Generalized Limit Theorem for Mellin Transform of the Riemann Zeta-Function

Abstract

In the paper, we prove a limit theorem in the sense of the weak convergence of probability measures for the modified Mellin transform $\mathcal{Z}\left(s\right)$, $s=\sigma +it$, with fixed $1/2<\sigma <1$, of the square ${\left|\zeta (1/2+it)\right|}^2$ of the Riemann zeta-function. We consider probability measures defined by means of $\mathcal{Z}(\sigma +i\phi (t\left)\right)$, where $\phi \left(t\right)$, $t⩾t_0>0$, is an increasing to $+\infty$ differentiable function with monotonically decreasing derivative ${\phi }^{\prime }\left(t\right)$ satisfying a certain normalizing estimate related to the mean square of the function $\mathcal{Z}(\sigma +i\phi (t\left)\right)$. This allows us to extend the distribution laws for $\mathcal{Z}\left(s\right)$.

1. Introduction

Let $s=\sigma +it$ be a complex variable. One of the most important objects of the classical analytic number theory is the Riemann zeta-function $\zeta \left(s\right)$, which is defined, for $\sigma >1$, by the Dirichlet series

$$\zeta \left(s\right)=\sum _{m=1}^{\infty }\frac{1}{m^s}.$$

Moreover, the function $\zeta \left(s\right)$ has analytic continuation to the region $\mathbb{C}\backslash \left\{1\right\}$, and the point $s=1$ is its simple pole with residue 1. The first value distribution results for $\zeta \left(s\right)$ with real s were obtained by Euler. Riemann was the first mathematician who began to study [1] $\zeta \left(s\right)$ with complex variables, proved the functional equation for $\zeta \left(s\right)$, obtained its analytic continuation, proposed a means of using $\zeta \left(s\right)$ for the investigation of the asymptotic prime number distribution law

$$\pi \left(x\right)=\sum _{p⩽x}1,x\to \infty ,$$

and stated some hypotheses on $\zeta \left(s\right)$. The most important hypothesis, now called the Riemann hypothesis, states that all zeros of $\zeta \left(s\right)$ in the region $\sigma ⩾0$ are located on the line $\sigma =1/2$. Riemann’s ideas concerning $\pi \left(x\right)$ were correct, and Hadamard [2] and de la Vallée Poussin [3], using them, independently proved that

$$\underset{x\to \infty }{lim}\pi \left(x\right){\left(\underset{2}{\overset{x}{\int }}\frac{\mathrm{d}u}{\log u}\right)}^{-1}=1.$$

However, the Riemann hypothesis remains open at present; it is among the seven Millennium Problems of mathematics [4]. In the theory of $\zeta \left(s\right)$, there are other important problems. One of them is connected to the asymptotics of moments

$$M_k(\sigma ,T)\stackrel{\mathrm{def}}{=}\underset{0}{\overset{T}{\int }}{\left|\zeta (\sigma +it)\right|}^{2k}\mathrm{d}t,k>0,\sigma ⩾\frac{1}{2},$$

as $T\to \infty$. For example, at the moment the asymptotics of $M_k(\sigma ,T)$, $\sigma =1/2$ is known only for $k=1$ and $k=2$; see [5]. For the investigation of $M_k(\sigma ,T)$, Motohashi proposed (see [6,7]) to use the modified Mellin transforms

$${\mathcal{Z}}_k\left(s\right)=\underset{1}{\overset{\infty }{\int }}{\left|\zeta \left(\frac{1}{2}+ix\right)\right|}^{2k}{x}^{-s}\mathrm{d}x,k\in \mathbb{N}.$$

Let $g\left(x\right)$ be a certain function, e.g., $g\left(x\right)x^{\sigma -1}\in L(0,\infty )$, and

$$G\left(s\right)=\underset{0}{\overset{\infty }{\int }}g\left(x\right)x^{s-1}\mathrm{d}x.$$

Then, using the Mellin inverse formula leads to the following equality (see [8]):

$$\underset{1}{\overset{\infty }{\int }}g\left(\frac{x}{T}\right){\left|\zeta \left(\frac{1}{2}+ix\right)\right|}^{2k}\mathrm{d}x=\frac{1}{2\pi i}\underset{c-i\infty }{\overset{c+i\infty }{\int }}G\left(s\right)T^s{\mathcal{Z}}_k\left(s\right)\mathrm{d}s$$

with a certain $c>1$. This shows that a suitable choice of the function $g\left(x\right)$ reduces investigations of $M_k(1/2,T)$ to those of properties of ${\mathcal{Z}}_k\left(s\right)$. The latter assertion inspired the creation of the analytic theory of the functions ${\mathcal{Z}}_k\left(s\right)$.

In this paper, we limit ourselves to the probabilistic value distribution of the function $\mathcal{Z}\left(s\right)\stackrel{\mathrm{def}}{=}{\mathcal{Z}}_1\left(s\right)$ only. Before this, we recall some known results of the function $\mathcal{Z}\left(s\right)$.

Let $\gamma =0.577\dots$ denote the Euler constant and $E\left(T\right)$ be defined by

$$\underset{0}{\overset{T}{\int }}{\left|\zeta (\left(\frac{1}{2}+it\right)\right|}^2\mathrm{d}t=T\log \frac{T}{2\pi }+(2\gamma -1)T+E\left(T\right).$$

Moreover, let

$$F\left(t\right)=\underset{1}{\overset{T}{\int }}E\left(t\right)\mathrm{d}t-\pi T\mathrm{and}{F}_1\left(T\right)=\underset{1}{\overset{T}{\int }}F\left(t\right)\mathrm{d}t.$$

The analytic behavior of the function $\mathcal{Z}\left(s\right)$ was described in [9] and forms the following theorem.

Theorem 1 

([9]). The function $\mathcal{Z}\left(s\right)$ is analytically continuable to the region $\sigma >-3/4$, except the point $s=1$, which is a double pole, and

$$\mathcal{Z}\left(s\right)=\frac{1}{{(s-1)}^2}+\frac{2\gamma -\log 2\pi }{s-1}-E\left(1\right)+\pi (s+1)+s(s+1)(s+2)\underset{1}{\overset{\infty }{\int }}{F}_1\left(x\right)x^{-s-3}\mathrm{d}x.$$

Moreover, the estimates

$$\mathcal{Z}(\sigma +it){\ll }_{\epsilon }{t}^{1-\sigma +\epsilon },0⩽\sigma ⩽1,t⩾t_0>0,$$

and

$$\underset{1}{\overset{T}{\int }}{\left|\mathcal{Z}(\sigma +it)\right|}^2\mathrm{d}t{\ll }_{\epsilon }\left\{\begin{array}{cc}{T}^{3-4\sigma +\epsilon }\hfill & \mathit{if}0⩽\sigma ⩽1/2,\hfill \\ T^{2-2\sigma +\epsilon }\hfill & \mathit{if}1/2⩽\sigma ⩽1,\hfill \end{array}\right.$$

(1)

are valid.

Here and in what follows, $\epsilon$ is an arbitrary fixed positive number that is not always the same, and the notation $x{\ll }_{\epsilon }y$, $x\in \mathbb{C}$, $y>0$, means that there is a constant $c=c\left(\epsilon \right)>0$ such that $\left|x\right|⩽cy$.

In [10], Bohr proposed to characterize the asymptotic behavior of the Riemann zeta-function by using a probabilistic approach. This idea is acceptable because the value distribution of $\zeta \left(s\right)$ is quite chaotic. Denote by $\mathrm{J}A$ the Jordan measure of the set $A\subset \mathbb{R}$. Then, Bohr, jointly with Jessen, roughly speaking, obtained in [11,12] that, for $\sigma >1/2$ and every rectangle $R\subset \mathbb{C}$ with edges parallel to the axes, there exists a limit

$$\underset{T\to \infty }{lim}\mathrm{J}\{t\in [0,T]:\zeta (\sigma +it)\in R\}.$$

In modern terminology, the Bohr–Jessen theorem is stated as a limit theorem on weakly convergent probability measures. Let $\mathcal{B}\left(\mathbb{X}\right)$ stand for the Borel $\sigma$-field of the space $\mathbb{X}$ (in general, topological), and let $P_n$, $n\in \mathbb{N}$, and P be probability measures defined on $(\mathbb{X},\mathcal{B}(\mathbb{X}\left)\right)$. By this definition, $P_n$ converges weakly to P as $n\to \infty$ ($P_n\underset{n\to \infty }{\overset{\mathrm{w}}{\to }}P$) if

$$\underset{n\to \infty }{lim}\underset{\mathbb{X}}{\int }g\mathrm{d}{P}_n=\underset{\mathbb{X}}{\int }g\mathrm{d}P$$

for every real continuous bounded function g on $\mathbb{X}$. Let $\mathbf{L}A$ stand for the Lebesgue measure of a measurable set $A\subset \mathbb{R}$. Then, the modern version of the Bohr–Jessen theorem is of the following form: for every fixed $\sigma >1/2$, there exists a probability measure $P_{\sigma }$ on $(\mathbb{C},\mathcal{B}(\mathbb{C}\left)\right)$ such that

$$\frac{1}{T}\mathbf{L}\{t\in [0,T]:\zeta (\sigma +it)\in A\},A\in \mathcal{B}\left(\mathbb{C}\right),$$

converges weakly to $P_{\sigma }$ as $T\to \infty$.

The first probabilistic limit theorems for the function $\mathcal{Z}\left(s\right)$ were discussed in [13]. For $A\in \mathcal{B}\left(\mathbb{C}\right)$, set

$$Q_{T,\sigma }\left(A\right)=\frac{1}{T}\mathbf{L}\{t\in [0,T]:\mathcal{Z}(\sigma +it)\in A\}.$$

Assuming that $\sigma >1/2$, it was obtained that there is a probability measure $Q_{\sigma }$ on $(\mathbb{C},\mathcal{B}(\mathbb{C}\left)\right)$ such that $Q_{T,\sigma }\underset{T\to \infty }{\overset{\mathrm{w}}{\to }}{Q}_{\sigma }$. On the other hand, for every $\kappa >0$, we have

$$\frac{1}{T}\mathbf{L}\{t\in [0,T]:|\mathcal{Z}(\sigma +it)|⩾\kappa \}⩽\frac{1}{\kappa T}\underset{0}{\overset{T}{\int }}\left|\mathcal{Z}(\sigma +it)\right|\mathrm{d}t⩽\frac{1}{\kappa }{\left(\frac{1}{T}\underset{0}{\overset{T}{\int }}{\left|\mathcal{Z}(\sigma +it)\right|}^2\mathrm{d}t\right)}^{1/2}.$$

This, together with Theorem 1, implies that, for $1/2<\sigma <1$,

$$\underset{T\to \infty }{lim}\frac{1}{T}\mathbf{L}\{t\in [0,T]:|\mathcal{Z}(\sigma +it)|⩾\kappa \}=0.$$

The latter equality remains valid also for $\sigma >1$. Thus, the limit measure $Q_{\sigma }$ is degenerated at the point $s=0$. In order to avoid this situation, we propose to consider $\mathcal{Z}(\sigma +i\phi (t\left)\right)$ with a certain function $\phi \left(t\right)$. Moreover, it is more convenient to deal with $t\in [T,2T]$ because, in this case, additional restrictions for $\phi \left(t\right)$ with $t=0$ are not needed.

Denote

$$I_{\sigma }\left(T\right)=\underset{1}{\overset{T}{\int }}{\left|\mathcal{Z}(\sigma +it)\right|}^2\mathrm{d}t.$$

We suppose that $\phi \left(t\right)$ is a positive increasing to $+\infty$ differentiable function with a monotonically decreasing derivative, such that

$$\frac{I_{\sigma -\epsilon }\left(\phi \left(T\right)\right)}{{\phi }^{\prime }\left(T\right)}\ll T,T\to \infty .$$

The class of such functions $\phi \left(t\right)$ is denoted by $W_{\sigma }$. Consider the weak convergence for

$$P_{T,\sigma }\left(A\right)=\frac{1}{T}\mathbf{L}\{t\in [T,2T]:\mathcal{Z}(\sigma +i\phi \left(\tau \right))\in A\},A\in \mathcal{B}\left(\mathbb{C}\right).$$

In this case, we have, by $\epsilon \to 0$, that

$$\frac{I_{\sigma }\left(\phi \left(T\right)\right)}{{\phi }^{\prime }\left(T\right)}\ll T,$$

and

$$\frac{1}{T}\underset{T}{\overset{2T}{\int }}{\left|\mathcal{Z}(\sigma +i\phi \left(t\right))\right|}^2\mathrm{d}t=\frac{1}{T}\underset{\phi \left(T\right)}{\overset{\phi \left(2T\right)}{\int }}\frac{1}{{\phi }^{\prime }\left(t\right)}{\left|\mathcal{Z}(\sigma +iu)\right|}^2\mathrm{d}u⩽\frac{1}{T{\phi }^{\prime }\left(2T\right)}{I}_{\sigma }\left(\phi \left(2T\right)\right)\ll 1$$

(2)

for $\phi \left(t\right)\in W_{\sigma }$. Thus, we cannot claim that the limit measure for $P_{T,\sigma }$ is degenerated at zero.

Now, we state a limit theorem for $P_{T,\sigma }$.

Theorem 2. 

Assume that $\sigma \in (1/2,1)$ is a given fixed number, and $\phi \left(t\right)\in W_{\sigma }$. Then, on $(\mathbb{C},\mathcal{B}(\mathbb{C}\left)\right)$, there exists a probability measure $P_{\sigma }$ such that $P_{T,\sigma }\underset{T\to \infty }{\overset{\mathrm{w}}{\to }}{P}_{\sigma }$.

In virtue of Theorem 1, we see that

$$I_{\sigma -\epsilon }\left(T\right)\ll T^{{\alpha }_{\sigma }}$$

with certain $0<{\alpha }_{\sigma }<1$. Take $\phi \left(t\right)={\left(\log t\right)}^{{\beta }_{\sigma }}$, $t⩾2$, ${\beta }_{\sigma }>0$. Then, ${\phi }^{\prime }\left(t\right)$ is decreasing, and

$$\frac{I_{\sigma -\epsilon }\left(\phi \left(T\right)\right)}{{\phi }^{\prime }\left(T\right)}\ll T{\left(\log T\right)}^{{\alpha }_{\sigma }{\beta }_{\sigma }+1-{\beta }_{\sigma }}\ll T$$

if we choose

$${\beta }_{\sigma }={(1-{\alpha }_{\sigma })}^{-1}.$$

This shows that ${\left(\log T\right)}^{{\beta }_{\sigma }}$ is an element of the class $W_{\sigma }$.

Theorem 2 shows that the asymptotic behavior of the function $\mathcal{Z}\left(s\right)$ on vertical lines is governed by a certain probabilistic law, and this confirms the chaos in its value distribution. Moreover, Theorem 2 is an example of the application of probability methods in analysis. Thus, it continues a tradition initiated in works [11,12] and developed by Selberg [14], Joyner [15], Bagchi [16], Korolev [17,18], Kowalski [19], Lamzouri, Lester and Radziwill [20,21], Steuding [22], and others; see also a survey paper [23]. We note that a generalization of Theorem 2 for the functional spaces can be applied for approximation problems of some classes of functions.

We divide the proof of Theorem 2 into several parts. First, we discuss weak convergence on a certain group. The second part is devoted to the case related to a integral. Further, we consider a measure defined by an absolutely convergent improper integral. In the last part, Theorem 2 is proven. For proofs of all assertions on weak convergence, the notions of relative compactness as well as of tightness and convergence in distribution are employed.

2. Fourier Transform Method

Let $b>1$ be a fixed finite number, and

$${\mathbb{I}}_b=\prod _{x\in [1,b]}\{s\in \mathbb{C}:|s|=1\}.$$

The Cartesian product ${\mathbb{I}}_b$ consists of all functions $\mathfrak{i}:[1,b]\to \{s\in \mathbb{C}:|s|=1\}$. On ${\mathbb{I}}_b$, the product topology and operation of pointwise multiplication can be defined. This reduces ${\mathbb{I}}_b$ to a compact topological group. We will give a limit lemma for probability measures on $({\mathbb{I}}_b,\mathcal{B}\left({\mathbb{I}}_b\right))$.

For $A\in \mathcal{B}\left({\mathbb{I}}_b\right)$, put

$$V_{T,b}\left(A\right)=\frac{1}{T}\mathbf{L}\left\{t\in [T,2T]:\left(x^{-i\phi \left(t\right)}:x\in [1,b]\right)\in A\right\}.$$

Lemma 1. 

Suppose that the function $\phi \left(t\right)$ has a monotonically decreasing derivative ${\phi }^{\prime }\left(t\right)$ such that

$${\left({\phi }^{\prime }\left(T\right)\right)}^{-1}=o\left(T\right),T\to \infty .$$

(3)

Then $V_{T,b}$ converges weakly to a certain probability measure $V_b$ as $T\to \infty$.

Proof. 

We use the Fourier transform approach. Denote the elements of ${\mathbb{I}}_b$ by $\mathfrak{i}=\{i_x:x\in [1,b]\}$. Then, the Fourier transform $f_{T,b}\left(\underline{k}\right)$, $\underline{k}=(k_x:k_x\in \mathbb{Z},x\in [1,b])$ of the measure $V_{T,b}$ is the integral

$$f_{T,b}\left(\underline{k}\right)=\underset{{\mathbb{I}}_b}{\int }\left(\prod _{x\in [1,b]}{i}_x^{k_x}\right)\mathrm{d}{V}_{T,b},$$

where only a finite number of integers $k_x$ are not zeros. Therefore, the definition of $V_{T,b}$ yields

$$f_{T,b}\left(\underline{k}\right)=\frac{1}{T}\underset{T}{\overset{2T}{\int }}\left(\prod _{x\in [1,b]}{x}^{-i{k}_x\phi \left(t\right)}\right)\mathrm{d}t=\frac{1}{T}\underset{T}{\overset{2T}{\int }}exp\left\{-i\phi \left(t\right)\sum _{x\in [1,b]}{k}_x\log x\right\}\mathrm{d}t.$$

(4)

For brevity, let $A_b\left(\underline{k}\right)={\sum }_{k\in [1,b]}{k}_x\log x$. Then, the second mean value theorem, (4), and (3) give

$$\begin{array}{cc}\hfill \mathrm{Re}{f}_{T,b}\left(\underline{k}\right)& =\frac{1}{T}\underset{T}{\overset{2T}{\int }}cos\left(\phi \left(t\right)A_b\left(\underline{k}\right)\right)\mathrm{d}t=\frac{1}{A_b\left(\underline{k}\right)T}\underset{T}{\overset{2T}{\int }}\frac{1}{{\phi }^{\prime }\left(t\right)}\mathrm{d}sin\left(\phi \left(t\right)A_b\left(\underline{k}\right)\right)\hfill \\ & \ll \frac{1}{|A_b\left(\underline{k}\right)|}\frac{1}{{\phi }^{\prime }\left(2T\right)T}=o\left(1\right),T\to \infty ,\hfill \end{array}$$

provided that $A_b\left(\underline{k}\right)\ne 0$. Clearly, the same estimate holds for $\mathrm{Im}{f}_{T,b}\left(\underline{k}\right)$. Hence, for $A_b\left(\underline{k}\right)\ne 0$,

$$\underset{T\to \infty }{lim}{f}_{T,b}\left(\underline{k}\right)=0.$$

(5)

Obviously,

$$f_{T,b}\left(\underline{k}\right)=1$$

if $A_b\left(\underline{k}\right)=0$. This and (5) show that

$$V_{T,b}\underset{T\to \infty }{\overset{\mathrm{w}}{\to }}{V}_b,$$

where $V_b$ is a probability measure on $({\mathbb{I}}_b,\mathcal{B}\left({\mathbb{I}}_b\right))$ defined by the Fourier transform

$$f_b\left(\underline{k}\right)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}{A}_b\left(\underline{k}\right)=0,\hfill \\ 0\hfill & \mathrm{if}{A}_b\left(\underline{k}\right)\ne 0.\hfill \end{array}\right.$$

□

Now, we will apply Lemma 1 for the measure defined by means of a certain finite sum.

Let $\theta >1/2$ be a fixed number, and, for $x,y\in [1,\infty )$,

$$u(x,y)=exp\left\{-{\left(\frac{x}{y}\right)}^{\theta }\right\}.$$

Moreover, we use the notation $\widehat{\zeta }\left(t\right)={\left|\zeta (1/2+it)\right|}^2$. Consider the nth integral sum

$$U_{n,b,y}(\sigma +i\phi \left(t\right))=\frac{b-1}{n}\sum _{l=1}^n\widehat{\zeta }\left(a_l\right)u(a_l,y)a_l^{-\sigma -i\phi \left(t\right)},n\in \mathbb{N},$$

where $a_l\in [x_{l-1},x_l]$ and $x_l=1+((b-1)/n)l$.

For $A\in \mathcal{B}\left(\mathbb{C}\right)$, set

$$P_{T,n,b,y}\left(A\right)=\frac{1}{T}\mathbf{L}\left\{t\in [T,2T]:U_{n,b,y}(\sigma +i\phi \left(t\right))\in A\right\}.$$

For simplicity, here and in the following, we omit the dependence on $\sigma$ of some objects. Before the statement of the limit lemma for $P_{T,n,b,y}$, we will present some lower estimates for the mean square $I_{\sigma }\left(T\right)$. For this, we will apply the following general lemma from [8]. Let $\mathcal{F}\left(s\right)$ be the modified Mellin transform of $f\left(x\right)$, i.e.,

$$\mathcal{F}\left(s\right)=\underset{1}{\overset{\infty }{\int }}f\left(x\right)x^{-s}\mathrm{d}x.$$

Lemma 2 

([8], Lemma 5). Let $f\left(x\right)\in C^{\infty }[2,\infty ]$ be a real-valued function such that

$1^{\circ }$

$$\underset{1}{\overset{X}{\int }}\left|f^{\left(k\right)}\left(x\right)\right|\mathrm{d}x{\ll }_{\epsilon ,k}{X}^{1+\epsilon },k\in {\mathbb{N}}_0;$$

$2^{\circ }$$\mathcal{F}\left(s\right)$ has analytic continuation to the half-plane $\sigma >1/2$, except for a pole of order l at the point $s=1$;

$3^{\circ }$ For $\sigma >1/2$, $\mathcal{F}\left(s\right)$ is of polynomial growth in $\left|t\right|$.

Then, for $1/2<\sigma <1$ and any fixed $\epsilon >0$,

$$\underset{T}{\overset{2T}{\int }}{f}^2\left(x\right)\mathrm{d}x{\ll }_{\epsilon }{\log }^{l-1}T\underset{T/2}{\overset{5T/2}{\int }}\left|f\left(x\right)\right|\mathrm{d}x+T^{2\sigma -1}\underset{0}{\overset{T^{1+\epsilon }}{\int }}{\left|\mathcal{F}(\sigma +it)\right|}^2\mathrm{d}t.$$

Lemma 3. 

For $1/2<\sigma <1$, and any $\epsilon >0$, the estimate

$$I_{\sigma }\left(T\right){\gg }_{\epsilon }{T}^{2-2\sigma -\epsilon }$$

holds.

Proof. 

As usual, denote by $Z\left(t\right)$, $t\in \mathbb{R}$, the Hardy function, i.e.,

$$Z\left(t\right)=\zeta \left(\frac{1}{2}+it\right){\chi }^{-1/2}\left(\frac{1}{2}+it\right),$$

where

$$\chi \left(s\right)=\frac{\zeta \left(s\right)}{\zeta (1-s)}.$$

It is well known that $Z\left(t\right)$ is a real-valued function satisfying $\left|Z\right(t\left)\right|=\left|\zeta \right(1/2+it\left)\right|$. Moreover, the estimate [8]

$$Z^{\left(k\right)}\left(t\right){\ll }_k{t}^{-1/4}{\left(\log T\right)}^{k+1}+\sum _{m⩽\sqrt{t/\left(2\pi \right)}}{m}^{-1/2}{\left(\log \frac{\sqrt{t/\left(2\pi \right)}}{m}\right)}^k$$

(6)

holds. Take $f\left(x\right)=Z^2\left(x\right)$. Then, we have

$$\mathcal{F}\left(s\right)=\underset{1}{\overset{\infty }{\int }}{Z}^2\left(x\right)x^{-s}\mathrm{d}x=\underset{1}{\overset{\infty }{\int }}{\left|\zeta \left(\frac{1}{2}+ix\right)\right|}^2{x}^{-s}\mathrm{d}x=\mathcal{Z}\left(s\right).$$

In view of Theorem 1 and (6), the function satisfies the hypotheses of Lemma 1 with $l=2$. Thus, for $1/2<\sigma <1$,

$$\underset{T}{\overset{2T}{\int }}{f}^2\left(t\right)\mathrm{d}t=\underset{T}{\overset{2T}{\int }}{\left|\zeta \left(\frac{1}{2}+it\right)\right|}^4\mathrm{d}t{\ll }_{\epsilon }\log T\underset{T/2}{\overset{5T/2}{\int }}{\left|\zeta \left(\frac{1}{2}+it\right)\right|}^2\mathrm{d}t+T^{2\sigma -1}\underset{0}{\overset{T^{1+\epsilon }}{\int }}{\left|\mathcal{Z}(\sigma +it)\right|}^2\mathrm{d}t.$$

Since [5]

$$\underset{0}{\overset{T}{\int }}{\left|\zeta \left(\frac{1}{2}+it\right)\right|}^4\mathrm{d}t=\frac{1}{2{\pi }^2}T{\log }^{4}T+O\left(T{\log }^{3}T\right)$$

and

$$\underset{0}{\overset{T}{\int }}{\left|\zeta \left(\frac{1}{2}+it\right)\right|}^2\mathrm{d}t\ll T\log T,$$

this implies

$$T{\log }^{4}T{\ll }_{\epsilon }{T}^{2\sigma -1}\underset{0}{\overset{T^{1+\epsilon }}{\int }}{\left|\mathcal{Z}(\sigma +it)\right|}^2\mathrm{d}t.$$

Consequently,

$$I_{\sigma }\left(T\right){\gg }_{\epsilon }{T}^{(2-2\sigma )/(1+\epsilon )}{\gg }_{\epsilon }{T}^{2-2\sigma -\epsilon }.$$

□

Lemma 4. 

Assume that $\sigma \in (1/2,1)$ is a given fixed number, and $\phi \left(t\right)\in W_{\sigma }$. Then, on $(\mathbb{C},\mathcal{B}(\mathbb{C}\left)\right)$, there exists a probability measure $P_{n,b,y}$ such that $P_{T,n,b,y}\underset{T\to \infty }{\overset{\mathrm{w}}{\to }}{P}_{n,b,y}$.

Proof. 

Lemma 3 implies that, for $\sigma \in (1/2,1)$, $I_{\sigma }\left(T\right)\to \infty$ as $T\to \infty$. Therefore, if $\phi \left(t\right)\in W_{\sigma }$, then

$$\frac{1}{{\phi }^{\prime }\left(T\right)}\ll T{I}_{\sigma }^{-1}\left(\phi \left(T\right)\right)=o\left(T\right)$$

as $T\to \infty$. Thus, the application of Lemma 1 is possible.

Consider the mapping $v_{n,b}:{\mathbb{I}}_b\to \mathbb{C}$ defined by

$$v_{n,b}\left(\mathfrak{i}\right)=\frac{b-1}{n}\sum _{l=1}^n\widehat{\zeta }\left(a_l\right)u(a_l,y)a_l^{-\sigma }{i}_{a_l}.$$

(7)

Since the latter sum is finite, and ${\mathbb{I}}_b$ is equipped with the product topology, the mapping $v_{n,b}$ is continuous. Moreover, in view of (7),

$$v_{n,b}\left(x^{-i\phi \left(t\right)}:x\in [1,b]\right)=\frac{b-1}{n}\sum _{l=1}^n\widehat{\zeta }\left(a_l\right)u(a_l,y)a_l^{-\sigma -i\phi \left(t\right)}=U_{n,b,y}(\sigma +i\phi \left(t\right)).$$

Hence, for $A\in \mathcal{B}\left(\mathbb{C}\right)$,

$$\begin{array}{cc}\hfill P_{T,n,b,y}\left(A\right)& =\frac{1}{T}\mathbf{L}\left\{t\in [T,2T]:v_{n,b}\left(x^{-i\phi \left(t\right)}:x\in [1,b]\right)\in A\right\}\hfill \\ & =\frac{1}{T}\mathbf{L}\left\{t\in [T,2T]:\left(x^{-i\phi \left(t\right)}:x\in [1,b]\right)\in v_{n,b}^{-1}A\right\}=V_{T,b}\left(v_{n,b}^{-1}A\right),\hfill \end{array}$$

(8)

where $V_{T,b}$ is from Lemma 1. The continuity of the mapping $v_{n,b}$ implies its $(\mathcal{B}\left({\mathbb{I}}_b\right),\mathcal{B}\left(\mathbb{C}\right))$-measurability. Therefore, the mapping $v_{n,b}$ and any probability measure P on $({\mathbb{I}}_b,\mathcal{B}\left({\mathbb{I}}_b\right))$ define the unique probability measure $P{v}_{n,b}^{-1}$ on $(\mathbb{C},\mathcal{B}(\mathbb{C}\left)\right)$ given by

$$P{v}_{n,b}^{-1}\left(A\right)=P\left(v_{n,b}^{-1}A\right),A\in \mathcal{B}\left(\mathbb{C}\right).$$

See Section 2 of [24]. Thus, by (8), we have $P_{T,n,b,y}=V_{T,b}{v}_{n,b}^{-1}$. Therefore, Lemma 1, the continuity of $v_{n,b}$, and the principle of the preservation of week convergence under continuity mappings (Theorem 5.1 of [24]) show that

$$P_{T,n,b,y}\underset{T\to \infty }{\overset{\mathrm{w}}{\to }}{P}_{n,b,y},$$

where $P_{n,b,y}=V_b{v}_{n,b}^{-1}$, and $V_b$ is the limit measure in Lemma 1. □

3. Limit Lemma for Integral

Denote

$${\mathcal{Z}}_{b,y}(\sigma +i\phi \left(t\right))=\underset{1}{\overset{b}{\int }}\widehat{\zeta }\left(x\right)u(x,y)x^{-\sigma -i\phi \left(t\right)}\mathrm{d}x,$$

and, for $A\in \mathcal{B}\left(\mathbb{C}\right)$, set

$$P_{T,b,y}\left(A\right)=\frac{1}{T}\mathbf{L}\left\{t\in [T,2T]:{\mathcal{Z}}_{b,y}(\sigma +i\phi \left(t\right))\in A\right\}.$$

In this section, we will prove the weak convergence for $P_{T,b,y}$ as $T\to \infty$. Before this, we recall some known probabilistic results. Let $\left\{Q\right\}$ be a certain family of probability measures on $(\mathbb{X},\mathcal{B}(\mathbb{X}\left)\right)$. The family $\left\{Q\right\}$ is called tight if, for every $\delta >0$, there is a compact set $K\subset \mathbb{X}$ such that

$$Q\left(K\right)>1-\delta$$

for all $Q\in \left\{Q\right\}$. The family $\left\{Q\right\}$ is said to be relatively compact if every sequence contains a subsequence weakly convergent to a certain probability measure on $(\mathbb{X},\mathcal{B}(\mathbb{X}\left)\right)$. The Prokhorov theorem connects two above notions, and, for convenience, we state it as the following lemma.

Lemma 5. 

If a family of probability measures is tight, then it is relatively compact.

The proof of the lemma is given in [24], Theorem 5.1.

Moreover, we recall one useful property on convergence in distribution. Let ${\xi }_n$ and $\xi$ be $\mathbb{X}$-valued random elements defined on the probability space $(\Omega ,\mathcal{F},\mu )$ with distributions $P_n$ and P, respectively. Then, ${\xi }_n$ converges in distribution to $\xi$ as $n\to \infty$($\underset{n\to \infty }{\overset{\mathcal{D}}{\to }}$) if

$$P_n\underset{n\to \infty }{\overset{\mathrm{w}}{\to }}P.$$

Now, we state a lemma on convergence in distribution.

Lemma 6. 

Assume that the metric space $(\mathbb{X},d)$ is separable, and ${\xi }_{nk}$, ${\xi }_n$ are $\mathbb{X}$-valued random elements defined on the same probability space $(\Omega ,\mathcal{F},\mu )$. Let

$${\xi }_{nk}\underset{n\to \infty }{\overset{\mathcal{D}}{\to }}{\xi }_k$$

and

$${\xi }_k\underset{k\to \infty }{\overset{\mathcal{D}}{\to }}\xi .$$

If, for every $\delta >0$,

$$\underset{k\to \infty }{lim}\underset{n\to \infty }{lim\; sup}\mu \{d({\xi }_{nk},{\eta }_k)⩾\delta \}=0,$$

then

$${\eta }_n\underset{n\to \infty }{\overset{\mathcal{D}}{\to }}\xi .$$

The lemma is proven in [24], Theorem 3.2.

Lemma 7. 

Assume that $\sigma \in (1/2,1)$ is a given fixed number, and $\phi \left(t\right)\in W_{\sigma }$. Then, on $(\mathbb{C},\mathcal{B}(\mathbb{C}\left)\right)$, there exists a probability measure $P_{b,y}$ such that $P_{T,b,y}\underset{T\to \infty }{\overset{\mathrm{w}}{\to }}{P}_{b,y}$.

Proof. 

First, we will show that ${\mathcal{Z}}_{b,y}(\sigma +i\phi \left(t\right))$ is close in a certain sense to $U_{n,b,y}(\sigma +i\phi \left(t\right))$.

Let

$$J_{T,n}\stackrel{\mathrm{def}}{=}\frac{1}{T}\underset{T}{\overset{2T}{\int }}\left|{\mathcal{Z}}_{b,y}(\sigma +i\phi \left(t\right))-U_{n,b,y}(\sigma +\phi \left(t\right))\right|\mathrm{d}t.$$

Clearly,

$$J_{T,n}^2⩽\frac{1}{T}\underset{T}{\overset{2T}{\int }}{\left|{\mathcal{Z}}_{b,y}(\sigma +i\phi \left(t\right))-U_{n,b,y}(\sigma +\phi \left(t\right))\right|}^2\mathrm{d}t.$$

(9)

We have

$$\begin{array}{cc}\hfill \underset{T}{\overset{2T}{\int }}& {\left|{\mathcal{Z}}_{b,y}(\sigma +i\phi \left(t\right))\right|}^2\mathrm{d}t\hfill \\ & =\underset{T}{\overset{2T}{\int }}\left(\underset{1}{\overset{b}{\int }}\widehat{\zeta }\left(x\right)u(x,y)x^{-\sigma -i\phi \left(t\right)}\mathrm{d}x\right)\left(\underset{1}{\overset{b}{\int }}\widehat{\zeta }\left(x\right)u(x,y)x^{-\sigma +i\phi \left(t\right)}\mathrm{d}x\right)\mathrm{d}t\hfill \\ & =T\underset{x_1=x_2}{\underset{1}{\overset{b}{\int }}\underset{1}{\overset{b}{\int }}}\widehat{\zeta }\left(x_1\right)\widehat{\zeta }\left(x_2\right)u(x_1,y)u(x_2,y)x_1^{-\sigma }{x}_2^{-\sigma }\mathrm{d}{x}_1\mathrm{d}{x}_2\hfill \\ & +\underset{x_1\ne x_2}{\underset{1}{\overset{b}{\int }}\underset{1}{\overset{b}{\int }}}\widehat{\zeta }\left(x_1\right)\widehat{\zeta }\left(x_2\right)u(x_1,y)u(x_2,y)x_1^{-\sigma }{x}_2^{-\sigma }\left(\underset{T}{\overset{2T}{\int }}{\left(\frac{x_1}{x_2}\right)}^{i\phi \left(t\right)}\mathrm{d}t\right)\mathrm{d}{x}_1\mathrm{d}{x}_2.\hfill \end{array}$$

(10)

Since

$$\mathrm{Re}\underset{T}{\overset{2T}{\int }}{\left(\frac{x_1}{x_2}\right)}^{i\phi \left(t\right)}\mathrm{d}t={\left(\log \left(\frac{x_1}{x_2}\right)\right)}^{-1}\underset{T}{\overset{2T}{\int }}\frac{1}{{\phi }^{\prime }\left(t\right)}\mathrm{d}sin\left(\phi \left(t\right)\log \left(\frac{x_1}{x_2}\right)\right)\ll {\left|\log \frac{x_1}{x_2}\right|}^{-1}\frac{1}{{\phi }^{\prime }\left(2T\right)},$$

and the same bound is true for the imaginary part of the latter integral, we obtain by (10) that

$$\underset{T}{\overset{2T}{\int }}{\left|{\mathcal{Z}}_{b,y}(\sigma +i\phi \left(t\right))\right|}^2\mathrm{d}t=o\left(T\right),T\to \infty .$$

(11)

Reasoning similarly, we find

$$\begin{array}{cc}\hfill \underset{T}{\overset{2T}{\int }}& {\left|U_{n,b,y}(\sigma +i\phi \left(t\right))\right|}^2\mathrm{d}t=T{\left(\frac{b-1}{n}\right)}^2\sum _{l=1}^n{\widehat{\zeta }}^2\left(a_l\right)u^2(a_l,y)a_l^{-2}\hfill \\ & +O\left({\left(\frac{b-1}{n}\right)}^2\underset{l_1\ne l_2}{\sum _{l_1=1}^n\sum _{l_2=1}^n}\widehat{\zeta }\left(a_{l_1}\right)\widehat{\zeta }\left(a_{l_2}\right)u(a_{l_1},y)u(a_{l_2},y)a_{l_1}^{-\sigma }{a}_{l_2}^{-\sigma }{\left|\log \frac{a_{l_1}}{a_{l_2}}\right|}^{-1}\right).\hfill \end{array}$$

(12)

Thus,

$$\underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\frac{1}{T}\underset{T}{\overset{2T}{\int }}{\left|U_{n,b,y}(\sigma +i\phi \left(t\right))\right|}^2\mathrm{d}t=0.$$

(13)

By (9),

$$\begin{array}{cc}\hfill J_{T,n}^2\ll & \frac{1}{T}\left(\underset{T}{\overset{2T}{\int }}{\left|{\mathcal{Z}}_{b,y}(\sigma +i\phi \left(t\right))\right|}^2\mathrm{d}t+{\left(\underset{T}{\overset{2T}{\int }}{\left|{\mathcal{Z}}_{b,y}(\sigma +i\phi \left(t\right))\right|}^2\mathrm{d}t\underset{T}{\overset{2T}{\int }}{\left|U_{n,b,y}(\sigma +i\phi \left(t\right))\right|}^2\mathrm{d}t\right)}^{1/2}\right.\hfill \\ & +\left.\underset{T}{\overset{2T}{\int }}{\left|U_{n,b,y}(\sigma +i\phi \left(t\right))\right|}^2\mathrm{d}t\right).\hfill \end{array}$$

Therefore, (11) and (13) yield

$$\underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}{J}_{T,n}=0.$$

(14)

Now, we will deal with the sequence $\{P_{n,b,y}:n\in \mathbb{N}\}$. By (12), we have

$$\begin{array}{cc}\hfill \underset{s\in \mathbb{N}}{sup}& \underset{T\to \infty }{lim\; sup}\frac{1}{T}\underset{T}{\overset{2T}{\int }}\left|U_{n,b,y}(\sigma +i\phi \left(t\right))\right|\mathrm{d}t\hfill \\ & \ll \underset{s\in \mathbb{N}}{sup}\underset{T\to \infty }{lim\; sup}{\left(\frac{1}{T}\underset{T}{\overset{2T}{\int }}{\left|U_{n,b,y}(\sigma +i\phi \left(t\right))\right|}^2\mathrm{d}t\right)}^{1/2}\hfill \\ & \ll \underset{n\in \mathbb{N}}{sup}\frac{b-1}{n}{\left(\sum _{l=1}^n{\widehat{\zeta }}^2\left(a_l\right)u^2(a_l,y)a_l^{-2\sigma }\right)}^{1/2}⩽C_{b,y,\sigma }<\infty \hfill \end{array}$$

(15)

because

$$\underset{n\to \infty }{lim}\frac{b-1}{n}\sum _{l=1}^n{\widehat{\zeta }}^2\left(a_l\right)u^2(a_l,y)a_l^{-2\sigma }=\underset{1}{\overset{b}{\int }}{\widehat{\zeta }}^2\left(x\right)u^2(x,y)x^{-2\sigma }\mathrm{d}x.$$

Take a random variable ${\theta }_T$ given on the probability space $(\Omega ,\mathcal{F},\mu )$ that is uniformly distributed on $[T,2T]$. Consider the complex-valued random variables

$$x_{T,n.,b,y}=x_{T,n,b,y}\left(\sigma \right)=U_{n,b,y}(\sigma +i\phi \left({\theta }_T\right)),$$

and $x_{n,b,y}\left(\sigma \right)$ with the distribution $P_{n,b,y,\sigma }$. Then, rewrite the assertion of Lemma 4 in the form

$$x_{T,n,b,y}\underset{T\to \infty }{\overset{\mathcal{D}}{\to }}{x}_{n,b,y}.$$

(16)

Fix $\delta >0$. Then, in view of (15) and (16),

$$\begin{array}{cc}\hfill \mu \left\{\left|x_{n,b,y}\left(\sigma \right)\right|>{\delta }^{-1}{C}_{b,y,\sigma }\right\}& ⩽\underset{n\in \mathbb{N}}{sup}\underset{T\to \infty }{lim\; sup}\mu \left\{\left|x_{T,n,b,y}\left(\sigma \right)\right|>{\delta }^{-1}{C}_{b,y,\sigma }\right\}\hfill \\ & ⩽\underset{n\in \mathbb{N}}{sup}\underset{T\to \infty }{lim\; sup}\frac{\delta }{C_{b,y,\sigma }}\underset{T}{\overset{2T}{\int }}\left|U_{n,b,y}(\sigma +i\phi \left(t\right))\right|\mathrm{d}t⩽\delta .\hfill \end{array}$$

(17)

The set $K=\{s\in \mathbb{C}:|s|⩽{\delta }^{-1}{C}_{b,y,\sigma }\}$ is compact in $\mathbb{C}$. Moreover, by (17),

$$\mu \left\{x_{n,b,y}\in K\right\}=1-\mu \left\{x_{n,b,y}\notin K\right\}>1-\delta$$

for all $n\in \mathbb{N}$. This and the definition of $x_{n,b,y}$ show that, for all $n\in \mathbb{N}$,

$$P_{n,b,y,\sigma }\left(K\right)>1-\delta .$$

This means that the sequence $\{P_{n,b,y,\sigma }:n\in \mathbb{N}\}$ is tight. Therefore, by Lemma 5, it is relatively compact. Hence, there exists a subsequence $\left\{P_{n_l,b,y,\sigma }\right\}\subset \left\{P_{n,b,y,\sigma }\right\}$ and a probability measure $P_{b,y,\sigma }$ on $(\mathbb{C},\mathcal{B}(\mathbb{C}\left)\right)$ such that $P_{n_l,b,y,\sigma }\underset{l\to \infty }{\overset{\mathrm{w}}{\to }}{P}_{b,y,\sigma }$. In other words,

$$x_{n_l,b,y}\underset{l\to \infty }{\overset{\mathcal{D}}{\to }}{P}_{b,y,\sigma }.$$

This, (16), and (14) show that all hypotheses of Lemma 6 for $x_{T,n,b,y}$, $x_{n_l,b,y}$ and

$$y_{T,b,y}=y_{T,b,y}\left(\sigma \right)={\mathcal{Z}}_{b,y}(\sigma +i\phi \left({\theta }_T\right))$$

are satisfied. Thus, we have

$$y_{T,b,y}\underset{T\to \infty }{\overset{\mathcal{D}}{\to }}{P}_{b,y,\sigma },$$

which proves the lemma. □

4. Case of Improper Integral

This section is devoted to a limit lemma for the integral

$${\mathcal{Z}}_y(\sigma +i\phi \left(t\right))=\underset{1}{\overset{\infty }{\int }}\widehat{\zeta }\left(x\right)u(x,y)x^{-\sigma -i\phi \left(t\right)}\mathrm{d}x.$$

It is well known that $\zeta (1/2+ix)\ll {(1+|x\left|\right)}^{1/6}$. Therefore, the integral for $\mathcal{Z}(\sigma +i\phi (t\left)\right)$ converges absolutely for $\sigma >\widehat{\sigma }$ with every finite $\widehat{\sigma }$.

For $A\in \mathcal{B}\left(\mathbb{C}\right)$, let

$$P_{T,y,\sigma }\left(A\right)=\frac{1}{T}\mathbf{L}\left\{t\in [T,2T]:{\mathcal{Z}}_y(\sigma +i\phi \left(t\right))\in A\right\}.$$

Lemma 8. 

Assume that $\sigma \in (1/2,1)$ is a given fixed number, and $\phi \left(t\right)\in W_{\sigma }$. Then, there is a probability measure $P_{y,\sigma }$ on $\left(\mathbb{C}\right(\mathcal{B}\left(\mathbb{C}\right))$ such that $P_{T,y,\sigma }\underset{T\to \infty }{\overset{\mathrm{w}}{\to }}{P}_{y,\sigma }$.

Proof. 

We use a similar method as in the proof of Lemma 7. We begin with a mean value

$$J_{T,y}\stackrel{\mathrm{def}}{=}\frac{1}{T}\underset{0}{\overset{T}{\int }}\left|{\mathcal{Z}}_y(\sigma +i\phi \left(t\right))-{\mathcal{Z}}_{b,y}(\sigma +i\phi \left(t\right))\right|\mathrm{d}t.$$

Clearly, the absolute convergence of the integral for ${\mathcal{Z}}_y(\sigma +i\phi \left(t\right))$ shows that, for every fixed $y>0$,

$$\begin{array}{cc}\hfill {\mathcal{Z}}_y(\sigma +i\phi \left(t\right))-{\mathcal{Z}}_{b,y}(\sigma +i\phi \left(t\right))& =\underset{b}{\overset{\infty }{\int }}\widehat{\zeta }\left(x\right)u(x,y)x^{-\sigma -i\phi \left(t\right)}\mathrm{d}x\hfill \\ & \ll \underset{b}{\overset{\infty }{\int }}\widehat{\zeta }\left(x\right)u(x,y)x^{-\sigma }\mathrm{d}x=o_y\left(1\right)\hfill \end{array}$$

as $b\to \infty$. Hence, we obtain

$$\underset{b\to \infty }{lim}\underset{T\to \infty }{lim\; sup}{J}_{T,y}=0.$$

(18)

Let $y_{b,y}\left(\sigma \right)$ be the complex-valued random variable with distribution $P_{b,y,\sigma }$, and, in the notation of Lemma 7,

$$y_{T,b,y}=y_{T,b,y}\left(\sigma \right)={\mathcal{Z}}_{b,y}(\sigma +i\phi \left({\theta }_T\right)).$$

Then, by Lemma 7,

$$y_{T,b,y}\underset{T\to \infty }{\overset{\mathcal{D}}{\to }}{y}_{b,y}.$$

(19)

Moreover, in virtue of (11),

$$\underset{b⩾1}{sup}\underset{T\to \infty }{lim\; sup}\frac{1}{T}\underset{T}{\overset{2T}{\int }}\left|{\mathcal{Z}}_{b,y}(\sigma +i\phi \left(t\right))\right|\mathrm{d}t⩽C_{y,\sigma }<\infty .$$

This together with (19) gives, for $\delta >0$,

$$\begin{array}{cc}\hfill \mu \left\{\left|y_{b,y}\right|>{\delta }^{-1}{C}_{y,\sigma }\right\}& ⩽\underset{b⩾1}{sup}\underset{T\to \infty }{lim\; sup}\mu \left\{\left|y_{b,y}\right|>{\delta }^{-1}{C}_{y,\sigma }\right\}\hfill \\ & ⩽\underset{b⩾1}{sup}\underset{T\to \infty }{lim\; sup}\frac{\delta }{C_{y,\sigma }}\underset{T}{\overset{2T}{\int }}\left|{\mathcal{Z}}_{b,y}(\sigma +i\phi \left(t\right))\right|\mathrm{d}t⩽\delta .\hfill \end{array}$$

Taking a set $K=\{s\in \mathbb{C}:|s|⩽{\delta }^{-1}{C}_{y,\sigma }\}$, from this, we deduce that

$$\mu \left\{y_{b,y}\in K\right\}>1-\delta .$$

This implies that the family $\{P_{b,y,\sigma }:b⩾1\}$ is tight. Therefore, in view of Lemma 5, it is relatively compact. Thus, there is a sequence $\left\{P_{b_l,y,\sigma }\right\}$ and a probability measure $P_{y,\sigma }$ on $(\mathbb{C},\mathcal{B}(\mathbb{C}\left)\right)$ such that

$$y_{b_l,y,\sigma }\underset{l\to \infty }{\overset{\mathcal{D}}{\to }}{P}_{y,\sigma }.$$

This, (19), (18), and the application of Lemma 6 complete the proof of the lemma. □

5. Proof of Theorem 2

We recall that

$$u(x,y)=exp\left\{-{\left(\frac{x}{y}\right)}^{\theta }\right\},x,y\in [1,\infty ),$$

with a fixed $\theta >1/2$. For brevity, set

$$f(s,y)=\frac{1}{\theta }\Gamma \left(\frac{s}{\theta }\right)y^s,$$

where $\Gamma \left(s\right)$ is the Euler gamma-function. For the approximation of $\mathcal{Z}(\sigma +i\phi (t\left)\right)$ by ${\mathcal{Z}}_y(\sigma +\phi \left(t\right))$, we use the representation

$${\mathcal{Z}}_y\left(s\right)=\frac{1}{2\pi i}\underset{\theta -i\infty }{\overset{\theta +i\infty }{\int }}\mathcal{Z}(s+z)f(z,y)\mathrm{d}z,\frac{1}{2}<\sigma <1,$$

(20)

obtained in [25], Lemma 9.

Lemma 9. 

Under the hypotheses of Theorem 2,

$$\underset{y\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\frac{1}{T}\underset{T}{\overset{2T}{\int }}\left|\mathcal{Z}(\sigma +i\phi \left(t\right))-{\mathcal{Z}}_y(\sigma +i\phi \left(t\right))\right|\mathrm{d}t=0.$$

Proof. 

Let ${\theta }_1=-\epsilon$ and $\theta =1/2+\epsilon$. The integrand in (20) has a double pole $z=1-s$ and a simple pole $z=0$ lying in ${\theta }_1<\mathrm{Re}z<\theta$. Therefore, by the residue theorem and (20), we have

$${\mathcal{Z}}_y\left(s\right)-\mathcal{Z}\left(s\right)=\frac{1}{2\pi i}\underset{{\theta }_1-i\infty }{\overset{{\theta }_1+i\infty }{\int }}\mathcal{Z}(s+z)f(z,y)\mathrm{d}z+r_y\left(s\right),$$

where

$$r_y\left(s\right)=\underset{z=1-s}{\mathrm{Res}}\mathcal{Z}\left(s\right)f(s,y).$$

(21)

From this, we obtain

$$\begin{array}{cc}\hfill {\mathcal{Z}}_y& (\sigma +i\phi (t\left)\right)-\mathcal{Z}(\sigma +i\phi (t\left)\right)\hfill \\ & =\frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}\mathcal{Z}\left(\sigma -\epsilon +i\phi \left(t\right)+i\tau \right)f\left(-\epsilon +i\tau ,y\right)\mathrm{d}\tau +r_y(\sigma +i\phi \left(t\right))\hfill \\ & \ll \underset{-\infty }{\overset{\infty }{\int }}\left|\mathcal{Z}\left(\sigma -\epsilon +i\phi \left(t\right)+i\tau \right)\right|\left|f\left(-\epsilon +i\tau ,y\right)\right|\mathrm{d}\tau +\left|r_y(\sigma +i\phi \left(t\right))\right|.\hfill \end{array}$$

Thus,

$$\frac{1}{T}\underset{T}{\overset{2T}{\int }}\left|\mathcal{Z}(\sigma +i\phi \left(t\right))-{\mathcal{Z}}_y(\sigma +i\phi \left(t\right))\right|\mathrm{d}t\ll I_{T,y},$$

where

$$\begin{array}{cc}\hfill I_{T,y}\stackrel{\mathrm{def}}{=}& \underset{-\infty }{\overset{\infty }{\int }}\left(\frac{1}{T}\underset{T}{\overset{2T}{\int }}\left|\mathcal{Z}\left(\sigma -\epsilon +i\phi \left(t\right)+i\tau \right)\right|\mathrm{d}t\right)\left|f\left(-\epsilon +i\tau ,y\right)\right|\mathrm{d}\tau \hfill \\ & +\frac{1}{T}\underset{T}{\overset{2T}{\int }}\left|r_y(\sigma +i\phi \left(t\right))\right|\mathrm{d}t=I_{T,y}^{\left(1\right)}+I_{T,y}^{\left(2\right)}.\hfill \end{array}$$

(22)

To estimate $I_{T,y}^{\left(1\right)}$, we observe that

$$\begin{array}{cc}\hfill \frac{1}{T}\underset{T}{\overset{2T}{\int }}\left|\mathcal{Z}\left(\sigma -\epsilon +i\phi \left(t\right)+i\tau \right)\right|\mathrm{d}t& ⩽{\left(\frac{1}{T}\underset{T}{\overset{2T}{\int }}{\left|\mathcal{Z}\left(\sigma -\epsilon +i\phi \left(t\right)+i\tau \right)\right|}^2\mathrm{d}t\right)}^{1/2}\hfill \\ & ={\left(\frac{1}{T}\underset{T}{\overset{2T}{\int }}{\left|\mathcal{Z}\left(\sigma -\epsilon +i\phi \left(t\right)+i\tau \right)\right|}^2\frac{{\phi }^{\prime }\left(t\right)\mathrm{d}t}{{\phi }^{\prime }\left(t\right)}\right)}^{1/2}\hfill \\ & \ll \left(\frac{1}{T{\phi }^{\prime }(2T+|\tau \left|\right)}\underset{0}{\overset{\phi (2T+|\tau \left|\right)}{\int }}{\left|\mathcal{Z}\left(\sigma -\epsilon +iu\right)\right|}^2\mathrm{d}u\right)\hfill \\ & \ll {\left(\frac{I_{\sigma -\epsilon }\phi (2T+|\tau \left|\right)}{T{\phi }^{\prime }(2T+|\tau \left|\right)}\right)}^{1/2}\hfill \\ & \ll {\left(\frac{2T+\left|\tau \right|}{T}\right)}^{1/2}\ll {(1+|\tau \left|\right)}^{1/2}.\hfill \end{array}$$

(23)

For the gamma-function, the estimate

$$\Gamma (\sigma +it)\ll exp\{-c|t\left|\right\},c>0,$$

(24)

is valid. Therefore,

$$f\left(-\epsilon +i\tau ,y\right)\ll y^{-\epsilon }exp\{-c_1\left|\tau \right|\},c_1>0.$$

This together with (23) leads to the bound

$$I_{T,y}^{\left(1\right)}\ll y^{-\epsilon }\underset{-\infty }{\overset{\infty }{\int }}{(1+|\tau \left|\right)}^{1/2}exp\{-c_1\left|\tau \right|\}\mathrm{d}\tau \ll y^{-\epsilon }.$$

(25)

Let $a=2\gamma -\log 2\pi$. In view of the formula for $\mathcal{Z}\left(s\right)$ in Theorem 1,

$$\begin{array}{cc}\hfill r_y\left(s\right)& =f^{\prime }(1-s,y)+af(1-s,y)\hfill \\ & =\frac{1}{{\theta }^2}{\Gamma }^{\prime }\left(\frac{1-s}{\theta }\right)y^{1-s}+\frac{1}{\theta }\Gamma \left(\frac{1-s}{\theta }\right)y^{1-s}\log y+\frac{a}{\theta }\Gamma \left(\frac{1-s}{\theta }\right)y^{1-s}\hfill \\ & =\frac{y^{1-s}}{\theta }\Gamma \left(\frac{1-s}{\theta }\right)\left(\frac{1}{\theta }\frac{{\Gamma }^{\prime }}{\Gamma }\left(\frac{1-s}{\theta }\right)+\log y+a\right).\hfill \end{array}$$

Hence, the estimates (24) and

$$\frac{{\Gamma }^{\prime }}{\Gamma }(\sigma +it)\ll \log \left(\right|t|+2)$$

yield

$$\begin{array}{cc}\hfill I_{T,y}^{\left(2\right)}& {\ll }_{\theta }{y}^{1-\sigma }\log y\frac{1}{T}\underset{T}{\overset{2T}{\int }}exp\left\{-\frac{c}{\theta }\phi \left(t\right)\right\}\log \phi \left(t\right)\mathrm{d}t\hfill \\ & {\ll }_{\theta }{y}^{1-\sigma }\log yexp\left\{-\frac{c}{2\theta }\phi \left(T\right)\right\}.\hfill \end{array}$$

This, (25), and (22) show that

$$I_{T,y}{\ll }_{\delta }{y}^{-\epsilon }+y^{1-\sigma }\log yexp\left\{-\frac{c}{2\theta }\phi \left(T\right)\right\}.$$

Therefore,

$$\underset{y\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\frac{1}{T}\underset{T}{\overset{2T}{\int }}\left|\mathcal{Z}(\sigma +i\phi \left(t\right))-{\mathcal{Z}}_y(\sigma +i\phi \left(t\right))\right|\mathrm{d}t=0$$

(26)

because $\phi \left(T\right)\to \infty$ as $T\to \infty$. □

Now, we return to the limit measure $P_{y,\sigma }$ of Lemma 8.

Lemma 10. 

Under the hypotheses of Theorem 2, the family $\{P_{y,\sigma }:y\in [1,\infty )\}$ is tight.

Proof. 

We have

$$\frac{1}{T}\underset{T}{\overset{2T}{\int }}\left|{\mathcal{Z}}_y(\sigma +i\phi \left(t\right))\right|\mathrm{d}t⩽\frac{1}{T}\underset{T}{\overset{2T}{\int }}\left|\mathcal{Z}(\sigma +i\phi \left(t\right))-{\mathcal{Z}}_y(\sigma +i\phi \left(t\right))\right|\mathrm{d}t+\frac{1}{T}\underset{T}{\overset{2T}{\int }}\left|\mathcal{Z}(\sigma +i\phi (t\left)\right)\right|\mathrm{d}t.$$

Therefore, by (2) and (26),

$$\underset{y⩾1}{sup}\underset{T\to \infty }{lim\; sup}\frac{1}{T}\underset{T}{\overset{2T}{\int }}\left|{\mathcal{Z}}_y(\sigma +i\phi \left(t\right))\right|\mathrm{d}t⩽C<\infty .$$

(27)

Let

$$z_{T,y}=z_{T,y}\left(\sigma \right)={\mathcal{Z}}_y(\sigma +i\phi \left({\theta }_T\right)),$$

and $z_y=z_y\left(\sigma \right)$ be the complex-valued random variable with the distribution $P_{y,\sigma }$. Then, the statement of Lemma 8 can be written as

$$z_{T,y}\underset{T\to \infty }{\overset{\mathcal{D}}{\to }}{z}_y.$$

(28)

From this and (27), we obtain that, for every $\delta >0$,

$$\mu \left\{\left|z_y\right|>{\delta }^{-1}C\right\}⩽\underset{y⩾1}{sup}\underset{T\to \infty }{lim\; sup}\mu \left\{\left|z_{T,y}\right|>{\delta }^{-1}C\right\}⩽\frac{\delta }{TC}\underset{T}{\overset{2T}{\int }}\left|{\mathcal{Z}}_y(\sigma +i\phi \left(t\right))\right|\mathrm{d}t⩽\delta .$$

This shows that, for $K=\{s\in \mathbb{C}:|s|⩽{\delta }^{-1}C\}$,

$$P_{y,\sigma }\left(K\right)⩾1-\delta ,$$

and the lemma is proven. □

Proof of Theorem 2. 

Lemma 10 together with Lemma 5 implies that the family $\left\{P_{y,\sigma }\right\}$ is relatively compact. Therefore, there is a sequence $\left\{P_{y_k,\sigma }\right\}\subset \left\{P_{y,\sigma }\right\}$ weakly convergent to a certain probability measure $P_{\sigma }$ on $(\mathbb{C},\mathcal{B}(\mathbb{C})$ as $k\to \infty$. This also can be written as

$$z_{y_k,\sigma }\underset{k\to \infty }{\overset{\mathcal{D}}{\to }}{P}_{\sigma }.$$

(29)

Define one more random variable,

$$z_T=z_T\left(\sigma \right)=\mathcal{Z}(\sigma +i\phi \left({\theta }_T\right)).$$

Then, Lemma 9 implies, for every $\delta >0$,

$$\begin{array}{cc}\hfill \underset{k\to \infty }{lim}& \underset{T\to \infty }{lim\; sup}\mu \left\{\left|z_T-z_{T,y_k}\right|>\delta \right\}\hfill \\ & ⩽\underset{k\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\frac{1}{\delta T}\underset{T}{\overset{2T}{\int }}\left|\mathcal{Z}(\sigma +i\phi \left(t\right))-{\mathcal{Z}}_{y_k}(\sigma +i\phi \left(t\right))\right|\mathrm{d}t=0.\hfill \end{array}$$

This, (28), and (29) together with Lemma 6 prove that

$$z_T\underset{T\to \infty }{\overset{\mathcal{D}}{\to }}{P}_{\sigma }.$$

The theorem is proven. □

6. Conclusions

In the paper, we considered the asymptotic behavior of the modified Mellin transform of the square of the Riemann zeta-function by using a probabilistic approach. We proved a limit theorem on the weak convergence of probability measures defined by shifts $\mathcal{Z}(\sigma +i\phi (t\left)\right)$, $1/2<\sigma <1$, where $\phi \left(t\right)$ is a differentiable increasing to infinity function with a monotonically decreasing derivative ${\phi }^{\prime }\left(t\right)$ satisfying a certain estimate connected to the mean square of the function $\mathcal{Z}\left(s\right)$. We expect that such normalization of the function $\mathcal{Z}\left(s\right)$ extends the class of limit distributions for $\mathcal{Z}\left(s\right)$. Our future plans are related to a similar theorem in the space of analytic functions.