On the Approximation by Mellin Transform of the Riemann Zeta-Function

Abstract

This paper is devoted to the approximation of a certain class of analytic functions by shifts $\mathcal{Z}(s+i\tau )$, $\tau \in \mathbb{R}$, of the modified Mellin transform $\mathcal{Z}\left(s\right)$ of the square of the Riemann zeta-function $\zeta (1/2+it)$. More precisely, we prove the existence of a closed non-empty set F such that there are infinitely many shifts $\mathcal{Z}(s+i\tau )$, which approximate a given analytic function from F with a given accuracy. In the proof, the weak convergence of measures in the space of analytic functions is applied. Then, the set F coincides with the support of a limit measure.

1. Introduction

Recall that the Riemann zeta-function $\zeta \left(s\right)$, $s=\sigma +it$, is defined, for $\sigma >1$, by the Dirichlet series

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

and is analytically continuable to the whole complex plane, except for the point $s=1$, which is a simple pole with residue 1. It is well known that the function $\zeta \left(s\right)$ has good approximation properties, its shifts $\zeta (s+i\tau )$, $\tau \in \mathbb{R}$, approximate every non-vanishing analytic function defined on the strip $\{s\in \mathbb{C}:1/2<\sigma <1\}$. On the other hand, in the theory of the function $\zeta \left(s\right)$, there exists several important unsolved problems. One of them is the moment problem on the asymptotic behavior as $T\to \infty$ for

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

Y. Motohashi introduced [1], see also [2], the modified Mellin transforms

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

and applied them for investigation of the latter problem. He first considered the case $k=2$. The integral for ${\mathcal{Z}}_k\left(s\right)$ [3] is absolutely convergent for $\sigma >1$ if $0⩽k⩽2$, and for $\sigma >(k+2)/4$ if $2⩽k⩽6$. Hence, the function ${\mathcal{Z}}_k\left(s\right)$ is analytic in the corresponding half-planes. Later, the Mellin transforms ${\mathcal{Z}}_k\left(s\right)$ with applications were studied in [4,5,6].

We give one example from [4]. Define $E_2\left(T\right)$ by

$\underset{0}{\overset{T}{\int }}{\left|\zeta \left(\frac{1}{2}+it\right)\right|}^4 \mathrm{d}t=T{P}_4(logT)+E_2\left(T\right)$

with

$P_4\left(x\right)=\sum _{j=0}^4{a}_j{x}^j, a_4=\frac{1}{2{\pi }^2}.$

There exists a problem to estimate $E_2\left(T\right)$. In [4], using the mean square estimates for $\mathcal{Z}\left(s\right)$, it was obtained that, for $\epsilon >0$,

$E_2\left(T\right){\ll }_{\epsilon }{T}^{2/3+\epsilon },$

and this estimate is the best up to $\epsilon$. The notation $a{\ll }_{\epsilon }b$, $a\in \mathbb{C}$, $b>0$, means that there exists a positive constant $c=c\left(\epsilon \right)$ such that $\left|a\right|⩽cb$.

In function theory, much attention is devoted to the approximation of analytic functions. We recall some results related to number theory. S.N. Mergelyan obtained [7] a very deep result connected to polynomials. Suppose that K is a compact set with a connected complement, and $f\left(s\right)$ a continuous function on K, which is analytic inside of K. Mergelyan proved [7] the existence of a polynomial sequence uniformly convergent on K to the function $f\left(s\right)$. From this, it follows that, for any $\epsilon >0$, we can find a polynomial $p_{f,\epsilon }\left(s\right)$ satisfying

$\underset{s\in K}{sup}\left|f\left(s\right)-p_{f,\epsilon }\left(s\right)\right|<\epsilon .$

Thus, a function satisfying the above hypotheses can be approximated by a polynomial.

In 1975, it turned out that there exist functions that approximate a whole class of analytic functions. The first example of such functions is the Riemann zeta-function. S.M. Voronin proved [8] that if $0<r<1/4$, the function $f\left(s\right)\ne 0$ is continuous on the disc $\left|s\right|⩽r$, and analytic inside that disc, then, for any $\epsilon >0$, there is a real number $\tau =\tau \left(\epsilon \right)$ satisfying the inequality

$\underset{\left|s\right|⩽r}{max}\left|f\left(s\right)-\zeta \left(s+\frac{3}{4}+i\tau \right)\right|<\epsilon .$

This shows that a set of non-vanishing analytic functions defined in the strip $\{s\in \mathbb{C}:1/2<\sigma <1\}$ is approximated by shifts $\zeta (s+i\tau )$ of one and the same function. In other words, $\zeta \left(s\right)$ is universal with respect to the approximation of analytic functions. The Voronin universality theorem was reinforced and extended for other zeta-functions. We recall its last form, see [9,10,11,12]. Suppose that $K\subset D$ is a compact set having a connected complement, $f\left(s\right)$ is continuous, having no zeros on K and analytic in inside of K function. Then, for any positive $\epsilon$,

$\underset{T\to \infty }{lim\; inf}\frac{1}{T}\mu \left\{\tau \in [0,T]:\underset{s\in K}{sup}|\zeta (s+i\tau )-f\left(s\right)|<\epsilon \right\}>0.$

Here, $\mu$ stands for the Lebesgue measure on the line $\mathbb{R}$.

The proof of the Voronin theorem in [8] is based on the rearrangement theorem for series in Hilbert space. B. Bagchi proposed [12] a new original probabilistic method that uses weak convergence of measures in the space of analytic functions. The Bagchi method was developed in [9,10]. Other results on the universality of zeta-functions are discussed in a survey paper [13]. We notice that an idea of application probabilistic methods in the theory of $\zeta \left(s\right)$ was proposed by H. Bohr and B. Jessen. In [14,15], they obtained the existence of the limit

$\underset{T\to \infty }{lim}\frac{1}{T}\mathrm{J}\{t\in [0,T]:\zeta (\sigma +it)\in \mathrm{R}\}$

for every rectangle $\mathrm{R}\subset \mathbb{C}$ with edges parallel to the axis and $\sigma >1/2$. Here, $\mathrm{J}$ denotes the Jordan measure on $\mathbb{R}$. A modern version of the Bohr–Jessen theorem in terms of weak convergence is presented in [9].

In general, for description value distribution of $\zeta \left(s\right)$, various methods and terms are used. For example, it was observed in [16] that the distribution of a-points of $\zeta \left(s\right)$, $a\ne 0$ (the solution of $\zeta \left(s\right)=a$), has a certain relation to a Julia line [17] with respect to the essential singularity of $\zeta \left(s\right)$ at infinity.

In the present paper, we are connected to a new problem—the approximation of analytic functions by the function $\mathcal{Z}\left(s\right)\stackrel{def}{=}{\mathcal{Z}}_1\left(s\right)$. We need some results from [3]. The function $\mathcal{Z}\left(s\right)$ is analytic in the half-plane $\sigma >-3/4$, except for a double pole at the point $s=1$, and has simple poles at the points $s=-(2k-1)$, $k\in \mathbb{N}$. Let ${\gamma }_0$ be the Euler constant, $E\left(T\right)$ be defined by

$\underset{0}{\overset{T}{\int }}{\left|\zeta \left(\frac{1}{2}+it\right)\right|}^2 \mathrm{d}t=Tlog\frac{T}{2\pi }+(2{\gamma }_0-1)T+E\left(t\right),$

and

$G\left(T\right)=\underset{1}{\overset{T}{\int }}E\left(t\right) \mathrm{d}t-\pi T, G_1\left(T\right)=\underset{1}{\overset{T}{\int }}G\left(t\right) \mathrm{d}t.$

Then, it was obtained that

(1)$\begin{array}{cc}\hfill \mathcal{Z}\left(s\right)=& \frac{1}{{(s-1)}^2}+\frac{2{\gamma }_0-log2\pi }{s-1}-E\left(1\right)+\pi (s+1)\hfill \\ & +s(s+1)(s+2)\underset{1}{\overset{\infty }{\int }}{G}_1\left(x\right)x^{-s-3} \mathrm{d}x, \sigma >-\frac{3}{4}.\hfill \end{array}$

Moreover, for $0⩽\sigma ⩽1$, $t⩾t_0>0$, and fixed $\epsilon >0$,

$\mathcal{Z}(\sigma +it){\ll }_{\epsilon }{t}^{1-\sigma +\epsilon },$

and

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

Let $D=\{s\in \mathbb{C}:1/2<\sigma <1\}$, and $H\left(D\right)$ be the space of analytic on D functions equipped with the topology of uniform convergence on compacts. The main result of the paper is the following theorem.

Theorem 1.

There is a non-empty closed subset $F\subset H\left(D\right)$ such that, for arbitrary compact set $K\subset D$, $f\left(s\right)\in F$, and every $\epsilon >0$,

$\underset{T\to \infty }{lim\; inf}\frac{1}{T} \mu \left\{\tau \in [0,T]:\underset{s\in K}{sup}\left|\mathcal{Z}(s+i\tau )-f\left(s\right)\right|<\epsilon \right\}>0.$

Moreover, except for at most a countably set of values of $\epsilon >0$, “lim inf” can be replaced by “lim”.

Theorem 1 implies that there are infinitely many shifts $\mathcal{Z}(s+i\tau )$ approximating a function from the set F. Theorem 1 is a certain version of the modern form of the Voronin universality theorem [8] for the Riemann zeta-function, see, for example, [9,10]. In the case of $\zeta \left(s\right)$, $F=\{g\in H(D):g(s)\ne 0 \mathrm{or} g(s)\equiv 0\}$.

Unfortunately, in the case of Theorem 1, the set F cannot be explicit described. Let $\mathcal{B}\left(\mathcal{X}\right)$ stand for Borel $\sigma$-field of the space $\mathcal{X}$. We will show that F is the support of a probability measure on $\left(H\right(D),\mathcal{B}(H\left(D\right)\left)\right)$. Theorem 1 is a corollary of a limit theorem for weakly convergent measures in the space $\left(H\right(D),\mathcal{B}(H\left(D\right)\left)\right)$. For $A\in \mathcal{B}\left(H\right(D\left)\right)$, set

$P_T\left(A\right)=\frac{1}{T} \mu \{\tau \in [0,T]:\mathcal{Z}(s+i\tau )\in A\}.$

Denote by $\underset{ }{\overset{W}{\to }}$ the weak convergence.

Theorem 2.

On the space $\left(H\right(D),\mathcal{B}(H\left(D\right)\left)\right)$, there is a probability measure P such that $P_T\underset{T\to \infty }{\overset{W}{\to }}P$.

For the proof of Theorem 2, the auxiliary function

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

where

$v(x,y)=exp\left\{-{\left(\frac{x}{y}\right)}^{{\sigma }_0}\right\}, x,y\in (1,\infty ),$

with a fixed ${\sigma }_0>0$, will be useful.

2. Case of Finite Interval

Let $a>1$, and

${\mathcal{Z}}_{a,y}\left(s\right)=\underset{1}{\overset{a}{\int }}{\left|\zeta \left(\frac{1}{2}+ix\right)\right|}^{2}v(x,y)x^{-s} \mathrm{d}x,$

For $A\in \mathcal{B}\left(H\right(D\left)\right)$, set

$P_{T,a,y}\left(A\right)=\frac{1}{T} \mu \left\{\tau \in [0,T]:{\mathcal{Z}}_{a,y}(s+i\tau )\in A\right\}.$

Lemma 1.

On the space $\left(H\right(D),\mathcal{B}(H\left(D\right)\left)\right)$, there is a probability measure $P_{a,y}$ such that $P_{T,a,y}\underset{T\to \infty }{\overset{W}{\to }}{P}_{a,y}$.

The proof of Lemma 1 is divided into parts. In the first part, we will deal with weak convergence on a certain Cartesian product. Let $\gamma$ be the unit circle on $\mathbb{C}$, and

${\Omega }_a=\prod _{u\in [1,a]}\gamma .$

In virtue of the Tikhonov theorem, the set ${\Omega }_a$ with the product topology is a compact topological Abeliam group. For $A\in \mathcal{B}\left({\Omega }_a\right)$, set

$Q_{T,a}\left(A\right)=\frac{1}{T} \mu \left\{\tau \in [0,T]:\left(u^{-i\tau }:u\in [1,a]\right)\in A\right\}.$

Lemma 2.

On the space $({\Omega }_a,\mathcal{B}\left({\Omega }_a\right))$, there is a probability measure $Q_a$ such that $Q_{T,a}\underset{T\to \infty }{\overset{W}{\to }}{Q}_a$.

Proof. 

The character group of ${\Omega }_a$ is isomorphic to $\underset{u\in [1,a]}{\oplus }{\mathbb{Z}}_u$, where ${\mathbb{Z}}_u=\mathbb{Z}$ for all $u\in [1,a]$. Therefore, the Fourier transform $g_{T,a}(k_u:u\in [1,a])$ of $Q_{T,a}$ is given by

$g_{T,a}(k_u:u\in [1,a])=\underset{{\Omega }_a}{\int }\prod _{u\in [1,a]}{x}_u^{k_u} \mathrm{d}{Q}_{T,a},$

where $x_u\in \gamma$, $k_u\in \mathbb{Z}$, and only a finite number of $k_u$ are not zeros. Thus,

$\begin{array}{cc}\hfill g_{T,a}(k_u:u\in [1,a])& =\frac{1}{T}\underset{0}{\overset{T}{\int }}exp\left\{-i\tau \sum _{u\in [1,a]}{k}_{u}logu\right\} \mathrm{d}\tau \hfill \\ & =\left\{\begin{array}{cc}1\hfill & \mathrm{if} {\displaystyle \sum _{u\in [1,a]}}{k}_{u}logu=0,\hfill \\ \frac{1-exp\left\{-i\tau \sum _{u\in [1,a]}{k}_{u}logu\right\}}{iT{\sum }_{u\in [1,a]}{k}_{u}logu}\hfill & \mathrm{if} {\displaystyle \sum _{u\in [1,a]}}{k}_{u}logu\ne 0.\hfill \end{array}\right.\hfill \end{array}$

Therefore, we have

$\underset{T\to \infty }{lim}{g}_{T,a}(k_u:u\in [1,a])\stackrel{def}{=}{g}_a(k_u:u\in [1,a])=\left\{\begin{array}{cc}1\hfill & \mathrm{if} {\displaystyle \sum _{u\in [1,a]}}{k}_{u}logu=0,\hfill \\ 0\hfill & \mathrm{if} {\displaystyle \sum _{u\in [1,a]}}{k}_{u}logu\ne 0.\hfill \end{array}\right.$

This shows that $Q_{T,a}\underset{T\to \infty }{\overset{W}{\to }}{Q}_a$ with $Q_a$ on $({\Omega }_a,\mathcal{B}\left({\Omega }_a\right))$ with the Fourier transform $g_a(k_u:u\in [1,a])$. □

Lemma 2 implies a certain limit lemma in the space $H\left(D\right)$. We recall that if $h:\mathcal{X}\to {\mathcal{X}}_1$ is a $(\mathcal{B}\left(\mathcal{X}\right),\mathcal{B}\left({\mathcal{X}}_1\right))$-measurable mapping, then a probability measure P on $(\mathcal{X},\mathcal{B}(\mathcal{X}\left)\right)$ defines the unique probability measure $P{h}^{-1}$ on $({\mathcal{X}}_1,\mathcal{B}\left({\mathcal{X}}_1\right))$ defined by $P{h}^{-1}\left(A\right)=P\left(h^{-1}A\right)$, $A\in {\mathcal{X}}_1$. Moreover, if the mapping h is continuous, then the weak convergence is preserved, i.e., if $P_n\underset{n\to \infty }{\overset{W}{\to }}P$ in $\mathcal{X}$, then also $P_n{h}^{-1}\underset{n\to \infty }{\overset{W}{\to }}P{h}^{-1}$ in ${\mathcal{X}}_1$. The latter remark is sometimes very useful.

Let

$S_{n,a,y}\left(s\right)=\frac{a-1}{n}\sum _{k=1}^n{\left|\zeta \left(\frac{1}{2}+i{\xi }_k\right)\right|}^{2}v({\xi }_k,y){\xi }_k^{-s},$

where ${\xi }_k\in [x_{k-1},x_k]$ and $x_k=1+((a-1)/n)k$. For $A\in \mathcal{B}\left(H\right(D\left)\right)$, let

$P_{T,n,a,y}\left(A\right)=\frac{1}{T} \mu \left\{\tau \in [0,T]:S_{n,a,y}(s+i\tau )\in A\right\}.$

Lemma 3.

On the space $\left(H\right(D),\mathcal{B}(H\left(D\right)\left)\right)$, there is a probability measure $P_{n,a,y}$ such that $P_{T,n,a,y}\underset{T\to \infty }{\overset{W}{\to }}{P}_{n,a,y}$.

Proof. 

Let the mapping $h_{n,a}:{\Omega }_a\to H\left(D\right)$ be given by the formula

$h_{n,a}\left(\underline{y}\right)=\frac{a-1}{n}\sum _{k=1}^n{\left|\zeta \left(\frac{1}{2}+i{\xi }_k\right)\right|}^{2}v({\xi }_k,y){\xi }_k^{-s}{y}_{{\xi }_k}, \underline{y}=\{y_u\in \gamma :u\in [1,a]\}.$

Then, $h_{n,a}$ is a continuous in the product topology, and $h_{n,a}\left(\{u^{-i\tau }:u\in [1,a]\}\right)=S_{n,a,y}(s+i\tau )$. Thus, $P_{T,n,a,y}=Q_{T,a}{h}_{n,a}^{-1}$, where $Q_{T,a}$ comes from Lemma 2. This equality, the continuity of $h_{n,a}$, Lemma 2 and the above remark on the preservation of weak convergence show that $P_{T,n,a,y}$ converges weakly to $P_{n,a,y}=Q_a{h}_{n,a}^{-1}$ as $T\to \infty$ with $Q_a$ defined in Lemma 2. □

In the sequel, we will use one lemma on the convergence in distribution ($\underset{ }{\overset{\mathcal{D}}{\to }}$). Recall that the random element $X_n$ converges in distribution to X as $n\to \infty$, if the distribution $P_n$ of $X_n$ converges weakly to that P of X as $n\to \infty$. In this case, we use the notation $X_n\underset{n\to \infty }{\overset{\mathcal{D}}{\to }}P$ as well.

Suppose that the metric space $(\mathcal{X},d)$ is separable and the $\mathcal{X}$-valued random elements X, $Y_n$ and $X_{nk}$ are defined on the same probability space with measure $\mathbb{P}$.

Lemma 4.

For $k\in \mathbb{N}$, let

$X_{nk}\underset{n\to \infty }{\overset{\mathcal{D}}{\to }}{X}_k$

and

$X_k\underset{k\to \infty }{\overset{\mathcal{D}}{\to }}X.$

If, for any $\epsilon >0$,

$\underset{k\to \infty }{lim}\underset{n\to \infty }{lim\; sup}\mathbb{P}\left\{d(X_{nk},Y_n)⩾\epsilon \right\}=0,$

then $Y_n\underset{n\to \infty }{\overset{\mathcal{D}}{\to }}X$.

The lemma is proved, for example, in [18], Theorem 4.2.

We note that the space $H\left(D\right)$ is separable and metrizable. It is known that there is a sequence $\{K_l:l\in \mathbb{N}\}\subset D$ of compact embedded sets such that D is the union of $K_l$, and every set $K\subset D$ lies in some set $K_l$. Then,

$\rho (g_1,g_2)=\sum _{l=1}^{\infty }{2}^{-l}\frac{{sup}_{s\in K_l}|g_1\left(s\right)-g_2\left(s\right)|}{1+{sup}_{s\in K_l}|g_1\left(s\right)-g_2\left(s\right)|}, g_1,g_2\in H\left(D\right),$

is a metric in $H\left(D\right)$ which induces its topology of uniform convergence on compacts.

Lemma 5.

The equality

$\underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\frac{1}{T}\underset{0}{\overset{T}{\int }}\rho \left(S_{n,a,y}(s+i\tau ),{\mathcal{Z}}_{a,y}(s+i\tau )\right) \mathrm{d}\tau =0$

holds.

Proof. 

The definition of the metric $\rho$ implies that it suffices to show that

(3)$\underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\frac{1}{T}\underset{0}{\overset{T}{\int }}\underset{s\in K}{sup}\left|S_{n,a,y}(s+i\tau )-{\mathcal{Z}}_{a,y}(s+i\tau )\right| \mathrm{d}\tau =0$

for every compact set $K\subset D$. Let L be a simple closed contour lying in D and enclosing a compact set K; suppose also that ${inf}_{s\in K}{inf}_{z\in L}|s-z|⩾c\left(L\right)>0$. Then, by the integral Cauchy formula, we have

$\underset{s\in K}{sup}\left|S_{n,a,y}(s+i\tau )-{\mathcal{Z}}_{a,y}(s+i\tau )\right|{\ll }_L\underset{L}{\int }\left|S_{n,a,y}(z+i\tau )-{\mathcal{Z}}_{a,y}(z+i\tau )\right|\left|\mathrm{d}z\right|.$

Therefore,

(4)$\begin{array}{cc}\hfill \frac{1}{T}& \underset{0}{\overset{T}{\int }}\underset{s\in K}{sup}\left|S_{n,a,y}(s+i\tau )-{\mathcal{Z}}_{a,y}(s+i\tau )\right| \mathrm{d}\tau \hfill \\ & {\ll }_L\underset{L}{\int }\left|\mathrm{d}z\right|\left(\frac{1}{T}\underset{0}{\overset{T}{\int }}\left|S_{n,a,y}(z+i\tau )-{\mathcal{Z}}_{a,y}(z+i\tau )\right|\mathrm{d}\tau \right).\hfill \end{array}$

Clearly,

(5)$\begin{array}{cc}\hfill \frac{1}{T}& \underset{0}{\overset{T}{\int }}\left|S_{n,a,y}(z+i\tau )-{\mathcal{Z}}_{a,y}(z+i\tau )\right| \mathrm{d}\tau \hfill \\ & ⩽{\left(\frac{1}{T}\underset{0}{\overset{T}{\int }}{\left|S_{n,a,y}(z+i\tau )-{\mathcal{Z}}_{a,y}(z+i\tau )\right|}^2 \mathrm{d}\tau \right)}^{1/2}\hfill \end{array}$

We have

(6)$\begin{array}{cc}\hfill & {\left|S_{n,a,y}(z+i\tau )-{\mathcal{Z}}_{a,y}(z+i\tau )\right|}^2\hfill \\ & =\left(S_{n,a,y}(z+i\tau )-{\mathcal{Z}}_{a,y}(z+i\tau )\right)\overline{\left(S_{n,a,y}(z+i\tau )-{\mathcal{Z}}_{a,y}(z+i\tau )\right)}\hfill \\ & =S_{n,a,y}(z+i\tau )\overline{S_{n,a,y}(z+i\tau )}-S_{n,a,y}(z+i\tau )\overline{{\mathcal{Z}}_{a,y}(z+i\tau )}\hfill \\ & -\overline{S_{n,a,y}(z+i\tau )}{\mathcal{Z}}_{a,y}(z+i\tau )+{\mathcal{Z}}_{a,y}(z+i\tau )\overline{{\mathcal{Z}}_{a,y}(z+i\tau )},\hfill \end{array}$

where $\overline{z}$ denotes the complex conjugate of $z\in \mathbb{C}$. By the definition of $S_{n,a,y}\left(s\right)$,

$\begin{array}{cc}\hfill S_{n,a,y}& (z+i\tau )\overline{S_{n,a,y}(z+i\tau )}={\left(\frac{a-1}{n}\right)}^2\sum _{k=1}^n{\left|\zeta \left(\frac{1}{2}+i{\xi }_k\right)\right|}^4{v}^2({\xi }_k,y){\xi }_k^{-2\mathrm{Re}z}\hfill \\ & +{\left(\frac{a-1}{n}\right)}^2\underset{k_1\ne k_2}{\sum _{k_1=1}^n\sum _{k_2=1}^n}{\left|\zeta \left(\frac{1}{2}+i{\xi }_{k_1}\right)\right|}^2{\left|\zeta \left(\frac{1}{2}+i{\xi }_{k_2}\right)\right|}^2\hfill \\ & \times v({\xi }_{k_1},y)v({\xi }_{k_2},y){\xi }_{k_1}^{-z}{\xi }_{k_2}^{-\overline{z}}{\left(\frac{{\xi }_{k_1}}{{\xi }_{k_2}}\right)}^{-i\tau }.\hfill \end{array}$

Therefore,

$\begin{array}{cc}\hfill \frac{1}{T}\underset{0}{\overset{T}{\int }}& S_{n,a,y}(z+i\tau )\overline{S_{n,a,y}(z+i\tau )} \mathrm{d}\tau \hfill \\ & ={\left(\frac{a-1}{n}\right)}^2\sum _{k=1}^n{\left|\zeta \left(\frac{1}{2}+i{\xi }_k\right)\right|}^4{v}^2({\xi }_k,y){\xi }_k^{-2\mathrm{Re}z}\hfill \\ & +O({\left(\frac{a-1}{n}\right)}^2\frac{1}{T}\underset{k_1\ne k_2}{\sum _{k_1=1}^n\sum _{k_2=1}^n}{\left|\zeta \left(\frac{1}{2}+i{\xi }_{k_1}\right)\right|}^2{\left|\zeta \left(\frac{1}{2}+i{\xi }_{k_2}\right)\right|}^2\hfill \\ & \times v({\xi }_{k_1},y)v({\xi }_{k_2},y){\xi }_{k_1}^{-\mathrm{Re}z}{\xi }_{k_2}^{\mathrm{Re}z}{\left|log\frac{{\xi }_{k_1}}{{\xi }_{k_2}}\right|}^{-1}).\hfill \end{array}$

Since

$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\frac{a-1}{n}\sum _{k=1}^n{\left|\zeta \left(\frac{1}{2}+i{\xi }_k\right)\right|}^4{v}^2({\xi }_k,y){\xi }_k^{-2\mathrm{Re}z}=\underset{1}{\overset{a}{\int }}{\left|\zeta \left(\frac{1}{2}+ix\right)\right|}^4{v}^2(x,y)x^{-2\mathrm{Re}z} \mathrm{d}x,\end{array}$

hence we obtain, for all $z\in L$,

(7)$\underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\frac{1}{T}\underset{0}{\overset{T}{\int }}{S}_{n,a,y}(z+i\tau )\overline{S_{n,a,y}(z+i\tau )} \mathrm{d}\tau =0.$

Similarly, by the definition of ${\mathcal{Z}}_{a,y}\left(s\right)$, for all $z\in L$,

(8)$\begin{array}{cc}\hfill & \underset{T\to \infty }{lim\; sup}\frac{1}{T}\underset{0}{\overset{T}{\int }}{\mathcal{Z}}_{a,y}(z+i\tau )\overline{{\mathcal{Z}}_{a,y}(z+i\tau )} \mathrm{d}\tau \hfill \\ & =\underset{T\to \infty }{lim\; sup}\frac{1}{T}\underset{0}{\overset{T}{\int }}(\underset{1}{\overset{a}{\int }}\underset{1}{\overset{a}{\int }}{\left|\zeta \left(\frac{1}{2}+i{x}_1\right)\right|}^2{\left|\zeta \left(\frac{1}{2}+i{x}_2\right)\right|}^2\hfill \\ & \times v(x_1,y)v(x_2,y)x_1^{-z-i\tau }{x}_2^{-\overline{z}+i\tau } \mathrm{d}{x}_1 \mathrm{d}{x}_2) \mathrm{d}\tau \hfill \\ & =\underset{T\to \infty }{lim\; sup}\frac{1}{T}\underset{x_1\ne x_2}{\underset{1}{\overset{a}{\int }}\underset{1}{\overset{a}{\int }}}{\left|\zeta \left(\frac{1}{2}+i{x}_1\right)\right|}^2{\left|\zeta \left(\frac{1}{2}+i{x}_2\right)\right|}^2\hfill \\ & \times v(x_1,y)v(x_2,y)x_1^{-z}{x}_2^{-\overline{z}}\left(e^{-iTlog(x_1/x_2)}-1\right){\left(log\frac{x_1}{x_2}\right)}^{-1} \mathrm{d}{x}_1 \mathrm{d}{x}_2=0.\hfill \end{array}$

The latter equality suggests that the set of values of the function ${\mathcal{Z}}_{a,y}\left(s\right)$ is not dense. On the other hand, this case is convenient for our investigations. Moreover,

$\begin{array}{cc}\hfill & \left|\frac{1}{T}\underset{0}{\overset{T}{\int }}{S}_{n,a,y}(z+i\tau )\overline{{\mathcal{Z}}_{a,y}(z+i\tau )} \mathrm{d}\tau \right|\hfill \\ & ⩽{\left(\frac{1}{T}\underset{0}{\overset{T}{\int }}{\left|S_{n,a,y}(z+i\tau )\right|}^2 \mathrm{d}\tau \right)}^{1/2}{\left(\frac{1}{T}\underset{0}{\overset{T}{\int }}{\left|{\mathcal{Z}}_{a,y}(z+i\tau )\right|}^2 \mathrm{d}\tau \right)}^{1/2},\hfill \end{array}$

and this is true for the integral of $\overline{S_{n,a,y}(z+i\tau )}{\mathcal{Z}}_{a,y}(z+i\tau )$. This and (4)–(8) show that

$\underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\frac{1}{T}\underset{0}{\overset{T}{\int }}\underset{s\in K}{sup}\left|S_{n,a,y}(s+i\tau )-{\mathcal{Z}}_{a,y}(s+i\tau )\right| \mathrm{d}\tau =0.$

□

Proof of Lemma 1.

Suppose that ${\theta }_T$ is a random variable uniformly distributed on $[0,T]$ and defined on a certain probability space with measure $\mathbb{P}$. Define the $H\left(D\right)$-valued random element

$X_{T,n,a,y}\left(s\right)=S_{n,a,y}(s+i{\theta }_T).$

In view of Lemma 3, we have

(9)$X_{T,n,a,y}\underset{T\to \infty }{\overset{\mathcal{D}}{\to }}{X}_{n,a,y},$

where $X_{n,a,y}$ is the $H\left(D\right)$-valued random element with the distribution $P_{n,a,y}$.

Now, we will prove that the sequence $\{P_{n,a,y}:n\in \mathbb{N}\}$ is tight, i.e., that, for every $\epsilon >0$, there exists a compact set $K=K\left(\epsilon \right)\subset H\left(D\right)$ such that

$P_{n,a,y}\left(K\right)>1-\epsilon$

for all $n\in \mathbb{N}$. Let $K_l$ be a compact set in the definition of the metric $\rho$. Then, (7) and the integral Cauchy formula imply

$\underset{n\in \mathbb{N}}{sup}\underset{T\to \infty }{lim\; sup}\frac{1}{T}\underset{0}{\overset{T}{\int }}\underset{s\in K_l}{sup}\left|S_{n,a,y}(s+i\tau )\right| \mathrm{d}\tau ⩽R_{l,a,y}<\infty .$

Let $\epsilon >0$ be a fixed, and $M_l=M_{l,a,y}=R_{l,a,y}{2}^l{\epsilon }^{-1}$. Then, in view of (9),

(10)$\mathbb{P}\left(\underset{s\in K_l}{sup}\left|X_{n,a,y}\left(s\right)\right|>M_l\right)⩽\underset{n\in \mathbb{N}}{sup}\underset{T\to \infty }{lim\; sup}\frac{1}{M_{l}T}\underset{0}{\overset{T}{\int }}\underset{s\in K_l}{sup}\left|S_{n,a,y}(s+i\tau )\right| \mathrm{d}\tau ⩽\frac{\epsilon }{2^l}$

for all n and $l\in \mathbb{N}$. Let

$K=K\left(\epsilon \right)=\left\{g\in H\left(D\right):\underset{s\in K_l}{sup}\left|g\left(s\right)\right|⩽M_l, l\in \mathbb{N}\right\}.$

Then, the set K is compact in the space $H\left(D\right)$, and, by (10),

$\mathbb{P}\left(X_{n,a,y}\in K\right)=1-\mathbb{P}\left(X_{n,a,y}\notin K\right)>1-\epsilon \sum _{l=1}^{\infty }{2}^{-l}=1-\epsilon$

for all $n\in \mathbb{N}$. This and the definition of $P_{n,a,y}$ prove the tightness of the sequence $\{P_{n,a,y}:n\in \mathbb{N}\}$. In the theory of weak convergence of probability measures, the Prokhorov theorem, see, for example, [18], occupies an important place. Let $\left\{P\right\}$ be a family of probability measures on $(\mathcal{X},\mathcal{B}(\mathcal{X}\left)\right)$. The Prokhorov theorem connects the tightness and relative compactness of $\left\{P\right\}$; namely, if the family $\left\{P\right\}$ is tight, then it is relatively compact.

Since the sequence $\left\{P_{n,a,y}\right\}$ is tight, by the Prokhorov theorem [18], it is relatively compact, i.e., every subsequence $\left\{P_{n_k,a,y}\right\}$ contains a subsequence weakly convergent to a certain probability measure on $\left(H\right(D),\mathcal{B}(H\left(D\right)\left)\right)$. Thus, there exists a probability measure $P_{a,y}$ on $\left(H\right(D),\mathcal{B}(H\left(D\right)\left)\right)$ and a sequence $\left\{P_{n_r,a,y}\right\}$ such that $P_{n_r,a,y}$ converges weakly to $P_{a,y}$ as $r\to \infty$. In other words,

(11)$X_{n_r,a,y}\underset{r\to \infty }{\overset{\mathcal{D}}{\to }}{P}_{a,y}.$

Now, we are in position to apply Lemma 4 for the random elements

$Y_{T,a,y}\left(s\right)={\mathcal{Z}}_{a,y}(s+i{\theta }_T),$

$X_{n_r,a,y}$ and $X_{a,y}$, where $X_{a,y}$ has the distribution $P_{a,y}$. By Lemma 5, we have, for every $\epsilon >0$,

$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}& \underset{T\to \infty }{lim\; sup}\mathbb{P}\left(\rho (Y_{T,a,y},X_{n_r,a,y})⩾\epsilon \right)\hfill \\ & ⩽\underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\frac{1}{T\epsilon }\underset{0}{\overset{T}{\int }}\rho \left(S_{n_r,a,y}(s+i\tau ),{\mathcal{Z}}_{a,y}(s+i\tau )\right) \mathrm{d}\tau =0.\hfill \end{array}$

This, (9) and (11) together with Lemma 5 prove Lemma 1, i.e., $P_{T,a,y}$ converges weakly to $P_{a,y}$ as $T\to \infty$. □

3. Case of Infinite Interval

In this section, we will prove a limit theorem for the function ${\mathcal{Z}}_y\left(s\right)$. Since $\zeta (1/2+it)\ll t^{1/6}$ as $t\to \infty$, and $v(x,y)$ with respect to x is decreasing exponentially, the integral for ${\mathcal{Z}}_y\left(s\right)$ is absolutely convergent for $\sigma >{\sigma }_0$ with every fixed ${\sigma }_0$ and $y>0$.

For $A\in \mathcal{B}\left(H\right(D\left)\right)$, define

$P_{T,y}\left(A\right)=\frac{1}{T} \mu \left\{\tau \in [0,T]:{\mathcal{Z}}_y(s+i\tau )\in A\right\}.$

Lemma 6.

On $\left(H\right(D),\mathcal{B}(H\left(D\right)\left)\right)$, there exists a probability measure $P_y$ such that $P_{T,y}\underset{T\to \infty }{\overset{W}{\to }}{P}_y$.

Proof. 

First, we observe that the equality

(12)$\underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\frac{1}{T}\underset{0}{\overset{T}{\int }}\rho \left({\mathcal{Z}}_y(s+i\tau ),{\mathcal{Z}}_{a,y}(s+i\tau )\right) \mathrm{d}\tau =0$

holds. As in the case of Lemma 5, it suffices to show that, for every compact set $K\subset D$,

(13)$\underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\frac{1}{T}\underset{0}{\overset{T}{\int }}\underset{s\in K}{sup}\left|{\mathcal{Z}}_y(s+i\tau )-{\mathcal{Z}}_{a,y}(s+i\tau )\right| \mathrm{d}\tau =0.$

It is easily seen that, for every fixed $y>0$ and $s\in K$,

$\begin{array}{cc}\hfill {\mathcal{Z}}_y(s+i\tau )-{\mathcal{Z}}_{a,y}(s+i\tau )& =\underset{a}{\overset{\infty }{\int }}{\left|\zeta \left(\frac{1}{2}+ix\right)\right|}^{2}v(x,y)x^{-s-i\tau } \mathrm{d}x\hfill \\ & {\ll }_y\underset{a}{\overset{\infty }{\int }}{\left|\zeta \left(\frac{1}{2}+ix\right)\right|}^{2}v(x,y)x^{-1/2} \mathrm{d}x=o_y\left(1\right)\hfill \end{array}$

as $a\to \infty$ in view of convergence of the integral. From this, equality (13) follows.

Let ${\theta }_T$ be the same random variable as in the proof of Lemma 1. Define

$Y_{T,y}\left(s\right)={\mathcal{Z}}_y(s+i{\theta }_T),$

and denote by $Y_{a,y}$ the $H\left(D\right)$-valued random element with the distribution $P_{a,y}$. Then, Lemma 1 implies the relation

(14)$Y_{T,a,y}\underset{T\to \infty }{\overset{\mathcal{D}}{\to }}{Y}_{a,y}.$

Let $K_l$, $l\in \mathbb{N}$ be a compact set from the definition of metric $\rho$. Then, (8) and the integral Cauchy formula give

$\underset{a⩾1}{sup}\underset{T\to \infty }{lim\; sup}\frac{1}{T}\underset{0}{\overset{T}{\int }}\underset{s\in K_l}{sup}\left|{\mathcal{Z}}_{a,y}(s+i\tau )\right| \mathrm{d}\tau ⩽R_{l,y}<\infty .$

Thus, taking ${\widehat{M}}_l={\widehat{M}}_{l,y}=R_{l,y}{2}^l{\epsilon }^{-1}$, we find by (14)

$\mathbb{P}\left(\underset{s\in K_l}{sup}\left|Y_{a,y}\left(s\right)\right|>{\widehat{M}}_l\right)⩽\underset{a⩾1}{sup}\underset{T\to \infty }{lim\; sup}\frac{1}{{\widehat{M}}_{l}T}\underset{0}{\overset{T}{\int }}\underset{s\in K_l}{sup}\left|{\mathcal{Z}}_{a,y}(s+i\tau )\right| \mathrm{d}\tau ⩽\frac{\epsilon }{2^l}.$

This shows that

$\mathbb{P}\left(Y_{a,y}\in K\right)>1-\epsilon ,$

for all $a⩾1$, where $K=\{g\in H\left(D\right):{sup}_{s\in K_l}|g\left(s\right)|⩽{\widehat{M}}_l, l\in \mathbb{N}\}$. Therefore, the family of probability measures $\{P_{a,y}:a⩾1\}$ is tight. Thus, there exists a sequence $P_{a_r,y}$ weakly convergent to a certain probability measure $P_y$ as $r\to \infty$, i.e.,

$Y_{a_r,y}\underset{r\to \infty }{\overset{\mathcal{D}}{\to }}{P}_y.$

This, (12), (14) and Lemma 4 prove that

(15)$Y_{T,y}\underset{T\to \infty }{\overset{\mathcal{D}}{\to }}{P}_y,$

and the lemma is proved. □

4. Formula for ${\mathcal{Z}}_{\mathbf{y}}\left(\mathbf{s}\right)$

As usual, denote by $\Gamma \left(s\right)$ the Euler gamma-function, and define

$l_y\left(s\right)=\frac{s}{{\sigma }_0}\Gamma \left(\frac{s}{{\sigma }_0}\right)y^s,$

where ${\sigma }_0$ is from definition of $v(x,y)$.

Lemma 7.

The integral representation, for $s\in D$,

${\mathcal{Z}}_y\left(s\right)=\frac{1}{2\pi i}\underset{{\sigma }_0-i\infty }{\overset{{\sigma }_0+i\infty }{\int }}\mathcal{Z}(s+z)l_y\left(z\right)\frac{\mathrm{d}z}{z}$

is valid.

Proof. 

We will apply the classical Mellin formula

$\frac{1}{2\pi i}\underset{a-i\infty }{\overset{a+i\infty }{\int }}\Gamma \left(s\right)b^{-s} \mathrm{d}s=e^{-b}, a,b>0,$

and obtain

(16)$\begin{array}{cc}\hfill \frac{1}{2\pi i}\underset{{\sigma }_0-i\infty }{\overset{{\sigma }_0+i\infty }{\int }}{l}_y\left(z\right)x^{-z}\frac{\mathrm{d}z}{z}& =\frac{1}{2\pi i}\underset{{\sigma }_0-i\infty }{\overset{{\sigma }_0+i\infty }{\int }}\frac{1}{{\sigma }_0}\Gamma \left(\frac{z}{{\sigma }_0}\right){\left(\frac{x}{y}\right)}^{-z} \mathrm{d}z\hfill \\ & =\frac{1}{2\pi i}\underset{1-i\infty }{\overset{1+i\infty }{\int }}\Gamma \left(z\right){\left(\frac{x}{y}\right)}^{-{\sigma }_{0}z}\mathrm{d}z\hfill \\ & =exp\left\{-{\left(\frac{x}{y}\right)}^{{\sigma }_0}\right\}=v(x,y).\hfill \end{array}$

Setting, for brevity,

$f(x,t)=\frac{1}{2\pi i}\frac{l_y({\sigma }_0+it)}{{\sigma }_0+it}{\left|\zeta \left(\frac{1}{2}+ix\right)\right|}^2{x}^{-s-{\sigma }_0-it}$

and applying theorem from ([19], §1.84), we obtain

(17)$\underset{-T}{\overset{T}{\int }} \mathrm{d}t\underset{1}{\overset{X}{\int }}f(x,t) \mathrm{d}x=\underset{1}{\overset{X}{\int }} \mathrm{d}x\underset{-T}{\overset{T}{\int }}f(x,t) \mathrm{d}t,$

for any $X,T>1$. Next, the well-known estimate

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

which is uniform in any fixed strip ${\sigma }_1<\sigma <{\sigma }_2$, together with the inequality

$\underset{1}{\overset{X}{\int }}{\left|\zeta \left(\frac{1}{2}+ix\right)\right|}^2 \mathrm{d}x\ll X(logX)$

imply

$\underset{T}{\overset{+\infty }{\int }} \mathrm{d}t\underset{1}{\overset{X}{\int }}\left(\left|f\right(x,t\left)\right|+\left|f\right(x,-t\left)\right|\right) \mathrm{d}x, \underset{1}{\overset{X}{\int }} \mathrm{d}x\underset{T}{\overset{+\infty }{\int }}\left(\left|f\right(x,t\left)\right|+\left|f\right(x,-t\left)\right|\right) \mathrm{d}t\ll R,$

where

(18)$\begin{array}{cc}\hfill R=R(X,T)& =y^{{\sigma }_0}\underset{T}{\overset{+\infty }{\int }}{e}^{-ct/{\sigma }_0} \mathrm{d}t\underset{1}{\overset{X}{\int }}{\left|\zeta \left(\frac{1}{2}+ix\right)\right|}^2{x}^{-\sigma -{\sigma }_0} \mathrm{d}x\hfill \\ & \ll y^{{\sigma }_0}exp\left\{-\frac{c}{{\sigma }_0}T\right\}\left(1+X^{1-\sigma -{\sigma }_0}\right){(logX)}^2.\hfill \end{array}$

Hence, using (17) and (18), we find that

$\begin{array}{cc}\hfill \underset{-\infty }{\overset{+\infty }{\int }} \mathrm{d}t\underset{1}{\overset{X}{\int }}f(x,t) \mathrm{d}x& =\left(\underset{-T}{\overset{T}{\int }}+\underset{T}{\overset{+\infty }{\int }}+\underset{-\infty }{\overset{-T}{\int }}\right) \mathrm{d}t\underset{1}{\overset{X}{\int }}f(x,t) \mathrm{d}x\hfill \\ & =\underset{1}{\overset{X}{\int }} \mathrm{d}x\underset{-T}{\overset{T}{\int }}f(x,t) \mathrm{d}t+O\left(R\right)\hfill \\ & =\underset{1}{\overset{X}{\int }} \mathrm{d}x\left(\underset{-\infty }{\overset{+\infty }{\int }}-\underset{T}{\overset{+\infty }{\int }}-\underset{-\infty }{\overset{-T}{\int }}\right)f(x,t) \mathrm{d}t+O\left(R\right)\hfill \\ & =\underset{1}{\overset{X}{\int }} \mathrm{d}x\underset{-\infty }{\overset{+\infty }{\int }}f(x,t) \mathrm{d}t+O\left(R\right).\hfill \end{array}$

Tending $T\to \infty$, we obtain

$\underset{-\infty }{\overset{+\infty }{\int }} \mathrm{d}t\underset{1}{\overset{X}{\int }}f(x,t) \mathrm{d}x=\underset{1}{\overset{X}{\int }} \mathrm{d}x\underset{-\infty }{\overset{+\infty }{\int }}f(x,t) \mathrm{d}t$

for any $X>1$. Therefore, the application of a theorem from ([19], §1.84) together with (16) yields

$\begin{array}{cc}\hfill \frac{1}{2\pi i}& \underset{{\sigma }_0-i\infty }{\overset{{\sigma }_0+i\infty }{\int }}\mathcal{Z}(s+z)l_y\left(z\right)\frac{\mathrm{d}z}{z}\hfill \\ & =\underset{-\infty }{\overset{+\infty }{\int }}\underset{1}{\overset{+\infty }{\int }}\frac{1}{2\pi i}{\left|\zeta \left(\frac{1}{2}+ix\right)\right|}^2\frac{l_y({\sigma }_0+it)}{{\sigma }_0+it} x^{-s-{\sigma }_0-it}\mathrm{d}x\hfill \\ & =\underset{-\infty }{\overset{+\infty }{\int }} \mathrm{d}t\underset{1}{\overset{+\infty }{\int }}f(x,t) \mathrm{d}x=\underset{1}{\overset{+\infty }{\int }} \mathrm{d}x\underset{-\infty }{\overset{+\infty }{\int }}f(x,t)\mathrm{d}t\hfill \\ & =\underset{1}{\overset{+\infty }{\int }}{\left|\zeta \left(\frac{1}{2}+ix\right)\right|}^2{x}^{-s}\frac{1}{2\pi i}\underset{{\sigma }_0-i\infty }{\overset{{\sigma }_0+i\infty }{\int }}{l}_y\left(z\right)x^{-z}\frac{\mathrm{d}z}{z} \mathrm{d}x\hfill \\ & =\underset{1}{\overset{+\infty }{\int }}{\left|\zeta \left(\frac{1}{2}+ix\right)\right|}^2{x}^{-s}v(x,y) \mathrm{d}x={\mathcal{Z}}_y\left(s\right).\hfill \end{array}$

□

5. Approximation of $\mathcal{Z}\left(\mathit{s}\right)$ by ${\mathcal{Z}}_{\mathit{y}}\left(\mathit{s}\right)$

Lemma 8.

The equality

$\underset{y\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\frac{1}{T}\underset{0}{\overset{T}{\int }}\rho \left(\mathcal{Z}(s+i\tau ),{\mathcal{Z}}_y(s+i\tau )\right) \mathrm{d}\tau =0$

holds.

Proof. 

Let K be an arbitrary fixed compact set of the strip D. Then, there exists a number $\epsilon >0$ such that, for all $s=\sigma +it\in K$, $1/2+2\epsilon ⩽\sigma ⩽1-\epsilon$. Denote

${\sigma }_1=\sigma -\epsilon -\frac{1}{2}, {\sigma }_0=\frac{1}{2}+\epsilon .$

Then, ${\sigma }_1>0$ for all $s\in K$. Since the point $z=1-s$ is a double pole, and $z=0$ is a simple pole of the function

$\mathcal{Z}(s+z)\frac{l_y\left(z\right)}{z},$

Lemma 7 and the residue theorem imply

(19)${\mathcal{Z}}_y\left(s\right)-\mathcal{Z}\left(s\right)=\frac{1}{2\pi i}\underset{-{\sigma }_1-i\infty }{\overset{-{\sigma }_1+i\infty }{\int }}\mathcal{Z}(s+z)l_y\left(z\right)\frac{\mathrm{d}z}{z}+R_y\left(s\right),$

where

$R_y\left(s\right)=\underset{z=1-s}{\mathrm{Res}}\mathcal{Z}(s+z)\frac{l_y\left(z\right)}{z}.$

Let $a_0=2{\gamma }_0-log2\pi$. Then, in view of (1),

(20)$R_y\left(s\right)={\left(\frac{l_y\left(z\right)}{z}\right)}^{\prime }{|}_{z=1-s}+a_0\frac{l_y(1-s)}{1-s}.$

By (19), for all $s\in K$, we have

$\begin{array}{cc}\hfill {\mathcal{Z}}_y(s+i\tau )-\mathcal{Z}(s+i\tau )=& \frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}\mathcal{Z}\left(\sigma +it-\sigma +\frac{1}{2}+\epsilon +i\tau +iv\right)\hfill \\ & \times \frac{l_y(1/2+\epsilon -\sigma +iv)}{1/2+\epsilon -\sigma +iv} \mathrm{d}v+R_y(s+i\tau ).\hfill \end{array}$

Hence, writing v in place of $t+v$, gives, for $s\in K$,

$\begin{array}{cc}\hfill {\mathcal{Z}}_y(s+i\tau )-\mathcal{Z}(s+i\tau )=& \frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}\mathcal{Z}\left(\frac{1}{2}+\epsilon +i\tau +iv\right)\frac{l_y(1/2+\epsilon -s+iv)}{1/2+\epsilon -s+iv} \mathrm{d}v\hfill \\ & +R_y(s+i\tau )\hfill \\ \hfill \ll & \underset{-\infty }{\overset{\infty }{\int }}\left|\mathcal{Z}\left(\frac{1}{2}+\epsilon +i\tau +iv\right)\right|\underset{s\in K}{sup}\left|\frac{l_y(1/2+\epsilon -s+iv)}{1/2+\epsilon -s+iv}\right| \mathrm{d}v\hfill \\ & +\underset{s\in K}{sup}\left|R_y(s+i\tau )\right|.\hfill \end{array}$

Thus,

(21)$\begin{array}{cc}\hfill \frac{1}{T}& \underset{0}{\overset{T}{\int }}\underset{s\in K}{sup}\left|{\mathcal{Z}}_y(s+i\tau )-\mathcal{Z}(s+i\tau )\right| \mathrm{d}\tau \hfill \\ & \ll \underset{-\infty }{\overset{\infty }{\int }}\left(\frac{1}{T}\underset{0}{\overset{T}{\int }}\left|\mathcal{Z}\left(\frac{1}{2}+\epsilon +i\tau +iv\right)\right| \mathrm{d}\tau \right)\underset{s\in K}{sup}\left|\frac{l_y(1/2+\epsilon -s+iv)}{1/2+\epsilon -s+iv}\right| \mathrm{d}v\hfill \\ & +\frac{1}{T}\underset{0}{\overset{T}{\int }}\underset{s\in K}{sup}\left|R_y(s+i\tau )\right| \mathrm{d}\tau \stackrel{def}{=}{I}_1+I_2.\hfill \end{array}$

First we estimate $I_1$. The definition of $l_y\left(s\right)$ and (17) imply that, for $s=\sigma +it\in K$,

(22)$\begin{array}{cc}\hfill \frac{l_y(1/2+\epsilon -s+iv)}{1/2+\epsilon -s+iv}& {\ll }_{{\sigma }_0}{y}^{1/2+\epsilon -\sigma }\left|\Gamma \left(\frac{1}{{\sigma }_0}\left(\frac{1}{2}+\epsilon -\sigma -it+iv\right)\right)\right|\hfill \\ & {\ll }_{{\sigma }_0}{y}^{1/2+\epsilon -\sigma }exp\left\{-\frac{c}{{\sigma }_0}|t-v|\right\}\hfill \\ & {\ll }_{{\sigma }_0}{y}^{1/2+\epsilon -\sigma }exp\left\{-\frac{c}{{\sigma }_0}|v-t|\right\}\hfill \\ & {\ll }_{{\sigma }_0}{y}^{1/2+\epsilon -\sigma }exp\left\{\frac{c}{{\sigma }_0}\left|t\right|\right\}exp\left\{-\frac{c}{{\sigma }_0}\left|v\right|\right\}.\hfill \end{array}$

Since $\sigma ⩾1/2+2\epsilon$, then $1/2+\epsilon -\sigma ⩽-\epsilon$ for any $s\in K$. Therefore,

$\underset{s\in K}{sup}\left|\frac{l_y(1/2+\epsilon +iv-s)}{1/2+\epsilon +iv-s}\right|{\ll }_{{\sigma }_0,K}{y}^{-\epsilon }exp\{-c_1\left|v\right|\}.$

Thus, we obtain

$I_1{\ll }_{{\sigma }_0,K}{y}^{-\epsilon }\underset{-\infty }{\overset{+\infty }{\int }}exp\left\{-\frac{c}{{\sigma }_0}\left|v\right|\right\} \frac{1}{T}\underset{0}{\overset{T}{\int }}\left|\mathcal{Z}\left(\frac{1}{2}+\epsilon +i\tau +iv\right)\right| \mathrm{d}\tau \mathrm{d}v.$

Given $0<{\epsilon }_1<\epsilon$, Cauchy inequality together with (2) imply

$\begin{array}{cc}\hfill \frac{1}{T}\underset{0}{\overset{T}{\int }}\left|\mathcal{Z}\left(\frac{1}{2}+\epsilon +i\tau +iv\right)\right| \mathrm{d}\tau & ⩽{\left(\frac{1}{T}\underset{0}{\overset{T}{\int }}{\left|\mathcal{Z}\left(\frac{1}{2}+\epsilon +i\tau +iv\right)\right|}^2 \mathrm{d}\tau \right)}^{1/2}\hfill \\ & ={\left(\frac{1}{T}\underset{v}{\overset{T+v}{\int }}{\left|\mathcal{Z}\left(\frac{1}{2}+\epsilon +i\tau \right)\right|}^2 \mathrm{d}\tau \right)}^{1/2}\hfill \\ & ⩽{\left(\frac{2}{T}\underset{0}{\overset{T+\left|v\right|}{\int }}{\left|\mathcal{Z}\left(\frac{1}{2}+\epsilon +i\tau \right)\right|}^2 \mathrm{d}\tau \right)}^{1/2}\hfill \\ & {\ll }_{\epsilon }{\left(\frac{1}{T}{(T+|v\left|\right)}^{2-2(1/2+\epsilon )+{\epsilon }_1}\right)}^{1/2}\hfill \\ & {\ll }_{\epsilon }{T}^{-\epsilon +{\epsilon }_1/2}+\frac{1}{\sqrt{T}}{\left|v\right|}^{1/2-\epsilon +{\epsilon }_1/2}\hfill \\ & {\ll }_{\epsilon }{T}^{-\epsilon /2}+\frac{{\left|v\right|}^{1/2}}{\sqrt{T}}.\hfill \end{array}$

Hence,

$\begin{array}{cc}\hfill I_1& {\ll }_{{\sigma }_0,K,\epsilon }{y}^{-\epsilon }\underset{-\infty }{\overset{+\infty }{\int }}{e}^{c_1\left|v\right|}\left(T^{-\epsilon /2}+\frac{{\left|v\right|}^{1/2}}{\sqrt{T}}\right) \mathrm{d}v{\ll }_{{\sigma }_0,K,\epsilon }{y}^{-\epsilon }\left(T^{-\epsilon /2}+T^{-1/2}\right)\hfill \\ & {\ll }_{{\sigma }_0,K,\epsilon }{y}^{-\epsilon }{T}^{-\epsilon /2}.\hfill \end{array}$

Passing to the estimate of $I_2$, and setting $g\left(z\right)=l_y\left(z\right)/z$, we obtain

$g^{\prime }\left(z\right)=\frac{y^z}{{\sigma }_0}\Gamma \left(\frac{z}{{\sigma }_0}\right)\left(\frac{1}{{\sigma }_0}\psi \left(\frac{z}{{\sigma }_0}\right)+logy\right)$

where $\psi \left(x\right)={\Gamma }^{\prime }\left(x\right)/\Gamma \left(x\right)$ denotes the digamma-function. Hence,

$\begin{array}{cc}\hfill R_y\left(s\right)& =g^{\prime }(1-s)+(2{\gamma }_0-log2\pi )g(1-s)\hfill \\ & =\frac{y^{1-s}}{{\sigma }_0} \Gamma \left(\frac{1-s}{{\sigma }_0}\right)\left(\frac{1}{{\sigma }_0}\psi \left(\frac{1-s}{{\sigma }_0}\right)+logy+2{\gamma }_0-log2\pi \right).\hfill \end{array}$

Thus, we find

$\begin{array}{cc}\hfill R_y(s+i\tau )& \ll y^{1-\sigma }\left|\Gamma \left(\frac{1-\sigma }{{\sigma }_0}+i\frac{t+\tau }{{\sigma }_0}\right)\right|\left(\left|\psi \left(\frac{1-\sigma }{{\sigma }_0}+i\frac{t+\tau }{{\sigma }_0}\right)\right|+logy+1\right)\hfill \\ & {\ll }_{{\sigma }_0}{y}^{1-\sigma }exp\left\{-\frac{c}{{\sigma }_0}|t+\tau |\right\}\left(log\left|\frac{t+\tau }{{\sigma }_0}\right|+logy+1\right)\hfill \end{array}$

and therefore

$\begin{array}{cc}\hfill \underset{s\in K}{sup}\left|R_y(s+i\tau )\right|& {\ll }_{{\sigma }_0,K}{y}^{1-(1/2+2\epsilon )}exp\left\{-\frac{c}{{\sigma }_0}\tau \right\}\left(log(\tau +2)+logy\right)\hfill \\ & {\ll }_{{\sigma }_0,K,\epsilon }{y}^{1/2-\epsilon }exp\{-c_2\left|\tau \right|\}.\hfill \end{array}$

Thus, we obtain

$I_2=\frac{1}{T}\underset{0}{\overset{T}{\int }}\underset{s\in K}{sup}\left|R_y(s+i\tau )\right| \mathrm{d}\tau {\ll }_{{\sigma }_0,K,\epsilon }\frac{y^{1/2-\epsilon }}{T}\underset{0}{\overset{T}{\int }}exp\{-c_2\tau \} \mathrm{d}\tau {\ll }_{{\sigma }_0,K,\epsilon }\frac{y^{1/2-\epsilon }}{T}.$

Consequently,

$\frac{1}{T}\underset{0}{\overset{T}{\int }}\underset{s\in K}{sup}\left|{\mathcal{Z}}_y(s+i\tau )-\mathcal{Z}(s+i\tau )\right| \mathrm{d}\tau =I_1+I_2{\ll }_{{\sigma }_0,K,\epsilon }{y}^{-\epsilon }{T}^{-\epsilon /2}+\frac{y^{1/2-\epsilon }}{T}.$

Tending $T\to \infty$, we find

$\underset{T\to \infty }{lim}\frac{1}{T}\underset{0}{\overset{T}{\int }}\underset{s\in K}{sup}\left|{\mathcal{Z}}_y(s+i\tau )-\mathcal{Z}(s+i\tau )\right| \mathrm{d}\tau =0.$

Then, the desired assertion follows. □

6. Proof of Theorem 2

We return to the limit measure $P_y$ in Lemma 6. We recall that $P_{T,y}\underset{T\to \infty }{\overset{W}{\to }}{P}_y$.

Lemma 9.

The family of probability measures $\{P_y:y>1\}$ is tight.

Proof. 

Let $K\subset D$ be a compact set. Then, we have

(23)$\begin{array}{cc}\hfill \frac{1}{T}\underset{0}{\overset{T}{\int }}\underset{s\in K}{sup}\left|{\mathcal{Z}}_y(s+i\tau )\right| \mathrm{d}\tau ⩽& \frac{1}{T}\underset{0}{\overset{T}{\int }}\underset{s\in K}{sup}\left|\mathcal{Z}(s+i\tau )-{\mathcal{Z}}_y(s+i\tau )\right| \mathrm{d}t\hfill \\ & +\frac{1}{T}\underset{0}{\overset{T}{\int }}\underset{s\in K}{sup}\left|\mathcal{Z}(s+i\tau )\right| \mathrm{d}\tau .\hfill \end{array}$

Let L be a simple closed contour lying in D, enclosing a compact K and such that

$\underset{z\in L}{inf}\underset{s\in K}{inf}|z-s|⩾c\left(L\right)>0.$

Then, the application of the integral Cauchy formula gives

$\left|\mathcal{Z}(s+i\tau )\right|⩽\frac{1}{2\pi }\underset{L}{\int }\frac{\left|\mathcal{Z}\right(z+i\tau \left)\right|}{|z-s|} \left|\mathrm{d}z\right|,$

and

$\underset{s\in K}{sup}\left|\mathcal{Z}(s+i\tau )\right|⩽\frac{1}{2\pi c}\underset{L}{\int }\left|\mathcal{Z}(z+i\tau )\right| \left|\mathrm{d}z\right|{\ll }_L\underset{L}{\int }\left|\mathcal{Z}(z+i\tau )\right| \left|\mathrm{d}z\right|.$

Setting $z=\sigma +it$ for $z\in L$, and using the inequality (2), we find

$\begin{array}{cc}\hfill \frac{1}{T}\underset{0}{\overset{T}{\int }}\underset{s\in K}{sup}\left|\mathcal{Z}(s+i\tau )\right| \mathrm{d}\tau & {\ll }_L\underset{L}{\int }\frac{1}{T}\underset{0}{\overset{T}{\int }}\left|\mathcal{Z}(z+i\tau )\right| \mathrm{d}\tau \left|\mathrm{d}z\right|\hfill \\ & {\ll }_L\underset{L}{\int }{\left(\frac{1}{T}\underset{0}{\overset{T}{\int }}{\left|\mathcal{Z}(z+i\tau )\right|}^2 \mathrm{d}\tau \right)}^{1/2} \left|\mathrm{d}z\right|\hfill \\ & {\ll }_L\underset{L}{\int }{\left(\frac{1}{T}\underset{0}{\overset{T+\left|t\right|+1}{\int }}{\left|\mathcal{Z}(\sigma +i\tau )\right|}^2 \mathrm{d}\tau \right)}^{1/2} \left|\mathrm{d}z\right|\hfill \\ & {\ll }_L\underset{L}{\int }{\left(\frac{1}{T}{(T+|t\left|\right)}^{2-2\sigma +{\epsilon }_1}\right)}^{1/2} \left|\mathrm{d}z\right|{\ll }_L{T}^{-\epsilon /2}.\hfill \end{array}$

Thus,

$\underset{T\to \infty }{lim\; sup}\frac{1}{T}\underset{0}{\overset{T}{\int }}\underset{s\in K}{sup}\left|\mathcal{Z}(s+i\tau )\right| \mathrm{d}t⩽C<\infty .$

Therefore, by (23) and Lemma 8,

$\underset{y⩾1}{sup}\underset{T\to \infty }{lim\; sup}\frac{1}{T}\underset{0}{\overset{T}{\int }}\underset{s\in K}{sup}\left|{\mathcal{Z}}_y(s+i\tau )\right| \mathrm{d}t⩽R<\infty .$

Fix $\epsilon >0$ and put $M_l=R{2}^l{\epsilon }^{-1}$, $l\in \mathbb{N}$. Let $Y_y\left(s\right)$ be a $H\left(D\right)$-valued random element with the distribution $P_y$. Then,

$\mathbb{P}\left(\underset{s\in K_l}{sup}\left|Y_y\left(s\right)\right|>M_l\right)⩽\underset{y⩾1}{sup}\underset{T\to \infty }{lim\; sup}\frac{1}{T{M}_l}\underset{0}{\overset{T}{\int }}\underset{s\in K_l}{sup}\left|{\mathcal{Z}}_y(s+i\tau )\right| \mathrm{d}\tau ⩽\frac{\epsilon }{2^l}.$

Hence, we have

$\mathbb{P}(Y_y\left(s\right)\in K)⩾1-\epsilon$

for all $y⩾1$, where

$K=\left\{g\in H\left(D\right):\underset{s\in K_l}{sup}\left|g\left(s\right)\right|⩽M_l, l\in \mathbb{N}\right\},$

and the lemma is proved. □

Proof of Theorem 2.

By Lemma 9 and the Prokhorov theorem, the family $\{P_y:y⩾1\}$ is relatively compact. Therefore, there exists a sequence $\left\{y_k\right\}$, $y_k\to \infty$ as $k\to \infty$ such that $\left\{P_{y_k}\right\}$ converges weakly to a certain probability measure P on $\left(H\right(D),\mathcal{B}(H\left(D\right)\left)\right)$ as $k\to \infty$. Since the distribution of $Y_{y_k}$ is $P_{y_k}$, we have

(24)$Y_{y_k}\underset{k\to \infty }{\overset{\mathcal{D}}{\to }}P.$

Define

$Z_T\left(s\right)=\mathcal{Z}(s+i{\theta }_T).$

Then, in view of Lemma 8, for every $\epsilon >0$,

$\begin{array}{cc}\hfill \underset{y\to \infty }{lim}& \underset{T\to \infty }{lim\; sup}\mathbb{P}\left(\rho (Z\left(s\right),Y_{T,Y}\left(s\right))⩾\epsilon \right)\hfill \\ & ⩽\underset{y\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\frac{1}{T\epsilon }\underset{0}{\overset{T}{\int }}\rho (\mathcal{Z}(s+i\tau ),{\mathcal{Z}}_y(s+i\tau )) \mathrm{d}\tau =0.\hfill \end{array}$

The latter equality, relations (15) and (24) show that all hypotheses of Lemma 4 are satisfied. Therefore, we obtain the relation

$Z_T\underset{T\to \infty }{\overset{\mathcal{D}}{\to }}P,$

and this is equivalent to the assertion of the theorem. □

7. Proof of Theorem 1

We derive Theorem 1 from Theorem 2 by applying properties of weak convergence of probability measures. We will approximate functions from the support of the limit measure P of $P_T$ in Theorem 2. The application of Theorem 2 is based on the following equivalents of weak convergence, see, for example, [18].

Lemma 10.

Let $P_n$, $n\in \mathbb{N}$, and P be probability measures on $(\mathcal{X},\mathcal{B}(\mathcal{X}\left)\right)$. Then, the following statements are equivalent.

• 1°. $P_n\underset{n\to \infty }{\overset{W}{\to }}P$.

• 2°. For every open set $G\subset \mathcal{X}$,

  $\underset{n\to \infty }{lim\; inf}{P}_n\left(G\right)⩾P\left(G\right).$

• 3°. For every continuity set A of the measure P (A is a continuity set of P if $P(\partial A)=0$, where $\partial A$ is a boundary of A),

  $\underset{n\to \infty }{lim}{P}_n\left(A\right)=P\left(A\right).$

Proof of Theorem 1.

Denote by F the support of the limit measure P in Theorem 2. The set F is a minimal closed subset of $H\left(D\right)$ such that $P\left(F\right)=1$. The set F consists of all elements $f\in H\left(D\right)$ such that, for every open neighborhood G of f, the inequality $P\left(G\right)>0$ is satisfied. Obviously, $F\ne ⌀$.

For $f\in F$, define

$G_{\epsilon }=\left\{g\in H\left(D\right):\underset{s\in K}{sup}|g\left(s\right)-f\left(s\right)|<\epsilon \right\}.$

Then, $G_{\epsilon }$ is an open neighborhood of the element f of the support of P. Therefore,

(25)$P\left(G_{\epsilon }\right)>0.$

Thus, by Theorem 2, and $1^{\circ }$ and $2^{\circ }$ of Lemma 10,

$\underset{T\to \infty }{lim\; inf}\frac{1}{T} \mu \left\{\tau \in [0,T]:\underset{s\in K}{sup}|\mathcal{Z}(s+i\tau )-f\left(s\right)|<\epsilon \right\}⩾P\left(G_{\epsilon }\right)>0.$

To prove the second assertion of the theorem, we notice that the boundary $\partial G_{\epsilon }$ lies in

$\left\{g\in H\left(D\right):\underset{s\in K}{sup}|g\left(s\right)-f\left(s\right)|=\epsilon \right\}.$

Hence, $\partial G_{{\epsilon }_1}\cap \partial G_{{\epsilon }_2}=⌀$ for ${\epsilon }_1\ne {\epsilon }_2$. Thus, $P(\partial G_{\epsilon })$ can be positive for at most countably many positive $\epsilon$, in other words, $G_{\epsilon }$ is a continuity set of P, except for all but at most a countable set of values $\epsilon >0$. Therefore, Theorem 2, $1^{\circ }$ and $3^{\circ }$ of Lemma 10, and (25) show that the limit

$\underset{T\to \infty }{lim}\frac{1}{T} \mu \left\{\tau \in [0,T]:\underset{s\in K}{sup}|\mathcal{Z}(s+i\tau )-f\left(s\right)|<\epsilon \right\}=P\left(G_{\epsilon }\right)>0.$

exists for all but at most countably many $\epsilon >0$. The theorem is proved. □

8. Conclusions

In this paper, we found that the shifts of the Mellin transform of the square of the Riemann zeta-function $\zeta \left(s\right)$

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

approximate a certain class F of analytic functions defined in the strip $\{s\in \mathbb{C}:1/2<\sigma <1\}$. The main ingredient of the proof of the above result is a limit theorem for the function $\mathcal{Z}\left(s\right)$ in the space of analytic functions. Note that a problem of approximation of analytic functions by shifts of the function $\mathcal{Z}\left(s\right)$ is new, and it is discussed for the first time. The main result and method are inspired by universality theorems for $\zeta \left(s\right)$. Unfortunately, the set F is not explicitly given. This is a complicated future problem. Additionally, we are planning to extend the results of the paper for Mellin transforms of other powers of $\zeta \left(s\right)$.

Author Contributions

Conceptualization, M.K. and A.L.; methodology, M.K. and A.L.; investigation, M.K. and A.L.; writing—original draft preparation, M.K. and A.L. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Data is contained within the article.

Conflicts of Interest

The authors declare no conflict of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

References

1. Motohashi, Y. A relation between the Riemann zeta-function and the hyperbolic Laplacian. Ann. Sc. Norm. Super. Pisa Cl. Sci. IV Ser.; 1995; 22, pp. 299-313.

2. Motohashi, Y. Spectral Theory of the Riemann Zeta-Function; Cambridge University Press: Cambridge, UK, 1997.

3. Ivič, A.; Jutila, M.; Motohashi, Y. The Mellin transform of powers of the zeta-function. Acta Arith.; 2000; 95, pp. 305-342. [DOI: https://dx.doi.org/10.4064/aa-95-4-305-342]

4. Ivič, A. On some conjectures and results for the Riemann zeta-function and Hecke series. Acta Arith.; 2001; 99, pp. 115-145. [DOI: https://dx.doi.org/10.4064/aa99-2-2]

5. Jutila, M. The Mellin transform of the square of Riemann’s zeta-function. Period. Math. Hung.; 2001; 42, pp. 179-190. [DOI: https://dx.doi.org/10.1023/A:1015213127383]

6. Lukkarinen, M. The Mellin Transform of the Square of Riemann’s Zeta-Function and Atkinson Formula. Ph.D. Thesis; University of Turku: Turku, Finland, 2004.

7. Mergelyan, S.N. Uniform approximations to functions of a complex variable. American Mathematical Society Translations; No. 101 American Mathematical Society: Providence, RI, USA, 1954.

8. Voronin, S.M. Theorem on the “universality” of the Riemann zeta-function. Math. USSR Izv.; 1975; 9, pp. 443-453. [DOI: https://dx.doi.org/10.1070/IM1975v009n03ABEH001485]

9. Laurinčikas, A. Limit Theorems for the Riemann Zeta-Function; Kluwer: Dordrecht, The Netherlands, Boston, MA, USA, London, UK, 1996.

10. Steuding, J. Value-Distribution of L-Functions; Lecture Notes Math. Springer: Berlin/Heidelberg, Germany, 2007; Volume 1877.

11. Gonek, S.M. Analytic Properties of Zeta and L-Functions. Ph.D. Thesis; University of Michigan: Ann Arbor, MI, USA, 1975.

12. Bagchi, B. The Statistical Behaviour and Universality Properties of the Riemann Zeta-Function and Other Allied Dirichlet Series. Ph.D. Thesis; Indian Statistical Institute: Calcutta, India, 1981.

13. Matsumoto, K. A survey on the theory of universality for zeta and L-functions. Number Theory: Plowing and Starring through High Wave Forms, Proceedings of the 7th China-Japan Seminar (Fukuoka 2013), Series on Number Theory and Its Applications, Fukuoka, Japan, 28 October–1 November 2013; Kaneko, M.; Kanemitsu, S.; Liu, J. World Scientific Publishing Co.: Hackensac, NJ, USA, London, UK, Singapore, Bejing, China, Shanghai, China, Hong Kong, China, Taipei, Taiwan, Chennai, India, 2015; pp. 95-144.

14. Bohr, H.; Jessen, B. Über die Wertwerteiling der Riemmanshen zeta funktion, Enste Mitteilung. Acta Math.; 1930; 54, pp. 1-35. [DOI: https://dx.doi.org/10.1007/BF02547516]

15. Bohr, H.; Jessen, B. Über die Wertwerteiling der Riemmanshen zeta funktion, Zweite Mitteilung. Acta Math.; 1932; 58, pp. 1-55. [DOI: https://dx.doi.org/10.1007/BF02547773]

16. Steuding, J.; Suriajaya, A.I. Value-distribution of the Riemann zeta-function along its Julia lines. Comp. Methods Funct. Theory; 2020; 20, pp. 389-401. [DOI: https://dx.doi.org/10.1007/s40315-020-00316-x]

17. Julia, G. Leçons sur les Fonctions Uniformes á Point Singulier Essentiel Isolé; Gauthier-Villars: Paris, France, 1924; FM 50, p. 254. BAMS 31, p. 59

18. Billingsley, P. Convergence of Probability Measures; Willey: New York, NY, USA, 1968.

19. Titchmarsh, E.C. The Theory of Functions; 2nd ed. Oxford University Press: Oxford, UK, 1939.