DISCREPANCY BOUNDS FOR THE DISTRIBUTION OF RANKIN–SELBERG L-FUNCTIONS

Abstract

We investigate the discrepancy between the distributions of the random variable $\log L (\sigma , f \times f, X)$ and that of $\log L(\sigma +it, f \times f)$, that is,

$$ \begin{align*} D_{\sigma} (T) := \sup_{\mathcal{R}} |\mathbb{P}_T(\log L(\sigma+it, f \times f) \in \mathcal{R}) - \mathbb{P}(\log L(\sigma, f \times f, X) \in \mathcal{R})|, \end{align*} $$

where the supremum is taken over rectangles $\mathcal {R}$ with sides parallel to the coordinate axes. For fixed $T>3$ and $2/3 <\sigma _0 < \sigma < 1$, we prove that

$$ \begin{align*} D_{\sigma} (T) \ll \frac{1}{(\log T)^{\sigma}}. \end{align*} $$

1 Introduction

Let $X(p)$ be independent random variables uniformly distributed on the unit circle, where p runs over the prime numbers. The random Euler product of the Riemann zeta-function is defined by $\zeta (\sigma ,X) = \prod _p (1- ({X(p)}/{p^{\sigma }}))^{-1}$ . The behaviour of $p^{-it}$ is almost like the independent random variables $X(p)$ , which indicates that $\zeta (\sigma , X)$ should be a good model for the Riemann zeta-function.

Bohr and Jessen [1] suggested that $\log \zeta (\sigma +it)$ converges in distribution to $\log \zeta (\sigma , X)$ for $\sigma> 1/2$ . In 1994, Harman and Matsumoto [4] studied the discrepancy between the distribution of the Riemann zeta-function and that of its random model. For fixed $\sigma $ with $1/2 <\sigma \leq 1$ and any $\varepsilon>0$ , they proved that the discrepancy

$$ \begin{align*}D_{\sigma, \zeta} (T) := \sup_{\mathcal{R}} |\mathbb{P}_T(\log \zeta(\sigma+it) \in \mathcal{R}) - \mathbb{P}(\log \zeta(\sigma, X) \in \mathcal{R})|,\end{align*} $$

where the supremum is taken over rectangles $\mathcal {R}$ with sides parallel to the coordinate axes, satisfies the bound $D_{\sigma , \zeta } (T) \ll {1}/{(\log T)^{(4\sigma -2)/(21+8\sigma )-\varepsilon }}$ . Here, $\mathbb {P}_T (f(t) \in \mathcal {R}) := T^{-1} \text {meas}\{T \leq t \leq 2T: f(t) \in \mathcal {R}\}$ . Lamzouri et al. [8] improved the result by showing that $D_{\sigma , \zeta } (T) \ll {1}/{(\log T)^{\sigma }}$ .

Dong et al. [3] analysed the discrepancy between the distribution of values of Dirichlet L-functions and the distribution of values of random models for Dirichlet L-functions in the q-aspect. Lee [9] investigated the upper bound on the discrepancy between the joint distribution of L-functions on the line $\sigma = 1/2 + 1/G(T), t \in [T, 2T]$ , and that of their random models, where $\log \log T \leq G(T) \leq (\log T)/(\log \log T)^2$ .

Let f be a primitive holomorphic cusp form of weight k for ${\mathrm {SL}}_2(\mathbb {Z})$ . The normalised Fourier expansion at the cusp $\infty $ is $f(z)=\sum _{n \geq 1}\lambda _f(n)n^{{(k-1)}/{2}}e^{2\pi inz}$ , where $\lambda _f(n) \in \mathbb {R}$ , $n = 1, 2, \ldots, $ are normalised eigenvalues of Hecke operators $T(n)$ with $\lambda _f(1)=1$ , that is, $T(n)f=\lambda _f(n)f$ .

According to Deligne [2], for all prime numbers p, there are complex numbers $\alpha _f(p)$ and $\beta _f(p)$ , satisfying

(1.1) $$ \begin{align} \left\{\! \begin{aligned} & |\alpha_f(p)| = \alpha_f(p)\beta_f(p) = 1, \\ & \lambda_f(p^{\nu}) = \sum_{0\leq j \leq \nu} \alpha_f(p)^{\nu -j} \beta_f(p)^{\,j} \quad (\nu \geq 1). \end{aligned} \right. \end{align} $$

The function $\lambda _f(n)$ is multiplicative. Moreover, $\lambda _f(p)$ is real and satisfies Deligne’s inequality $|\lambda _f(n)| \leq d(n)$ for $n \geq 1$ , where $d(n)$ is the divisor function. In particular, $|\lambda _f(p)| \leq 2$ . For $\mathrm {Re}\, s>1$ , the L-function attached to f is defined by

$$ \begin{align*} L(s, f) = \sum_{n \geq 1} \frac{\lambda_f(n)}{n^s} = \prod_p \bigg(1 - \frac{\alpha_f(p)}{p^s}\bigg)^{-1} \bigg(1 - \frac{\beta_f(p)}{p^s}\bigg)^{-1}. \end{align*} $$

For $\mathrm {Re}\, s>1$ , the Rankin–Selberg L-function associated to f is defined by

$$ \begin{align*} L(s, f \times f) := \prod_p \bigg(1 - \frac{\alpha_f(p)^2}{p^s}\bigg)^{-1} \bigg(1 - \frac{\beta_f(p)^2}{p^s}\bigg)^{-1} \bigg(1 - \frac{1}{p^s}\bigg)^{-2} = \zeta(2s) \sum_{n = 1}^{\infty} \frac{\lambda_f(n)^2}{n^s}. \end{align*} $$

According to [6], for $\mathrm {Re} s>1$ ,

$$ \begin{align*} \log L(s, f \times f) = \sum_{n=2}^{\infty} \frac{\Lambda_{f \times f}(n)}{n^s \log n}, \end{align*} $$

where

$$ \begin{align*}\Lambda_{f \times f}(n) := \begin{cases} |\alpha_f(p)^{\nu} + \beta_f(p)^{\nu}|^2 \log p & \text{for } n = p^{\nu}, \\ 0 & \text{otherwise}. \end{cases} \end{align*} $$

For automorphic L-functions, from [10],

(1.2) $$ \begin{align} \sum_{p \leq x} \lambda_f^4(p) \sim C_f \frac{x}{\log x}. \end{align} $$

Recently, Xiao and Zhai [12] studied the discrepancy between the distributions of $\log L(\sigma +it,f)$ and its corresponding random variable $\log L(\sigma , f, X)$ . In this article, we investigate the discrepancy between the distribution of the random variable $\log L (\sigma , f \times f, X)$ and that of $\log L(\sigma +it, f \times f)$ . Define the Euler product

$$ \begin{align*} L(\sigma, f \times f, X) = \prod_p \bigg(1 - \frac{\alpha_f(p)^2 X(p)}{p^{\sigma}}\bigg)^{-1} \bigg(1 - \frac{\beta_f(p)^2 X(p)}{p^{\sigma}}\bigg)^{-1} \bigg(1 - \frac{X(p)}{p^{\sigma}}\bigg)^{-2}, \end{align*} $$

which converges almost surely for $\sigma> \tfrac 12$ . Consider

$$ \begin{align*} D_{\sigma} (T) := \sup_{\mathcal{R}} |\mathbb{P}_T(\log L(\sigma+it, f \times f) \in \mathcal{R}) - \mathbb{P}(\log L(\sigma, f \times f, X) \in \mathcal{R})|, \end{align*} $$

where the supremum is taken over rectangles $\mathcal {R}$ with sides parallel to the coordinate axes. We prove the following theorem.

Theorem 1.1. Let $T>3$ and $2/3 <\sigma _0 < \sigma < 1$ , where T and $\sigma _0$ are fixed. Then,

$$ \begin{align*} D_{\sigma} (T) \ll \frac{1}{(\log T)^{\sigma}}, \end{align*} $$

where the implied constant depends on f and $\sigma $ .

The proof follows the method in [8]. The range of $\sigma $ depends on the zero density theorem of $L(s,f\times f)$ and $L(s, \mathrm {sym}^2f)$ by noticing that $L(s, f \times f) = \zeta (s) L(s, \mathrm {sym}^2f)$ . Unfortunately, the zero density of $L(s, \mathrm {sym}^2f)$ can only be obtained nontrivially when $2/3 < \sigma \leq 1$ (see [5]).

2 Preliminaries

This section gathers several preliminary results. Since several proofs are essentially the same as those in [8], we omit their details. For any prime number p and integer $\nu>0$ , we define $b_f(p^{\nu }) = |\alpha _f(p)^{\nu } + \beta _f(p)^{\nu }|^2$ . Thanks to (1.1),

$$ \begin{align*} |b_f(p^{\nu})| \leq 4. \end{align*} $$

From probability theory, if the characteristic functions of two real-valued random variables are close, then the corresponding probability distributions are also close. The key to proving Theorem 1.1 is to demonstrate that the joint distribution characteristic function of $\mathrm {Re} \log L(\sigma + it)$ and $\mathrm {Im} \log L(\sigma + it)$ can be well estimated. For u, $v \in \mathbb {R}$ , we define

(2.1) $$ \begin{align} \Phi_{\sigma, T}(u,v) := \frac{1}{T} \int_T^{2T} \exp (iu\,\mathrm{Re} \log L(\sigma + it, f \times f) + iv\,\mathrm{Im} \log L(\sigma + it, f \times f))\,dt \end{align} $$

and

(2.2) $$ \begin{align} &\Phi_{\sigma}^{\mathrm{rand}}(u,v) := {\mathbb{E}} (\exp (iu\,{\mathrm{Re}} \log L(\sigma, f \times f, X) + iv\,{\mathrm{Im}}\log L(\sigma, f\times f, X))). \end{align} $$

Lemma 2.1 [7, Lemma 4.3].

Let $y>2$ and $|t|\geq y+3$ be real numbers. Let $\tfrac 12< \sigma _0 < \sigma \leq 1$ and suppose that the rectangle $\{s: \sigma _0 < \mathrm {Re} (s) \leq 1, |\mathrm {Im} (s) - t|\leq y+2\}$ does not contain zeros of $L(s, f \times f)$ . Then,

$$ \begin{align*} \log L(s, f \times f) = \sum_{p^{\nu} \leq y} \frac{b_f(p^{\nu})}{\nu p^{\nu(\sigma+it)}} + O\bigg(\frac{\log |t|}{(\sigma_1 - \sigma_0)^2} y^{\sigma_1 - \sigma}\bigg), \end{align*} $$

where $\sigma _1 = \min (\sigma _0 + {1}/{\log y}, {(\sigma + \sigma _0)}/{2})$ .

Lemma 2.2. Define $N(\sigma _0,T)$ as the number of zeros $\rho _f = \beta _f + i \gamma _f$ of $L(s, f \times f)$ with $\sigma _0 \leq \beta _f \leq 1$ and $|\gamma _f| \leq T$ . Then,

$$ \begin{align*} N(\sigma_0,T) = \begin{cases} T^{{5(1-\sigma_0)}/{(3-2\sigma_0)}+\epsilon} & \text{for } 1/2 < \sigma_0 < 23/32, \\ T^{{26(1-\sigma_0)}/{(11-4\sigma_0)}+\epsilon} & \text{for } 23/32 \leq \sigma_0 < 3/4, \\ T^{{2(1-\sigma_0)}/{\sigma_0}+\epsilon} & \text{for } 3/4 \leq \sigma_0 < 1. \end{cases} \end{align*} $$

Proof. Here, $L(s, f \times f)$ can be written as $L(s, f \times f) = \zeta (s) L(s, \mathrm {sym}^2f)$ . The result is easily obtained from the zero density of the Riemann zeta-function [13] and symmetric square L-functions [5].

Lemma 2.3. Let $2/3 < \sigma <1$ and $3 \leq Y \leq T/2$ . Then, for all $t \in [T, 2T]$ ,

$$ \begin{align*} \log L(s, f \times f) = \sum_{p^{\nu} \leq Y} \frac{b_f(p^{\nu})}{\nu p^{\nu (\sigma +it)}} + O_f(Y^{-{(\sigma-2/3)}/{2}}\log^3 T) \end{align*} $$

except for a set $\mathcal {D}(T)$ with $\text {meas}(\mathcal {D}(T)) \ll _f T^{{(10/3-5/2\sigma )}/{(7/3-\sigma )}+\epsilon }Y$ .

Proof. Take $\sigma _0 = \tfrac 12 (\tfrac 23 + \sigma )$ in Lemma 2.1. The result follows easily from Lemma 2.2.

The details of the next three results can be found in [12].

Lemma 2.4. Let $2/3 < \sigma <1$ , $128 \leq y \leq z$ and $\{b(p)\}$ be any real sequence with $|b(p)| \leq 4$ . For any positive integer $k \leq {\log T}/{20 \log z}$ ,

$$ \begin{align*} \frac{1}{T} \int_{T}^{2T} \bigg| \sum_{y \leq p \leq z} \frac{b(p)}{p^{\sigma +it}} \bigg|^{2k} \,dt \ll k! \bigg(\sum_{y \leq p \leq z} \frac{(b(p))^2}{p^{2 \sigma}}\bigg)^k + T^{-{1}/{3}}. \end{align*} $$

Moreover,

$$ \begin{align*} \mathbb{E}\bigg ( \bigg|\sum_{y \leq p \leq z} \frac{b(p)X(p)}{p^{\sigma}} \bigg|^{2k}\bigg) \ll k! \bigg(\sum_{y \leq p \leq z} \frac{(b(p))^2}{p^{2\sigma}}\bigg)^k. \end{align*} $$

Proposition 2.5. Let ${2}/{3} < \sigma < 1$ and $Y = (\log T)^A$ for a fixed $A \geq 1$ . There exist $a_1 = a_1(\sigma , A)>0$ and $a_1' = a_1'(\sigma , A)>0$ such that

$$ \begin{align*} \mathbb{P}_T\bigg(\bigg|\sum_{p^{\nu}\leq Y}\frac{b_f(p^{\nu})}{\nu p^{\nu(\sigma+it)}}\bigg| \geq \frac{(\log T)^{1-\sigma}}{\log \log T}\bigg) \ll \exp \bigg(-a_1 \frac{\log T}{\log \log T}\bigg) \end{align*} $$

and

$$ \begin{align*} \mathbb{P}\bigg(\bigg|\sum_{p^{\nu}\leq Y}\frac{b_f(p^{\nu})X(p)^{\nu}}{\nu p^{\nu\sigma}}\bigg| \geq \frac{(\log T)^{1-\sigma}}{\log \log T}\bigg) \ll \exp \bigg(-a_1' \frac{\log T}{\log \log T}\bigg). \end{align*} $$

Lemma 2.6. Let Y be a large positive real number and $|z|\leq Y^{\sigma - 1/2}$ . Then,

$$ \begin{align*} \mathbb{E} (|L(\sigma, f \times f, X)|^z) & = \mathbb{E} \bigg( \exp \bigg( z \,\mathrm{Re} \bigg( \sum_{p^{\nu}\leq Y}\frac{b_f(p^{\nu})X(p)^{\nu}}{\nu p^{\nu\sigma}} \bigg) \bigg) \bigg)\\ &\quad + O \bigg( \mathbb{E} (|L(\sigma, f \times f, X)|^{\mathrm{Re} (z))}) \frac{|z|}{Y^{\sigma - 1/2}} \bigg). \end{align*} $$

Moreover, if u, v are real numbers such that $|u|+|v| \leq Y^{\sigma - 1/2}$ , then

$$ \begin{align*} \Phi_{\sigma}^{rand} (u,v) & = \mathbb{E} \bigg( \exp \bigg( iu\,\mathrm{Re} \bigg( \sum_{p^{\nu}\leq Y}\frac{b_f(p^{\nu})X(p)^{\nu}}{\nu p^{\nu\sigma}}\bigg) + iv \,\mathrm{Im} \bigg(\sum_{p^{\nu}\leq Y}\frac{b_f(p^{\nu})X(p)^{\nu}}{\nu p^{\nu\sigma}} \bigg) \bigg) \bigg) \\ &\quad + O\bigg(\frac{|u|+|v|}{Y^{\sigma - 1/2}}\bigg). \end{align*} $$

Lemma 2.7. Let $2/3 < \sigma <1$ and $Y = (\log T)^A$ for a fixed $A \geq 1$ . For any positive integer $k \leq \log T/(20 A \log \log z)$ , there exist $a_2(\sigma )> 0$ and $a_2'(\sigma )> 0$ such that

$$ \begin{align*} \frac{1}{T} \int_{T}^{2T} \bigg| \sum_{p^{\nu}\leq Y}\frac{b_f(p^{\nu})}{\nu p^{\nu(\sigma+it)}} \bigg|^{2k}\,dt \ll \bigg(\frac{a_2(\sigma)k^{1-\sigma}}{(\log k)^{\sigma}}\bigg)^{2k} \end{align*} $$

and

$$ \begin{align*} \mathbb{E} \bigg(\bigg|\sum_{p^{\nu}\leq Y}\frac{b_f(p^{\nu})X(p)^{\nu}}{\nu p^{\nu\sigma}}\bigg|^{2k}\bigg) \ll \bigg(\frac{a_2'(\sigma)k^{1-\sigma}}{(\log k)^{\sigma}}\bigg)^{2k}. \end{align*} $$

Here the implied constants are absolute.

Proof. By using Lemma 2.4, the lemma follows easily from the method in [8, Lemma 3.3].

Lemma 2.8 [11, Lemma 6].

Let ${2}/{3} < \sigma < 1$ and $Y = (\log T)^A$ for a fixed $A \geq 1$ . For any positive integers u, v such that $u+v \leq \log T/(6A \log \log T)$ ,

$$ \begin{align*} \begin{aligned} \frac{1}{T} & \int_{T}^{2T} \bigg(\sum_{p^{\nu}\leq Y}\frac{b_f(p^{\nu})}{\nu p^{\nu(\sigma+it)}}\bigg)^u \bigg(\sum_{p^{\nu}\leq Y}\frac{b_f(p^{\nu})}{\nu p^{\nu(\sigma-it)}}\bigg)^v \,dt \\ & = \mathbb{E}\bigg(\bigg(\sum_{p^{\nu}\leq Y}\frac{b_f(p^{\nu})X(p)^{\nu}}{\nu p^{\nu\sigma}}\bigg)^u \bigg(\sum_{p^{\nu}\leq Y}\frac{\overline{b_f(p^{\nu})X(p)^{\nu}}}{\nu p^{\nu\sigma}}\bigg)^v\bigg) + O\bigg(\frac{Y^{u+v}}{\sqrt{T}}\bigg), \end{aligned} \end{align*} $$

with an absolute implied constant.

Proposition 2.9. Let $2/3 < \sigma <1$ and $Y = (\log T)^A$ for a fixed $A \geq 1$ . For all complex numbers $z_1$ , $z_2$ , there exist positive constants $a_3 = a_3(\sigma ,A)>0$ and $a_4 = a_4(\sigma ,A)>0$ with $|z_1|$ , $|z_2| \leq a_3(\log T)^{\sigma }$ such that

$$ \begin{align*} \begin{aligned} \frac{1}{T} & \int_{\mathcal{A}(T)} \exp \bigg(z_1 \sum_{p^{\nu}\leq Y}\frac{b_f(p^{\nu})}{\nu p^{\nu(\sigma+it)}} + z_2 \sum_{p^{\nu}\leq Y}\frac{b_f(p^{\nu})}{\nu p^{\nu(\sigma-it)}}\bigg)\,dt \\ & = \mathbb{E} \bigg(\exp \bigg(z_1 \sum_{p^{\nu}\leq Y}\frac{b_f(p^{\nu})X(p)^{\nu}}{\nu p^{\nu\sigma}} + z_2 \sum_{p^{\nu}\leq Y}\frac{\overline{b_f(p^{\nu})X(p)^{\nu}}}{\nu p^{\nu\sigma}} \bigg) \bigg) + O \bigg(\exp \bigg(-a_4 \frac{\log T}{ \log \log T}\bigg)\bigg), \end{aligned} \end{align*} $$

with an absolute implied constant. Here, $\mathcal {A}(T)$ is the set of those $t \in [T, 2T]$ such that

$$ \begin{align*} \bigg|\sum_{p^{\nu}\leq Y}\frac{b_f(p^{\nu})}{\nu p^{\nu(\sigma+it)}}\bigg| \leq \frac{(\log T)^{1-\sigma}}{\log \log T}. \end{align*} $$

Proof. The proof is the same as that of [8, Proposition 2.3] by using Lemma 2.7, Proposition 2.5 and Lemma 2.8.

Proposition 2.10. Let $2/3 < \sigma _0 < \sigma <1$ and $A \geq 1$ be fixed. There exists a constant $a_5 = a_5(\sigma , A)$ such that for $|u|$ , $|v| \leq a_5 (\log T)^{\sigma }$ ,

$$ \begin{align*} \Phi_{\sigma, T}(u,v) = \Phi_{\sigma}^{\mathrm{rand}} (u,v) + O\bigg(\frac{1}{(\log T)^A}\bigg), \end{align*} $$

with the implied constant depending on $\sigma _0$ only.

Proof. Follow the general idea of the proof of [8, Theorem 2.1]. Let $B=B(A)$ be a large enough constant. Let $ Y = (\log T)^{B/(\sigma - 2/3)}$ . By Lemma 2.3,

$$ \begin{align*} \log L(s, f \times f) = \sum_{p^{\nu}\leq Y} \frac{b_f(p^{\nu})}{\nu p^{\nu (\sigma+it)}} + O\bigg(\frac{1}{(\log T)^{B/2-3}}\bigg) \end{align*} $$

for all $t \in [T, 2T]$ , except for a set $\mathcal {D}(T)$ of measure $T^{1-d(\sigma )}$ for some constant $d(\sigma )>0$ . Define $\mathcal {C}(T) = \{t \in [T, 2T], t \notin \mathcal {D} (T)\}$ . Then,

$$ \begin{align*} & \Phi_{\sigma,T} (u,v) \\& \quad = \frac{1}{T} \int_{\mathcal{C}(T)} \exp ( iu\, \mathrm{Re} \log L(\sigma+it, f \times f) + iv\, \mathrm{Im} \log L(\sigma+it, f \times f) ) \,dt + O (T^{-d(\sigma)} )\\& \quad = \frac{1}{T} \int_{\mathcal{C}(T)} \exp \bigg(iu\, \mathrm{Re} \sum_{p^{\nu}\leq Y} \frac{b_f(p^{\nu})}{\nu p^{\nu (\sigma+it)}} + iv\, \mathrm{Im} \sum_{p^{\nu}\leq Y} \frac{b_f(p^{\nu})}{\nu p^{\nu (\sigma+it)}} \bigg) \,dt + O \bigg(\frac{1}{(\log T)^{B/2-4}}\bigg)\\& \quad = \frac{1}{T} \int_T^{2T} \exp \bigg(iu\, \mathrm{Re} \sum_{p^{\nu}\leq Y} \frac{b_f(p^{\nu})}{\nu p^{\nu (\sigma+it)}} + iv\, \mathrm{Im} \sum_{p^{\nu}\leq Y} \frac{b_f(p^{\nu})}{\nu p^{\nu (\sigma+it)}} \bigg) \,dt + O\bigg(\frac{1}{(\log T)^{B/2-4}}\bigg). \end{align*} $$

Let $\mathcal {A}(T)$ be defined as in Proposition 2.9 and take $z_1={i}(u-iv)/2$ and $z_2 = i(u+iv)/2$ in Proposition 2.9. From Proposition 2.5 and Lemma 2.6, the integral above is

$$ \begin{align*} & = \frac{1}{T} \int_{\mathcal{A}(T)} \exp \bigg( iu \,\mathrm{Re} \sum_{p^{\nu}\leq Y} \frac{b_f(p^{\nu})}{\nu p^{\nu (\sigma+it)}} + iv\, \mathrm{Im} \sum_{p^{\nu}\leq Y} \frac{b_f(p^{\nu})}{\nu p^{\nu (\sigma+it)}} \bigg) \,dt + O\bigg(\frac{1}{(\log T)^B}\bigg)\\& = \mathbb{E} \bigg(\exp \bigg(iu\, \mathrm{Re} \sum_{p^{\nu}\leq Y} \frac{b_f(p^{\nu})X(p)^{\nu}}{\nu p^{\nu (\sigma+it)}} + iv\, \mathrm{Im} \sum_{p^{\nu}\leq Y} \frac{b_f(p^{\nu})X(p)^{\nu}}{\nu p^{\nu (\sigma+it)}}\bigg) \bigg) + O\bigg(\frac{1}{(\log T)^B}\bigg)\\& = \Phi_{\sigma}^{\mathrm{rand}} (u,v) + O\bigg(\frac{1}{(\log T)^{B-1}}\bigg).\\[-39pt] \end{align*} $$

Lemma 2.11 [8, Lemma 7.2].

Let $\lambda>0$ be a real number. Let $\chi (y)=1$ if $y>1$ and ${0}$ otherwise. For any $c>0$ ,

$$ \begin{align*} \chi(y) \leq \frac{1}{2 \pi i} \int_{(c)} y^s \frac{e^{\lambda s}-1}{\lambda s} \,\frac{ds}{s} \quad\text{for } y>0. \end{align*} $$

We cite the following smooth approximation [8] for the indicator function.

Lemma 2.12. Let $\mathcal {R} = \{z=x+iy \in \mathbb {C}: m_1 < x < m_2, n_1 < y < n_2\}$ for real numbers $m_1, m_2, n_1, n_2$ . Let $K>0$ be a real number. For any $z=x+iy \in \mathbb {C}$ , we denote the indicator function of $\mathcal {R}$ by

$$ \begin{align*} \mathbf{1}_{\mathcal{R}} (z) = W_{K, \mathcal{R}} (z) \,{+} \,&O \bigg( \frac{\sin^2 (\pi K(x-m_1))}{(\pi K(x-m_1))^2} + \frac{\sin^2 (\pi K(x-m_2))}{(\pi K(x-m_2))^2} \\ & {+}\, \frac{\sin^2 (\pi K(y-n_1))}{(\pi K(y-n_1))^2} + \frac{\sin^2 (\pi K(y-n_2))}{(\pi K(y-n_2))^2} \bigg), \end{align*} $$

where

$$ \begin{align*} W_{K, \mathcal{R}} (z) = \frac{1}{2} \mathrm{Re} \int_0^K \int_0^K &G \bigg(\frac{u}{K}\bigg) G\bigg(\frac{v}{K}\bigg) (e^{2\pi i (ux-vy)}f_{m_1,m_2}(u) \overline{f_{n_1,n_2}(v)} \\ & {-}\, e^{2\pi i (ux+vy)}f_{m_1,m_2}(u) f_{n_1,n_2}(v)) \,\frac{du}{u} \frac{dv}{v}. \end{align*} $$

Here,

$$ \begin{align*} G(u) = \frac{2u}{\pi} +2(1-u) u \cot (\pi u) \quad\text{for } u\in [0,1], \end{align*} $$

and

$$ \begin{align*} f_{\alpha,\beta}(u) = \frac{e^{-2\pi i \alpha u} - e^{-2\pi i \beta u}}{2} \quad\text{for } \alpha,\beta \in \mathbb{R}. \end{align*} $$

Lemma 2.13. Let $2/3 < \sigma <1$ . Let u be a large positive real number. There exist constants $a_6 = a_6(f, \sigma )$ and $a_6' = a_6'(f, \sigma )$ such that

$$ \begin{align*} \mathbb{E} (\exp (iu\, \mathrm{Re} \log L(\sigma, f \times f, X))) \ll \exp \bigg(-a_6\frac{u^{2/\sigma -2}}{\log u} \bigg) \end{align*} $$

and

$$ \begin{align*} \mathbb{E} (\exp (iu\, \mathrm{Im} \log L(\sigma, f \times f, X))) \ll \exp \bigg(-a_6'\frac{u^{2/\sigma -2}}{\log u} \bigg). \end{align*} $$

Proof. Follow the general idea of the proof of [8, Lemma 6.3]. We denote the Bessel function of order 0 by $J_0(s)$ for all $s \in \mathbb {R}$ . Note that for any prime p, $\mathbb {E} (e^{is \mathrm {Re} X(p)}) = \mathbb {E} (e^{is \mathrm {Im} X(p)}) = J_0(s)$ . Since $\log (1+t)=t+O(t^2)$ for $ |t|<1$ ,

$$ \begin{align*} |\mathbb{E} & (\exp (iu\, \mathrm{Re} \log L(\sigma, f \times f, X)))|\\ & = \bigg|\mathbb{E} \bigg(\exp \bigg(iu\, \mathrm{Re} \log \bigg( \prod_p \bigg(1 - \frac{\alpha_f(p)^2 X(p)}{p^{\sigma}}\bigg)^{-1} \bigg(1 - \frac{\beta_f(p)^2 X(p)}{p^{\sigma}}\bigg)^{-1} \bigg(1 - \frac{X(p)}{p^{\sigma}}\bigg)^{-2} \bigg) \bigg)\bigg)\bigg|\\ & \leq \prod_{p>u^{2/\sigma}} \mathbb{E} \bigg(\exp \bigg(\frac{iu \lambda_f^2(p)}{p^{\sigma}} \mathrm{Re} X(p) + O \bigg(\frac{u}{p^{2\sigma}}\bigg) \bigg) \bigg) = \exp (O(u^{2/\sigma -3})) \prod_{p>u^{2/\sigma}} \bigg|J_0 \bigg(\frac{u \lambda_f^2(p)}{p^{\sigma}}\bigg)\bigg|. \end{align*} $$

For $|s|<1$ , we have $J_0(s) = 1 - ({s}/{2})^2 + O(s^4)$ . By using (1.2), for some constant $a_6 = a_6(f, \sigma ), c>0 $ , the product above is

$$ \begin{align*} = \exp \bigg\{ -\frac{u^2}{4} \sum_{p>u^{2/\sigma}} \bigg(\frac{\lambda_f^4(p)}{p^{2\sigma}} + O\bigg(\frac{u^2}{p^{4\sigma}}\bigg) \bigg) \bigg\} \leq \exp \bigg(-a_6\frac{u^{2/\sigma -2}}{\log u} \bigg). \end{align*} $$

The second inequality can be derived similarly.

3 Proof of the main theorem

Let $\mathcal {R}$ be a rectangle with sides parallel to the coordinate axes. Define $\Psi _T(\mathcal {R}) = \mathbb {P}(\log L(\sigma +it, f \times f) \in \mathcal {R}) \text { and } \Psi (\mathcal {R}) = \mathbb {P}(\log L(\sigma , f \times f, X) \in \mathcal {R})$ . Let

$$ \begin{align*} \widetilde{\mathcal{R}} = \mathcal{R} \cap [-(\log T)^3, (\log T)^3] \times [-(\log T)^3, (\log T)^3]. \end{align*} $$

According to Lemma 2.3 and Proposition 2.5, for some constant $a_7>0$ ,

$$ \begin{align*} \Psi_T(\mathcal{R}) = \Psi_T(\widetilde{\mathcal{R}}) + O \bigg(\exp \bigg(-a_7\frac{\log T}{\log \log T} \bigg) \bigg). \end{align*} $$

Similarly to [12], by using Lemmas 2.6 and 2.11, we can obtain the relationship between $\Psi (\mathcal {R})$ and $\Psi (\widetilde {\mathcal {R}})$ : for some constant $a_7'>0$ ,

$$ \begin{align*} \Psi(\mathcal{R}) = \Psi(\widetilde{\mathcal{R}}) + O \bigg(\exp \bigg(-a_7'\frac{\log T}{\log \log T} \bigg) \bigg). \end{align*} $$

Let $\mathcal {S}$ be the set of rectangles $\mathcal {R} \subset [-(\log T)^3, (\log T)^3]\times [-(\log T)^3, (\log T)^3]$ with sides parallel to the coordinate axes. Then,

$$ \begin{align*} D_{\sigma}(T) = \sup_{\mathcal{R} \subset \mathcal{S}} |\Psi_T(\mathcal{R}) - \Psi(\mathcal{R})| + O \bigg(\exp \bigg(-a_7\frac{\log T}{\log \log T} \bigg) \bigg). \end{align*} $$

In light of Lemma 2.12, choose $K = a_8(\log T)^{\sigma }$ , for some $a_8> 0$ , and $|m_1|, |m_2|, |n_1|, |n_2| \leq (\log T)^3$ . Then it follows that

(3.1) $$ \begin{align} \Psi_T(\mathcal{R})= \frac{1}{T} \int_{T}^{2T} W_{K,\mathcal{R}} (\log L(\sigma+it, f \times f))\,dt + E_1 \end{align} $$

and, in addition,

$$ \begin{align*} E_1 \ll I_T(K, m_1) + I_T(K, m_2) + J_T(K, n_1) + J_T(K, n_2), \end{align*} $$

where

(3.2) $$ \begin{align} I_T(K, m) = \frac{1}{T} \int_{T}^{2T} \frac{\sin^2(\pi K(\mathrm{Re} \log L(\sigma+it, f \times f) - m))}{(\pi K (\mathrm{Re} \log L(\sigma+it, f \times f) - m))^2} \,dt \end{align} $$

and

$$ \begin{align*} J_T(K, n) = \frac{1}{T} \int_{T}^{2T} \frac{\sin^2(\pi K(\mathrm{Im} \log L(\sigma+it, f \times f) - n))}{(\pi K (\mathrm{Im} \log L(\sigma+it, f \times f) - n))^2} \,dt. \end{align*} $$

First, we treat the main term of (3.1):

$$ \begin{align*} \begin{aligned} \frac{1}{T} & \int_{T}^{2T} W_{K,\mathcal{R}} (\log L(\sigma+it, f \times f))\,dt = \frac{1}{2} \mathrm{Re} \int_0^K \int_0^K G \bigg(\frac{u}{K}\bigg) G \bigg(\frac{v}{K}\bigg)\\ & \times (\Phi_{\sigma,T}(2\pi u, -2\pi v)f_{m_1, m_2}(u) \overline{f_{n_1,n_2}(v)} - \Phi_{\sigma, T}(2\pi u, 2\pi v) f_{m_1, m_2}(u) f_{n_1, n_2}(v) ) \,\frac{du}{u}\, \frac{dv}{v}, \end{aligned} \end{align*} $$

where $\Phi _{\sigma ,T}$ is defined by (2.1). Since $0 \leq G(u) \leq 2/\pi $ and $|f_{\alpha , \beta }(u)| \leq \pi u |\beta - \alpha |$ , by Proposition 2.10,

$$ \begin{align*} \frac{1}{T} \int_{T}^{2T} W_{K,\mathcal{R}} (\log L(\sigma+it, f \times f))\,dt = \mathbb{E} (W_{K,\mathcal{R}} (\log L(\sigma, f \times f, X))) + O \bigg(\frac{1}{(\log T)^2} \bigg). \end{align*} $$

Moreover,

$$ \begin{align*} \Psi(\mathcal{R})= \mathbb{E} ( W_{K,\mathcal{R}} (\log L(\sigma, f \times f, X))\,dt) + E_2. \end{align*} $$

Here,

$$ \begin{align*} E_2 \ll I_{\mathrm{rand}}(K, m_1) + I_{\mathrm{rand}}(K, m_2) + J_{\mathrm{rand}}(K, n_1) + J_{\mathrm{rand}}(K, n_2), \end{align*} $$

where

$$ \begin{align*} I_{\mathrm{rand}}(K, m) = \mathbb{E} \bigg( \frac{\sin^2(\pi K(\mathrm{Re} \log L(\sigma, f \times f, X) - m))}{(\pi K (\mathrm{Re} \log L(\sigma, f \times f, X) - m))^2} \bigg), \end{align*} $$

and

$$ \begin{align*} J_{\mathrm{rand}}(K, n) = \mathbb{E} \bigg( \frac{\sin^2(\pi K(\mathrm{Im} \log L(\sigma, f \times f, X) - n))}{(\pi K (\mathrm{Im} \log L(\sigma, f \times f, X) - n))^2} \bigg). \end{align*} $$

Hence,

(3.3) $$ \begin{align} \Psi_T(\mathcal{R}) = \Psi(\mathcal{R}) + E_3, \end{align} $$

where

$$ \begin{align*} E_3 = E_1 + E_2 + O \bigg(\frac{1}{(\log T)^2} \bigg). \end{align*} $$

Notice that

(3.4) $$ \begin{align} \frac{\sin^2 (\pi K x)}{(\pi K x)^2} = \frac{2(1-\cos (2 \pi K x))}{K^2(2\pi x)^2} = \frac{2}{K^2} \int_0^K (K-v) \cos (2\pi xv) \,dv. \end{align} $$

To bound $E_1$ , we use (3.4) to rewrite (3.2):

$$ \begin{align*} \begin{aligned} I_T (K, m) & = \mathrm{Re} \bigg( \frac{1}{T} \int_T^{2T} \frac{2}{K^2} \int_0^K (K-v) \exp (2\pi i v (\mathrm{Re} \log L (\sigma+it, f \times f) - m) ) \,dv\,dt \bigg)\\ & = \mathrm{Re} \frac{2}{K^2} \int_0^K (K-v) e^{-2\pi ivm} \Phi_{\sigma, T} (2\pi v, 0)\,dv. \end{aligned} \end{align*} $$

From Proposition 2.10,

$$ \begin{align*} I_T (K, m) = \mathrm{Re} \frac{2}{K^2} \int_0^K (K-v) e^{-2\pi ivm} \Phi_{\sigma}^{rand} (2\pi v, 0)\,dv + O \bigg(\frac{1}{(\log T)^9} \bigg), \end{align*} $$

uniformly for all $m \in \mathbb {R}$ . Lemma 2.13 implies that

$$ \begin{align*} I_T(K, m) \ll \frac{1}{K}. \end{align*} $$

The bound $J_T(K, n) \ll {1}/{K}$ can be obtained using the same method. Therefore,

(3.5) $$ \begin{align} E_1 \ll \frac{1}{K}. \end{align} $$

Then, using (2.2), (3.4) and Lemma 2.13,

$$ \begin{align*} \begin{aligned} I_{\mathrm{rand}} (K, m) & = \mathbb{E} \bigg(\frac{2}{K^2} \int_0^K (K-v) \cos (2\pi v [ \mathrm{Re} \log L(\sigma, f \times f, X) - m] ) \bigg) \,dv \\ & = \mathrm{Re} \frac{2}{K^2} \int_0^K (K-v) e^{-2\pi ivm} \Phi_{\sigma}^{\mathrm{rand}}(2\pi v, 0) \,dv \ll \frac{1}{K}, \end{aligned} \end{align*} $$

uniformly for all $m \in \mathbb {R}$ . Similarly, we can obtain $J_{\mathrm {rand}} (K, n) \ll {1}/{K}$ , uniformly for all $n \in \mathbb {R}$ . Thus,

(3.6) $$ \begin{align} E_2 \ll \frac{1}{K}. \end{align} $$

Combining the estimates with (3.3), (3.5) and (3.6),

$$ \begin{align*} D_{\sigma}(T) \ll \frac{1}{(\log T)^{\sigma}}, \end{align*} $$

which completes the proof.