating, we find that the $m$th Dirichlet coefficient $\lambda _{\Pi ^{\star }}(m)$ of $L(s,\Pi ^{\star })$ is non-negative.

Lemma 8.6 If $\pi \in \mathfrak {F}_n$ and $\chi$ is a primitive non-trivial real Dirichlet character, then $L_{\chi }(1)>0$ and the Dirichlet coefficients of $L_{1}(s)L_{\chi }(s)$ are non-negative.

Proof. Let $K$ be the quadratic field associated to $\chi$. If $\pi _{\mathrm {BC}}$ is the base change of $\pi$ to an automorphic representation of $\mathrm {GL}_n(\mathbb {A}_K)$, then $L(s,\pi _{\mathrm {BC}}\times \widetilde {\pi }_{\mathrm {BC}})=L_{1}(s)L_{\chi }(s)$. Since $L(s,\pi _{\mathrm {BC}}\times \widetilde {\pi }_{\mathrm {BC}})$ is holomorphic on $\mathbb {C}-\{1\}$ apart from a simple pole at $s=1$, the same holds for $L_{1}(s)L_{\chi }(s)$. Since $\gcd (q,q_{\pi })=1$ (hence, $L_{\chi }(s)$ is entire), the residue of $L(s,\pi _{\mathrm {BC}}\times \widetilde {\pi }_{\mathrm {BC}})$ at $s=1$, which is positive, equals $L_{\chi }(1)\mathrm {Res}_{s=1}L_{1}(s)$. Since $\mathrm {Res}_{s=1}L_{1}(s)>0$, it follows that $L_{\chi }(1)>0$. The Dirichlet coefficients of $L_{1}(s)L_{\chi }(s)$ are non-negative per case 3 in the proof of Lemma 8.5.

Let $0<\varepsilon <1$, and let $\beta \in (1-\varepsilon,1)$. By (3.13) and the above discussion, we have

(8.16)\begin{equation} \begin{aligned} L({1}/{2}+it,\Pi^{\star}) & \ll_{\pi,\varepsilon} (q'q)^{{n^2}/{2}+\varepsilon}(3+|t|)^{n^2+\varepsilon},\\ \mathop{\mathrm{Res}}\limits_{s=1-\beta}L(s+\beta,\Pi^{\star})x^s\Gamma(s) & \ll_{\pi,\varepsilon}L_{\chi}(1)(q'q)^{\varepsilon}(1-\beta)^{-1}x^{1-\beta}. \end{aligned} \end{equation}

If $x\geq 3$, then since $\lambda _{\Pi ^{\star }}(1)=1$, we use (8.16) to deduce that

(8.17)\begin{align} \frac{1}{e}&\leq \sum_{m=1}^{\infty}\frac{\lambda_{\Pi^{\star}}(m)}{m^{\beta}}e^{-{m}/{x}}\nonumber\\ &=\frac{1}{2\pi i}\int_{3-i\infty}^{3+i\infty}L(s+\beta,\Pi^{\star})x^s\Gamma(s)\,ds\nonumber\\ &=\mathop{\mathrm{Res}}\limits_{s=1-\beta}L(s+\beta,\Pi^{\star})x^s\Gamma(s)+L(\beta,\Pi^{\star})+ \frac{1}{2\pi i}\int_{{1}/{2}-\beta-i\infty}^{{1}/{2}-\beta+i\infty}L(s+\beta,\Pi^{\star})x^s\Gamma(s)\,ds\nonumber\\ &\ll_{\pi,\varepsilon}L_{\chi}(1)(q'q)^{\varepsilon}(1-\beta)^{-1}x^{1-\beta}+L(\beta,\Pi^{\star})+(q'q)^{{n^2}/{2}+\varepsilon}x^{{1}/{2}-\beta}. \end{align}

Proposition 8.7 Recall the notation and hypotheses of Lemma 8.5. If $\beta _1$ exists for a primitive character $\chi \,(\mathrm {mod}\,\,q)$ such that $\gcd (q,q_{\pi })=1$, then for all $\varepsilon >0$, there exists an (ineffective) constant $c_{\pi }'(\varepsilon )>0$ such that $L_{\chi }(1)\geq c_{\pi }'(\varepsilon )q^{-\varepsilon }$.

Proof. It suffices to take $0<\varepsilon <1$. Let $\chi \,(\mathrm {mod}\,\,q)$ and $\chi '\,(\mathrm {mod}\,\,q')$ be primitive quadratic Dirichlet characters with, let $\psi$ be the primitive character that induces $\chi '\chi$, and recall the definition of $L(s,\Pi ^{\star })$ from (8.15). Our proof consists of two cases.

First, suppose that there exists no primitive quadratic Dirichlet character $\omega$ such that $L_{\omega }(s)$ does not vanish for $s\in (1- {\varepsilon }/{2},1)$. It then follows that there exists a constant $c_{19}=c_{19}(\pi )>0$ such that $L_{1}(s)L_{\chi }(s)\neq 0$ in the interval $s\in (1-c_{19}/\!\log q,1)$. Since the Dirichlet coefficients of $L_{1}(s)L_{\chi }(s)$ are non-negative (using Lemma 8.6) and the residue of $L_{1}(s)L_{\chi }(s)$ at $s=1$ is $L_{\chi }(1)\mathrm {Res}_{s=1}L_{1}(s)$, it follows from [HL94, Proposition 1.1] and (3.13) that $L_{\chi }(1){\mathrm {Res}}_{s=1}L_{1}(s)\gg _{\pi }(\log q)^{-1}$. Since each term on the left-hand side is positive (using Lemma 8.6), the desired result follows.

Second, suppose that there exists $\chi '\,(\mathrm {mod}\,\,q')$ and $\beta \in (1- {\varepsilon }/{2},1)$ such that $L_{\chi '}(\beta )=0$. We may assume that $q'$ is minimal, subject to this condition. Now, let $\chi$ be arbitrary. If $q< q'$, then the minimality of $q'$ ensures that $L_{\chi }(s)\neq 0$ for $s\in (1- {\varepsilon }/{2},1)$, and the preceding case implies the desired result. Suppose now that $q\geq q'$. If $L_{\chi }(s)$ has no real zero within a distance of $c_{17}/\!\log (3q'q C(\pi ))$ of $s=1$, then since $q\geq q'$, $L_{\chi }(s)$ has no real zero within a distance of $\frac {1}{2}c_{17}/\!\log (3qC(\pi ))$ of $s=1$. Again, the desired result follows by the preceding case. Finally, suppose that $L_{\chi }(s)$ has a real zero within a distance of $c_{17}/\!\log (3q'q C(\pi ))$ of $s=1$. At this stage, we assume that $\chi \neq \chi '$.

It follows from analysis nearly identical to the second case in Lemma 8.5 (with $L(s,\Pi ^{\star })$ replacing $D(s)$) that $L(s,\Pi ^{\star })$ has at most one real zero within distance $c_{17}/\!\log (3q'q C(\pi ))$ from 1. Since we have supposed that $L_{\chi }(s)$ has a real zero within distance $c_{17}/\!\log (3q'q C(\pi ))$ of $s=1$, the above discussion indicates that this is the sole real zero for $L(s,\Pi ^{\star })$ within a distance of $c_{17}/\!\log (3q'q C(\pi ))$ of $s=1$. It follows that the zero $\beta$ of $L_{\chi '}(s)$ must satisfy

\[ \beta\leq 1-\frac{c_{17}}{\log(3q'qC(\pi))},\quad \beta\in\bigg(1-\frac{\varepsilon}{2},1\bigg). \]

Since $L_{\chi '}(\beta )=0$, it follows that $L(\beta,\Pi ^{\star })=0$. Using (8.17), the above bounds on $\beta$, and the fact that $q\geq q'$, we find that

\[ 1\ll_{\pi,\varepsilon}L_{\chi}(1)q^{2\varepsilon}x^{1-\beta}+q^{n^2+2\varepsilon}x^{{1}/{2}-\beta} \ll_{\pi,\chi',\varepsilon}L_{\chi}(1)q^{2\varepsilon}x^{{\varepsilon}/{2}}+q^{n^2+2\varepsilon}x^{ {\varepsilon}/{2}-{1}/{2}}. \]

Choosing $x=q^{2n^2}/L_{\chi }(1)^2$ (which is at least 3 by (3.10) and (3.13)) and solving for $L_{\chi }(1)$, we find that for all $\chi \neq \chi '$ and all $0<\varepsilon <1$, there exists a constant $d_{\pi }(\varepsilon )>0$ such that $L_{\chi }(1)\geq d_{\pi }(\varepsilon )q^{-\varepsilon (n^2+2)/(1-\varepsilon )}$. Upon rescaling $\varepsilon$ to $\varepsilon /(n^2+2+\varepsilon )$, we have $L_{\chi }(1)\geq d_{\pi }(\varepsilon /(n^2+2+\varepsilon ))q^{-\varepsilon }$. As long as $\chi \neq \chi '$, the constant $d_{\pi }(\varepsilon /(n^2+2+\varepsilon ))$ is effective. Once we decrease $d_{\pi }(\varepsilon /(n^2+2+\varepsilon ))$ suitably to account for the case where $\chi =\chi '$, the claimed result holds for arbitrary $\chi$.

Corollary 8.8 Recall the notation and hypotheses of Lemma 8.5. If $\beta _1$ exists, then for all $\varepsilon >0$, there exists an (ineffective) constant $c_{\pi }(\varepsilon )>0$ such that $\beta _1\leq 1-c_{\pi }(\varepsilon )q^{-\varepsilon }$.

Proof. If $\beta _1$ exists, then there exists $\sigma \in [\beta _1,1]$ such that $L_{\chi }'(\sigma )(1-\beta _1)=L_{\chi }(1)\geq c_{\pi }'( {\varepsilon }/{2})q^{- {\varepsilon }/{2}}$ by Proposition 8.7 and the mean value theorem. The result follows once we establish the bound $L_{\chi }'(\sigma )\ll _{\pi,\varepsilon } q^{\frac {\varepsilon }{2}}$ for $\sigma \in [1-b_n/\!\log (3qC(\pi )),1]$, where $b_n>0$ is a suitable constant depending at most on $n$. To prove this, we observe via Cauchy's integral formula that

\[ L_{\chi}'(1)=\frac{1}{2\pi i}\int_{|z-1|={1}/{\!\log q}}\frac{L_{\chi}(z)}{(z-1)^2}\,dz\ll (\log q) \max_{|\xi-1|\leq{1}/{\!\log q}}|L_{\chi}(\xi)|, \]

in which case the desired bound follows from (3.13).

We show that among the primitive characters $\chi \,(\mathrm {mod}\,\,q)$ with $q\leq Q$, we encounter very few with the property that $L_{\chi }(s)$ has an exceptional zero.

Lemma 8.9 Let $Q\geq 3$. There exists an effectively computable constant $c_{20}=c_{20}(n)>0$ such that there is at most one real non-trivial primitive Dirichlet character $\chi _1\,(\mathrm {mod}\,\,q_1)$ with $q_1\leq Q$ such that $L_{\chi _1}(s)$ has a real zero $\beta _1$ satisfying $\beta _1>1-c_{20}/\!\log (C(\pi )Q)$.

Proof. Suppose to the contrary that $\chi \,(\mathrm {mod}\,\,q)$ and $\chi '\,(\mathrm {mod}\,\,q')$ are two distinct such characters with $q,q'\leq Q$. Let $\Pi = \pi \boxplus \pi \otimes \chi \boxplus \pi \otimes \chi '$, let $\psi$ be the primitive character that induces $\chi '\chi$, and let

\begin{align*} F(s) = L(s,\Pi\times\widetilde{\Pi}) = L_{1}(s)^3L_{\chi}(s)^2 L_{\chi'}(s)^2 L_{\psi}(s)^2. \end{align*}

By [HR95, Lemma a], the Dirichlet coefficients of $-(F'/F)(s)$ are non-negative. By Lemma 3.2, there exists a constant $c_{21}=c_{21}(n)\geq 1$ such that if $\omega$ runs through the nontrivial zeros of $F(s)$ and $1<\sigma <2$, then

(8.18)\begin{equation} \sum_{\omega}\text{Re}\bigg(\frac{1}{\sigma-\omega}\bigg)<\frac{3}{\sigma-1}+c_{21}\log(C(\pi)Q). \end{equation}

If $\sigma =1+ {1}/{2c_{21}\log (C(\pi )Q)}$ and $M$ is the number (necessarily an integer) of real zeros (counting multiplicity) of $F(s)$ that are at least $1- {1}/{14c_{21}\log (C(\pi )Q)}$, then it follows from (8.18) that

\begin{align*} & \frac{M}{1+{1}/{2c_{21}\log(C(\pi)Q)}-(1-{1}/{14c_{21}\log(C(\pi)Q)})} < \frac{3}{(1+{1}/{2c_{21}\log(C(\pi)Q)})-1} \\ & \quad +c_{21}\log(C(\pi)Q).\end{align*}

This implies that $M\leq 3$. However, if $L_{\chi }(s)$ and $L_{\chi '}(s)$ both have real zeros that are larger than $1- {1}/{14c_{21}\log (C(\pi )Q)}$, then $F(s)$ has four such zeros, a contradiction. The lemma now follows.

Proof of Theorem 8.2 This follows from Corollary 8.8 and Lemmas 8.4, 8.5, and 8.9.