Joint Approximation by the Riemann and Hurwitz Zeta-Functions in Short Intervals

Abstract

In this study, the approximation of a pair of analytic functions defined on the strip $\{s=\sigma +it\in \mathbb{C}:1/2<\sigma <1\}$ by shifts $\left(\zeta \right(s+i\tau ),\zeta (s+i\tau ,\alpha \left)\right)$, $\tau \in \mathbb{R}$, of the Riemann and Hurwitz zeta-functions with transcendental $\alpha$ in the interval $[T,T+H]$ with $T^{27/82}⩽H⩽T^{1/2}$ was considered. It was proven that the set of such shifts has a positive density. The main result was an extension of the Mishou theorem proved for the interval $[0,T]$, and the first theorem on the joint mixed universality in short intervals. For proof, the probability approach was applied.

1. Introduction

Denote by $s=\sigma +it$ is a complex variable and $0<\alpha ⩽1$ is a fixed parameter. The Riemann and Hurwitz zeta-functions $\zeta \left(s\right)$ and $\zeta (s,\alpha )$, for $\sigma >1$, are defined by the Dirichlet series as follows:

$$\zeta \left(s\right)=\sum _{m=1}^{\infty }{\displaystyle \frac{1}{m^s}}\mathrm{and}\zeta (s,\alpha )=\sum _{m=0}^{\infty }{\displaystyle \frac{1}{{(m+\alpha )}^s}},$$

The functions have analytic continuations to the whole complex plane, except for point $s=1$, which is a simple pole with residues 1. Moreover, the function $\zeta \left(s\right)$, for $\sigma >1$, can be defined by the Euler product as follows:

$$\zeta \left(s\right)=\prod _{p\in \mathbb{P}}{\left(1-{\displaystyle \frac{1}{p^s}}\right)}^{-1},$$

where $\mathbb{P}$ is the set of all prime numbers.

The functions $\zeta \left(s\right)$ and $\zeta (s,\alpha )$ are important tools for research in the analytic number theory. The function $\zeta \left(s\right)$ is the main tool for investigating the distribution of prime numbers in the set $\mathbb{N}$, while the function $\zeta (s,\alpha )$ with rational parameter $\alpha$ is applied for studying prime numbers in arithmetical progressions. However, the range of applications of the functions $\zeta \left(s\right)$ and $\zeta (s,\alpha )$ is significantly wider than the distribution of primes. They are used also in function theory, algebraic number theory, functional analysis, probability theory, and even in quantum mechanics, cosmology, and music [1,2,3,4,5].

One of the most interesting applications of the functions $\zeta \left(s\right)$ and $\zeta (s,\alpha )$ is connected to a very important problem of the function theory—the approximation of analytic functions. At present, it is known that analytic functions defined in the strip $D=\{s\in \mathbb{C}:1/2<\sigma <1\}$ can be approximated by shifts $\zeta (s+i\tau )$ (the case of non-vanishing analytic functions) or by shifts $\zeta (s+i\tau ,\alpha )$, $\tau \in \mathbb{R}$, for some classes of the parameter $\alpha$. The latter property of zeta-functions is called universality and, for the function $\zeta \left(s\right)$, was proved by S. M. Voronin in [6,7]. The initial form of the Voronin universality theorem was improved by various authors (see [8,9,10,11,12,13,14]), but its remains the same in essence: the set $\zeta (s+i\tau )$, $\tau \in \mathbb{R}$, is dense in the space of analytic functions. For the statement of a modern version of Voronin’s theorem, the following notation is convenient. The class of compact sets of the strip D with connected complements is denoted by $\mathcal{K}$, and the class of continuous functions that are analytic in the interior of K by $H_0\left(K\right)$ with $K\in \mathcal{K}$. Moreover, let $\mathrm{meas}A$ stand for the Lebesgue measure of a measurable set $A\subset \mathbb{R}$, and

$$L_T(\dots )={\displaystyle \frac{1}{T}}\mathrm{meas}\{\tau \in [0,T]:\dots \},$$

where in place of dots, a condition satisfied by $\tau$ is to be written. Then, we have the following statement [8,9,10,11,12,13,14]:

Theorem 1.

Let $K\in \mathcal{K}$ and $f\left(s\right)\in H_0\left(K\right)$. Then, for every $\epsilon >0$, 

$$\underset{T\to \infty }{lim\; inf}{L}_T\left(\underset{s\in K}{sup}|f\left(s\right)-\zeta (s+i\tau )|<\epsilon \right)>0.$$

Moreover, the limit 

$$\underset{T\to \infty }{lim}{L}_T\left(\underset{s\in K}{sup}|f\left(s\right)-\zeta (s+i\tau )|<\epsilon \right)$$

 exists and is positive for all but at most countably many $\epsilon >0$.

The problem of the approximation of analytic functions by shifts $\zeta (s+i\tau ,\alpha )$ is more complicated and depends on the arithmetic of the parameter $\alpha$. The simplest case is of transcendental $\alpha$, i.e., when $\alpha$ is not a root of any polynomial $p\left(s\right)\not\equiv 0$ with rational coefficients. In this case, the set $\{log(m+\alpha ):m\in {\mathbb{N}}_0\}$, ${\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$, is linearly independent over $\mathbb{Q}$, and we have a certain analogy with the function $\zeta \left(s\right)$, where the linear independence of the set $\{logp:p\in \mathbb{P}\}$ is applied. The case of rational parameter $\alpha =a/q$, $(a,q)=1$, in virtue of the following representation: 

$$\zeta \left(s,{\displaystyle \frac{a}{q}}\right)={\displaystyle \frac{q^s}{\phi \left(q\right)}}\sum _{\chi \left(\mathrm{mod}q\right)}\overline{\chi }\left(a\right)L(s,\chi ),$$

where a summing runs over all Dirichlet characters modulus q, $L(s,\chi )$ denotes the Dirichlet L-functions, and $\phi \left(q\right)$ is the Euler totient function, is reduced to the simultaneous approximation of a tuple of $\phi \left(q\right)$ analytic functions by shifts $(L(s+i\tau ,{\chi }_1),\dots ,L(s+i\tau ,{\chi }_{\phi \left(q\right)}))$. More precisely, the following result by different methods was obtained in [8,9,14] (see also [12,15]). The class of continuous on K functions that are analytic in the interior of K is denoted by $H\left(K\right)$ with $K\in \mathcal{K}$.

Theorem 2.

Suppose that the parameter α is transcendental, or rational $\ne 1,1/2$. Let $K\in \mathcal{K}$ and $f\left(s\right)\in H\left(K\right)$. Then, for every  $\epsilon >0$, 

$$\underset{T\to \infty }{lim\; inf}{L}_T\left(\underset{s\in K}{sup}|f\left(s\right)-\zeta (s+i\tau ,\alpha )|<\epsilon \right)>0.$$

Moreover, the limit 

$$\underset{T\to \infty }{lim}{L}_T\left(\underset{s\in K}{sup}|f\left(s\right)-\zeta (s+i\tau ,\alpha )|<\epsilon \right)$$

 exists and is positive for all but at most countably many $\epsilon >0$.

The cases $\alpha =1$ and $\alpha =1/2$ are excluded in Theorem 2 because $\zeta (s,1)=\zeta \left(s\right)$ and

$$\zeta \left(s,{\displaystyle \frac{1}{2}}\right)=\left(2^s-1\right)\zeta \left(s\right),$$

and, for them, the statement of Theorem 1 with class $H_0\left(K\right)$ is valid.

The most complicated case is of algebraic irrational parameter $\alpha$. This case was studied in [16]. The degree of $\alpha$ is denoted by d. Let $\theta =4{\left(27{\left(4.45\right)}^2\right)}^{-1}$ and $\beta =\theta d^{-2}$. Then, in [16], the following statement was proven to be true.

Theorem 3.

Suppose that the parameter α is algebraic irrational. Let $\gamma \in (0,\beta )$, $1-\beta +\gamma ⩽{\sigma }_0⩽1$, $s_0={\sigma }_0+i{t}_0$, and $f\left(s\right)$ be continuous functions on $|s-s_0|⩽r$, $r>0$ and analytic in the interior of that disc. Moreover, let $0<a<1$ and $\epsilon \in (0,|f\left(s_0\right)\left|\right)$. Then, for all but finitely many $\alpha \in [a,1]$, of degree at most $d_0-2{\theta }_1/d_0^2+\gamma$ with

$$d_0⩽{\left({\displaystyle \frac{\theta }{1-{\sigma }_0+\gamma }}\right)}^{1/2},$$

there exist $\tau \in [T,2T]$ and $\delta =\delta (\epsilon ,f,T)>0$ such that

$$\underset{|s-s_0|⩽\delta r}{max}|f\left(s\right)-\zeta (s+i\tau ,\alpha )|<3\epsilon ,$$

where $T=T(\epsilon ,f,\alpha )$ is explicitly given, the set of exceptional α is effectively described, and δ is also effectively computable.

Theorems 1–3 are devoted to the approximation of one function from a wide class of analytic functions. Also, there are the so-called joint universality theorems in which a tuple of analytic functions is approximated simultaneously by shifts of zeta-functions. The first joint universality result can also be found in Voronin [17] and deals with Dirichlet L-functions with pairwise non-equivalent characters (see also [9,18,19]). A joint universality theorem for a pair of Hurwitz zeta-functions was given in [20]. The joint approximation of analytic functions by shifts of Hurwitz zeta- functions involving imaginary parts of non-trivial zeros of the Riemann zeta-function was discussed in [21]. However, later, many joint universality theorems were obtained for functions of the same name (for more results, see [12]). For illustration purposes, we present one example. For $j=1,\dots ,r$, let $0<{\alpha }_j⩽1$, and

$$L({\alpha }_1,\dots ,{\alpha }_r)=\{log(m+{\alpha }_j):m\in {\mathbb{N}}_0,j=1,\dots ,r\}.$$

Theorem 4.

([15]). Suppose that the set $L({\alpha }_1,\dots ,{\alpha }_r)$ is linearly independent over $\mathbb{Q}$. For $j=1,\dots ,r$, let $K_j\in \mathcal{K}$ and $f_j\left(s\right)\in H\left(K_j\right)$. Then, for every $\epsilon >0$,

$$\underset{T\to \infty }{lim\; inf}{L}_T\left(\underset{1⩽j⩽r}{sup}\underset{s\in K_j}{sup}|f_j\left(s\right)-\zeta (s+i\tau ,{\alpha }_j)|<\epsilon \right)>0.$$

Also, some problems of joint universality for Hurwitz zeta-functions can be found in [22].

In [23], H. Mishou initiated a new type of joint mixed universality theorems; he proved a joint universality theorem for two functions of different types, for the Riemann zeta-function and Hurwitz zeta-function. Here, it is important to stress that $\zeta \left(s\right)$ has the Euler product, while $\zeta (s,\alpha )$ has no such a product for $\alpha \ne 1$ and $\alpha \ne 1/2$. Moreover, the function $\zeta \left(s\right)$ satisfies the symmetric functional equation

$$\xi \left(s\right)=\xi (1-s),s\in \mathbb{C},\xi \left(s\right)={\pi }^{-s/2}\mathrm{\Gamma }\left({\displaystyle \frac{s}{2}}\right)\zeta \left(s\right),$$

where $\mathrm{\Gamma }\left(s\right)$ is the Euler gamma-function, while, for $\zeta (s,\alpha )$, the following non-symmetric equations connecting s and $1-s$ are true:

$$\zeta (1-s,\alpha )={\displaystyle \frac{\mathrm{\Gamma }\left(s\right)}{{\left(2\pi \right)}^s}}\left({\mathrm{e}}^{-\pi is/2}\sum _{m=1}^{\infty }{\displaystyle \frac{{\mathrm{e}}^{2\pi im\alpha }}{m^s}}+{\mathrm{e}}^{\pi is/2}\sum _{m=1}^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-2\pi im\alpha }}{m^s}}\right),\sigma >1,$$

or

$$\zeta (s,\alpha )={\displaystyle \frac{2\mathrm{\Gamma }(1-s)}{{\left(2\pi \right)}^{1-s}}}\left(sin{\displaystyle \frac{\pi s}{2}}\sum _{m=1}^{\infty }{\displaystyle \frac{cos\left(2\pi m\alpha \right)}{m^{1-s}}}+cos{\displaystyle \frac{\pi s}{2}}\sum _{m=1}^{\infty }{\displaystyle \frac{sin\left(2\pi m\alpha \right)}{m^{1-s}}}\right),\sigma <0.$$

This is one of the causes of differences in the value distribution of $\zeta \left(s\right)$ and $\zeta (s,\alpha )$ and also reflects the approximate functional equation for $\zeta (s,\alpha )$, which is the main ingredient for the proof of the mean square estimate in short intervals [24].

Theorem 5.

([23]). Suppose that the parameter α is transcendental. Let $K_1,K_2\in \mathcal{K}$ and $f_1\left(s\right)\in H_0\left(K_1\right)$, $f_2\left(s\right)\in H\left(K_2\right)$. Then, for every $\epsilon >0$,

$$\begin{array}{c}\hfill \underset{T\to \infty }{lim\; inf}{L}_T\left(\underset{s\in K_1}{sup}|f_1\left(s\right)-\zeta (s+i\tau )|<\epsilon ,\underset{s\in K_2}{sup}|f_2\left(s\right)-\zeta (s+i\tau ,\alpha )|<\epsilon \right)>0.\end{array}$$

The thesis [25] is devoted to joint discrete universality for the Riemann and Hurwitz zeta-functions. Mixed joint universality is studied also for more general zeta-functions. We mention the works [26,27,28,29,30,31]. The weighted versions of the Mishou theorem are proven in [32]. Theorems 1, 2, 4, and 5 have one common shortcoming: they imply that the set of approximating shifts is infinite; however, they do not provide any algorithm to find at least one approximating shift. In this sense, these theorems are ineffective. Of course, it is difficult to discuss concrete approximation shifts; however, some additional information on the efficacy of universality theorems is always useful. In Theorem 3, the efficacy of approximation is described by indication of explicitly given interval $[T,2T]$ containing $\tau$ such that $\zeta (s+i\tau ,\alpha )$ is an approximating shift. This is a very good step in the effectivization direction.

In contrast to Theorem 3, the proofs of Theorems 1, 2, 4, and 5 are based on measure theory; thus, it is impossible to find an explicitly given interval containing $\tau$ with the approximation property. Therefore, there is another method to consider approximating shifts with $\tau$ in the interval of lengths shorter than T or, more precisely, $o\left(T\right)$ as $T\to \infty$. This method leads to universality theorems in short intervals. For the function $\zeta \left(s\right)$, the first universality theorem of such a type was obtained in [33]. Let

$$L_{T,H}(\dots )={\displaystyle \frac{1}{H}}\mathrm{meas}\{\tau \in [T,T+H]:\dots \},$$

where in place of dots, a condition satisfied by $\tau$ is to be written.

Theorem 6.

([33]). Suppose that $T^{1/3}{(logT)}^{26/15}⩽H⩽T$. Let $K\in \mathcal{K}$, $f\left(s\right)\in H_0\left(K\right)$. Then, for every $\epsilon >0$,

$$\underset{T\to \infty }{lim\; inf}{L}_{T,H}\left(\underset{s\in K}{sup}|f\left(s\right)-\zeta (s+i\tau )|<\epsilon \right)>0.$$

Moreover, “lim inf” can be replaced by “lim” for all but at most countably many $\epsilon >0$.

Recently, improvements in Theorem 6 were given in [34].

An analog of Theorem 6 for the Hurwitz zeta-function is given in [35].

Theorem 7.

([35]). Suppose that $T^{27/82}⩽H⩽T^{1/2}$, and the parameter α is transcendental. Let $K\in \mathcal{K}$ and $f\left(s\right)\in H\left(K\right)$. Then, for every $\epsilon >0$,

$$\underset{T\to \infty }{lim\; inf}{L}_{T,H}\left(\underset{s\in K}{sup}|f\left(s\right)-\zeta (s+i\tau ,\alpha )|<\epsilon \right)>0.$$

Moreover, the lower limit can be replaced by a limit for all but at most countably many $\epsilon >0$.

The aim of this study is to obtain a version of Theorem 5 in short intervals.

Theorem 8.

Suppose that $T^{27/82}⩽H⩽T^{1/2}$, and the parameter α is transcendental. Let $K_1,K_2\in \mathcal{K}$ and $f_1\left(s\right)\in H_0\left(K_1\right)$, $f_2\left(s\right)\in H\left(K_2\right)$. Then, for every $\epsilon >0$,

$$\underset{T\to \infty }{lim\; inf}{L}_{T,H}\left(\underset{s\in K_1}{sup}|f_1\left(s\right)-\zeta (s+i\tau )|<\epsilon ,\underset{s\in K_2}{sup}|f_2\left(s\right)-\zeta (s+i\tau ,\alpha )|<\epsilon \right)>0.$$

Moreover, the lower limit can be replaced by a limit for all but at most countably many $\epsilon >0$.

Using short intervals extends and improves the Mishou theorem on joint mixed universality for the functions $\zeta \left(s\right)$ and $\zeta (s,\alpha )$ and is the novel approach presented in this article.

For effectivization aims of approximation, the quantity of H must be as small as possible. On the other hand, H is closely connected to a very important but complicated problem of analytic number theory on the mean square estimates of the functions $\zeta \left(s\right)$ and $\zeta (s,\alpha )$ in short intervals. Unfortunately, at present, we only have a result of $H⩾T^{27/82}$ in the latter problem (see Lemmas 2 and 3 below).

Mean square estimates together with a joint probabilistic limit theorem for the pair $\left(\zeta \right(s),\zeta (s,\alpha \left)\right)$ in the space of analytic functions occupy a central place in the proof of Theorem 8 in short intervals for the functions $\zeta \left(s\right)$ and $\zeta (s,\alpha )$.

2. Mean Square Estimates

The first results for the Riemann zeta-function in short intervals were obtained by D. R. Heath-Brown, J.-M. Deshouillers, A. Ivič, H. Iwaniec, M. Jutila, A. A. Karatsuba, G. Kolesnik (for references, see [36]). We recall one mean square estimate from [36].

Lemma 1.

Let $(\kappa ,\lambda )$ be an exponential pair and $1/2<\sigma <1$ fixed. Then, for $T^{(\kappa +\lambda +1-2\sigma )/2(\kappa +1)}\times {(logT)}^{(2+\kappa )/(\kappa +1)}⩽H⩽T$, $1+\lambda -\kappa ⩾2\sigma$, we have uniformly in H

$${\int }_{T-H}^{T+H}{\left|\zeta (\sigma +it)\right|}^2\mathrm{d}t\ll H.$$

Lemma 2.

Suppose that $\alpha \ne 1/2,1$, and $1/2<\sigma ⩽7/12$ is fixed. Then, for $T^{27/82}⩽H⩽T$, uniformly in H

$${\int }_{T-H}^{T+H}{\left|\zeta (\sigma +it)\right|}^2\mathrm{d}t{\ll }_{\sigma }H.$$

Proof.  

The lemma follows from Lemma 1 by taking the exponential pair $(11/30,16/30)$. □

Lemma 3.

Suppose that $\alpha \ne 1/2,1$, and $1/2<\sigma ⩽7/12$ is fixed. Then, for $T^{27/82}⩽H⩽T^{\sigma }$, uniformly in H

$${\int }_{T-H}^{T+H}{\left|\zeta (\sigma +it,\alpha )\right|}^2\mathrm{d}t{\ll }_{\sigma ,\alpha }H.$$

Proof.  

The lemma is Theorem 2 from [24], where its proof is presented. □

Let $\theta >1/2$ be fixed, and, for $n\in \mathbb{N}$,

$$v_n\left(m\right)=exp\left\{-{\left({\displaystyle \frac{m}{n}}\right)}^{\theta }\right\}.$$

Define the series

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

which absolutely converges in any half-plane $\sigma >{\sigma }_0$ with finite ${\sigma }_0$.

Lemma 4.

Suppose that $K\subset D$ is a compact set and $T^{27/82}⩽H⩽T$. Then

$$\underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}{\displaystyle \frac{1}{H}}{\int }_T^{T+H}\underset{s\in K}{sup}|\zeta (s+i\tau )-{\zeta }_n(s+i\tau )|\mathrm{d}\tau =0.$$

Proof.  

Let

$$l_n\left(s\right)={\displaystyle \frac{1}{\theta }}\mathrm{\Gamma }\left({\displaystyle \frac{s}{\theta }}\right)n^s.$$

We use the following integral representation [10]:

$${\zeta }_n\left(s\right)={\displaystyle \frac{1}{2\pi i}}{\int }_{\theta -i\infty }^{\theta +i\infty }\zeta (s+z)l_n\left(z\right)\mathrm{d}z$$

(1)

which is a result of the classical Mellin formula that yields

$$v_n\left(m\right)={\displaystyle \frac{1}{2\pi i}}{\int }_{\theta -i\infty }^{\theta +i\infty }{\displaystyle \frac{1}{\theta }}\mathrm{\Gamma }\left({\displaystyle \frac{s}{\theta }}\right){\left({\displaystyle \frac{m}{n}}\right)}^{-s}\mathrm{d}s.$$

Let $K\subset D$ be a fixed compact set. Then, K is closed and bounded; hence, there exists a positive number $\delta <1/12$ such that $1/2+2\delta ⩽\sigma ⩽1-\delta$ for all $s=\sigma +it\in K$. We take $\theta =1/2+\delta$ and $\widehat{\theta }=1/2+\delta -\sigma$. Then, $\widehat{\theta }<0$ and $\widehat{\theta }⩾1/2+\delta -1+\delta =2\delta -1/2>-1/2-\delta =-\theta$. The integrand in (1) has only two simple poles in the strip $\widehat{\theta }<\mathrm{Re}z<\theta$, i.e., a pole at the point $z=0$ of the function $\mathrm{\Gamma }(s/\theta )$ and a pole at the point $z=1-s$ of the function $\zeta (s+z)$. Therefore, using the well-known estimate

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

(2)

which is uniform in $\sigma \in ({\sigma }_1,{\sigma }_2)$ with every ${\sigma }_1<{\sigma }_2$, and replacing $\theta$ by $\widehat{\theta }$ in the line of integration in (1), via the residue theorem, we obtain, for $s\in K$,

$${\zeta }_n\left(s\right)-\zeta \left(s\right)={\displaystyle \frac{1}{2\pi i}}{\int }_{\widehat{\theta }-i\infty }^{\widehat{\theta }+i\infty }\zeta (s+z)l_n\left(z\right)\mathrm{d}z+l_n(1-s).$$

This gives, for $s\in K$,

$${\zeta }_n\left(s\right)-\zeta \left(s\right)={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }\zeta \left(\sigma +it+{\displaystyle \frac{1}{2}}+\delta -\sigma +iu\right)l_n\left({\displaystyle \frac{1}{2}}+\delta -\sigma +iu\right)\mathrm{d}u+l_n(1-s).$$

Hence, for $s\in K$,

$$\begin{array}{cc}\hfill {\zeta }_n(s+i\tau )-\zeta (s+i\tau )\ll & {\int }_{-\infty }^{\infty }\left|\zeta \left({\displaystyle \frac{1}{2}}+\delta +it+i\tau +iu\right)\right|\left|l_n\left({\displaystyle \frac{1}{2}}+\delta -\sigma +iu\right)\right|\mathrm{d}u\hfill \\ & +\underset{s\in K}{sup}\left|l_n(1-s-i\tau )\right|,\hfill \end{array}$$

and, after change $t+u$ by u, we obtain

$$\begin{array}{cc}\hfill \underset{s\in K}{sup}\left|{\zeta }_n(s+i\tau )-\zeta (s+i\tau )\right|\ll & {\int }_{-\infty }^{\infty }\left|\zeta \left({\displaystyle \frac{1}{2}}+\delta +i\tau +iu\right)\right|\underset{s\in K}{sup}\left|l_n\left({\displaystyle \frac{1}{2}}+\delta -s+iu\right)\right|\mathrm{d}u\hfill \\ & +\underset{s\in K}{sup}\left|l_n(1-s-i\tau )\right|.\hfill \end{array}$$

(3)

In view of (2), we have, for $s\in K$,

$$l_n\left({\displaystyle \frac{1}{2}}+\delta -s+iu\right){\ll }_{\delta }{n}^{1/2+\delta -\sigma }exp\left\{-{\displaystyle \frac{c}{\theta }}|u-t|\right\}{\ll }_{\delta ,K}{n}^{-\delta }exp\{-c_1\left|u\right|\}$$

(4)

with $c_1>0$. Moreover, it is well known that, for $\sigma ⩾1/2$,

$$\zeta (\sigma +it)\ll t^{1-\sigma },t⩾2.$$

This and (4) imply that

$$\begin{array}{cc}\hfill \left({\int }_{-\infty }^{-{log}^{2}T}+{\int }_{{log}^{2}T}^{\infty }\right)& \left|\zeta \left({\displaystyle \frac{1}{2}}+\delta +i\tau +iu\right)\right|\underset{s\in K}{sup}\left|l_n\left({\displaystyle \frac{1}{2}}+\delta -s+iu\right)\right|\mathrm{d}u\hfill \\ & {\ll }_{\delta ,K}{n}^{-\delta }\left({\int }_{-\infty }^{-{log}^{2}T}+{\int }_{{log}^{2}T}^{\infty }\right)\left(\right|\tau |+{\left|u\right|)}^{1/2}exp\{-c_1\left|u\right|\}\mathrm{d}u\hfill \\ & {\ll }_{\delta ,K}{n}^{-\delta }{\left(\right|\tau |+1)}^{1/2}exp\{-c_2{log}^{2}T\},c_2>0.\hfill \end{array}$$

Therefore, via (3), we find

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{H}}& {\int }_T^{T+H}\underset{s\in K}{sup}|\zeta (s+i\tau )-{\zeta }_n(s+i\tau )|\mathrm{d}\tau \hfill \\ & {\ll }_{\delta ,K}{\int }_{-{log}^{2}T}^{{log}^{2}T}\left({\displaystyle \frac{1}{H}}{\int }_T^{T+H}\left|\zeta \left({\displaystyle \frac{1}{2}}+\delta +i\tau +iu\right)\right|\mathrm{d}\tau \right)\underset{s\in K}{sup}\left|l_n\left({\displaystyle \frac{1}{2}}+\delta -s+iu\right)\right|\mathrm{d}u\hfill \\ & +{\displaystyle \frac{1}{H}}{\int }_T^{T+H}\underset{s\in K}{sup}|l_n(1-s-i\tau )|\mathrm{d}\tau +{\displaystyle \frac{n^{-\delta }exp\{-c_2{log}^{2}T\}}{H}}{\int }_T^{T+H}{\left(\right|\tau |+1)}^{1/2}\mathrm{d}\tau \hfill \\ & \stackrel{\mathrm{def}}{=}{J}_1+J_2+J_3.\hfill \end{array}$$

(5)

Using the Cauchy–Schwarz inequality gives

$$\begin{array}{cc}\hfill {\int }_T^{T+H}\left|\zeta \left({\displaystyle \frac{1}{2}}+\delta +i\tau +iu\right)\right|\mathrm{d}\tau & \ll {\left({\int }_T^{T+H}{\left|\zeta \left({\displaystyle \frac{1}{2}}+\delta +i\tau +iu\right)\right|}^2\mathrm{d}\tau \right)}^{1/2}\hfill \\ & \ll {\left({\displaystyle \frac{1}{H}}{\int }_{T-H-\left|u\right|}^{T+H+\left|u\right|}{\left|\zeta \left({\displaystyle \frac{1}{2}}+\delta +i\tau \right)\right|}^2\mathrm{d}\tau \right)}^{1/2}.\hfill \end{array}$$

(6)

For $\left|u\right|⩽{log}^{2}T$ and large T,

$$H+\left|u\right|⩽T^{1/2}+{log}^{2}T⩽T.$$

Therefore, (6) and Lemma 2 show that, for $\left|u\right|⩽{log}^{2}T$,

$${\displaystyle \frac{1}{H}}{\int }_T^{T+H}\left|\zeta \left({\displaystyle \frac{1}{2}}+\delta +i\tau +iu\right)\right|\mathrm{d}\tau {\ll }_{\delta }{\left({\displaystyle \frac{1}{H}}(H+|u\left|\right)\right)}^{1/2}{\ll }_{\delta }{(1+|u\left|\right)}^{1/2}.$$

This and (4) give the estimate

$$J_1{\ll }_{\delta ,K}{n}^{-\delta }{\int }_{-{log}^{2}T}^{{log}^{2}T}exp\{-c_1\left|u\right|\left\}\right(1+{\left|u\right|)}^{1/2}\mathrm{d}u{\ll }_{\delta ,K}{u}^{-\delta }.$$

(7)

Similarly to (4), we obtain that, for $s\in K$,

$$l_n(1-s-i\tau )\ll n^{1-\sigma }exp\{-c_1|t+\tau |\}{\ll }_K{n}^{1/2-2\delta }exp\{-c_3\left|\tau \right|\},c_3>0.$$

Thus,

$$J_2{\ll }_K{n}^{1/2-2\delta }{\displaystyle \frac{1}{H}}{\int }_T^{T+H}exp\{-c_3\left|\tau \right|\}\mathrm{d}\tau {\ll }_K{\displaystyle \frac{n^{1/2-2\delta }}{H}}.$$

(8)

It is easily seen that

$$J_3\ll n^{-\delta }exp\{-c_2{log}^{2}T\}{T}^{1/2}.$$

This, (5), (7), and (8) lead to the following estimate:

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{H}}{\int }_T^{T+H}\underset{s\in K}{sup}|\zeta (s+i\tau )-{\zeta }_n(s+i\tau )|\mathrm{d}\tau {\ll }_{\delta ,K}& n^{-\delta }+n^{1/2-2\delta }{H}^{-1}\hfill \\ & +n^{-\delta }exp\{-c_2{log}^{2}T\}{T}^{1/2}.\hfill \end{array}$$

Taking $T\to \infty$, and then $n\to \infty$, gives the equality of the lemma. □

Recall a metric in $H\left(D\right)$, inducing its topology [37]. There exists a sequence $\left\{K_l\right\}$ of embedded compact sets lying in D such that

$$\underset{l=1}{\overset{\infty }{\cup }}{K}_l=D,$$

and every compact set $K\subset D$ lies in some set $K_l$. Then, for $g_1,g_2\in H\left(D\right)$, denoting

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

we have the metric $\rho$ that induces the topology of $H\left(D\right)$.

The latter formula with Lemma 4 yields the following statement.

Lemma 5.

Suppose that $T^{27/82}⩽H⩽T$. Then the equality

$$\underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}{\displaystyle \frac{1}{H}}{\int }_T^{T+H}\rho \left(\zeta (s+i\tau ),{\zeta }_n(s+i\tau )\right)\mathrm{d}\tau =0$$

holds.

A similar lemma for the Hurwitz zeta-function was obtained in [35]. For the same $\theta$ as above, define

$$v_n(m,\alpha )=exp\left\{-{\left({\displaystyle \frac{m+\alpha }{n}}\right)}^{\theta }\right\}.$$

and

$${\zeta }_n(s,\alpha )=\sum _{m=0}^{\infty }{\displaystyle \frac{v_n(m,\alpha )}{{(m+\alpha )}^s}}$$

Then, the latter series, as ${\zeta }_n\left(s\right)$, is absolutely convergent for $\sigma >{\sigma }_0$, with arbitrary finite ${\sigma }_0$.

Lemma 6.

Suppose that $T^{27/82}⩽H⩽T^{1/2}$, and $\alpha \ne 1/2$ or 1. Then

$$\underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}{\displaystyle \frac{1}{H}}{\int }_T^{T+H}\rho \left(\zeta (s+i\tau ,\alpha ),{\zeta }_n(s+i\tau ,\alpha )\right)\mathrm{d}\tau =0.$$

Proof.  

The lemma is Lemma 10 from [35], where its proof is given. □

For

$${\underline{g}}_k=(g_{k1},g_{k2})\in H^2\left(D\right),k=1,2,$$

set

$${\rho }_2({\underline{g}}_1,{\underline{g}}_2)=max\left(\rho (g_{11},g_{12}),\rho (g_{21},g_{22})\right).$$

Then ${\rho }_2$ is a metric that induces the topology of $H^2\left(D\right)$. This definition of ${\rho }_2$ and Lemmas 5 and 6 imply the following lemma. For brevity, let

$$\underline{\zeta }(s,\alpha )=\left(\zeta \left(s\right),\zeta (s,\alpha )\right)$$

and

$${\underline{\zeta }}_n(s,\alpha )=\left({\zeta }_n\left(s\right),{\zeta }_n(s,\alpha )\right).$$

Lemma 7.

Suppose that $T^{27/82}⩽H⩽T^{1/2}$, and $\alpha \ne 1/2$ or 1. Then

$$\underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}{\displaystyle \frac{1}{H}}{\int }_T^{T+H}{\rho }_2\left(\underline{\zeta }(s+i\tau ,\alpha ),{\underline{\zeta }}_n(s+i\tau ,\alpha )\right)\mathrm{d}\tau =0.$$

3. Limit Theorem

In this section, we will consider the weak convergence for

$$P_{T,H,\alpha }\left(A\right)=L_{T,H}\left(\underline{\zeta }(s+i\tau ,\alpha )\in A\right),A\in \mathcal{B}\left(H^2\left(D\right)\right),$$

as $T\to \infty$, with H restricted in Lemma 7, and $\mathcal{B}\left(\mathbb{X}\right)$ denotes the Borel $\sigma$-field of the space $\mathbb{X}$.

We start with the weak convergence of probability measures on a certain topological group. Let

$${\mathsf{\Omega }}_1=\prod _{p\in \mathbb{P}}\{s\in \mathbb{C}:|s|=1\},{\mathsf{\Omega }}_2=\prod _{m\in {\mathbb{N}}_0}\{s\in \mathbb{C}:|s|=1\},\mathrm{and}\mathsf{\Omega }={\mathsf{\Omega }}_1\times {\mathsf{\Omega }}_2.$$

Since ${\mathsf{\Omega }}_1$ and ${\mathsf{\Omega }}_2$ with the product topology and pointwise multiplication are compact topological groups, the Tikhonov theorem implies that $\mathsf{\Omega }$ is again a compact topological group. Thus, on $({\mathsf{\Omega }}_1,\mathcal{B}\left({\mathsf{\Omega }}_1\right))$, $({\mathsf{\Omega }}_2,\mathcal{B}\left({\mathsf{\Omega }}_2\right))$, and $(\mathsf{\Omega },\mathcal{B}(\mathsf{\Omega }\left)\right)$, the probability Haar measures $m_{1H}$, $m_{2H}$, and $m_H$, respectively, can be defined. We notice that $m_H=m_{1H}\times m_{2H}$, i. e., if $A=A_1\times A_2$, $A_1\in \mathcal{B}\left({\mathsf{\Omega }}_1\right)$, $A_2\in \mathcal{B}\left({\mathsf{\Omega }}_2\right)$, then

$$m_H\left(A\right)=m_{1H}\left(A_1\right)m_{2H}\left(A_2\right).$$

For $\omega \in \mathsf{\Omega }$, we have $\omega =({\omega }_1,{\omega }_2)$ with ${\omega }_1=({\omega }_1\left(p\right):p\in \mathbb{P})$ and ${\omega }_2=({\omega }_2\left(m\right):m\in {\mathbb{N}}_0)$.

For $A\in \mathcal{B}\left(\mathsf{\Omega }\right)$, set

$$P_{T,H,\alpha }^{\mathsf{\Omega }}\left(A\right)=L_{T,H}\left(\left(\left(p^{-i\tau }:p\in \mathbb{P}\right),\left({(m+\alpha )}^{-i\tau }:m\in {\mathbb{N}}_0\right)\right)\in A\right).$$

Lemma 8.

Suppose that $T^{27/82}⩽H⩽T^{1/2}$, and α is transcendental. Then, $P_{T,H,\alpha }^{\mathsf{\Omega }}$ converges weakly to the Haar measure $m_H$ as $T\to \infty$.

Proof.  

We use similar arguments as in the case $\tau \in [0,T]$. Let $F_{T,H,\alpha }({\underline{k}}_1,{\underline{k}}_2)$, ${\underline{k}}_1=(k_{1p}:k_{1p}\in \mathbb{Z},p\in \mathbb{P})$, ${\underline{k}}_2=(k_{2m}:k_{2m}\in \mathbb{Z},m\in {\mathbb{N}}_0)$, be the Fourier transform of $P_{T,H,\alpha }^{\mathsf{\Omega }}$, i. e.,

$$F_{T,H,\alpha }({\underline{k}}_1,{\underline{k}}_2)={\int }_{\mathsf{\Omega }}\underset{p\in \mathbb{P}}{{\prod }^{*}}{\omega }^{k_{1p}}\left(p\right)\underset{m\in {\mathbb{N}}_0}{{\prod }^{*}}{\omega }^{k_{2m}}\left(m\right)\mathrm{d}{P}_{T,H,\alpha }^{\mathsf{\Omega }},$$

where the star ∗ shows that only a finite number of integers $k_{1p}$ and $k_{2m}$ are not zeroes. Thus, taking into account the definition of the measure $P_{T,H,\alpha }^{\mathsf{\Omega }}$, we have

$$\begin{array}{cc}\hfill F_{T,H,\alpha }({\underline{k}}_1,{\underline{k}}_2)& ={\displaystyle \frac{1}{H}}{\int }_T^{T+H}\underset{p\in \mathbb{P}}{{\prod }^{*}}{p}^{-i\tau k_{1p}}\underset{m\in {\mathbb{N}}_0}{{\prod }^{*}}{(m+\alpha )}^{-i\tau k_{2m}}\mathrm{d}\tau \hfill \\ & ={\displaystyle \frac{1}{H}}{\int }_T^{T+H}exp\left\{-i\tau \left(\underset{p\in \mathbb{P}}{{\sum }^{*}}{k}_{1p}logp+\underset{m\in {\mathbb{N}}_0}{{\sum }^{*}}{k}_{2m}log(m+\alpha )\right)\right\}\mathrm{d}\tau .\hfill \end{array}$$

(9)

We have to show that

$$\underset{T\to \infty }{lim}{F}_{T,H,\alpha }({\underline{k}}_1,{\underline{k}}_2)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}({\underline{k}}_1,{\underline{k}}_2)=(\underline{0},\underline{0}),\hfill \\ 0\hfill & \mathrm{if}({\underline{k}}_1,{\underline{k}}_2)\ne (\underline{0},\underline{0}),\hfill \end{array}\right.$$

(10)

where $\underline{0}=(0,0,\dots )$. Obviously, by (9),

$$F_{T,H,\alpha }(\underline{0},\underline{0})=1.$$

(11)

Therefore, only the case $({\underline{k}}_1,{\underline{k}}_2)\ne (\underline{0},\underline{0})$ remains for consideration. Since $\alpha$ is transcendental, the set $L\left(\alpha \right)=\{log(m+\alpha ):m\in {\mathbb{N}}_0\}$ is linearly independent over $\mathbb{Q}$. The set $\{logp:p\in \mathbb{P}\}$ is also linearly independent over $\mathbb{Q}$. The linear independence over $\mathbb{Q}$ for the set $L(\mathbb{P},\alpha )\stackrel{\mathrm{def}}{=}\{(logp:p\in \mathbb{P}),(log(m+\alpha ):m\in {\mathbb{N}}_0)\}$ is easily seen. Actually, if, for some non-zeroes $k_{1p_1},\dots ,k_{1p_r},k_{2m_1},\dots ,k_{2m_v}$,

$$k_{1p_1}log{p}_1+\dots k_{1p_r}log{p}_r+k_{2m_1}log(m_1+\alpha )+\dots k_{2m_v}log(m_v+\alpha )=0,$$

then

$${(m_1+\alpha )}^{k_{21}}\dots {(m_v+\alpha )}^{k_{2v}}=p_1^{-k_{11}}\dots p_r^{-k_{1r}}.$$

From this, it follows that there exists a polynomial $p\left(s\right)$ with rational coefficients such that $p\left(\alpha \right)=0$, and this contradicts the transcendence of $\alpha$.

The linear independence of the set $L(\mathbb{P},\alpha )$ shows that, in the case $({\underline{k}}_1,{\underline{k}}_2)\ne (\underline{0},\underline{0})$,

$$A({\underline{k}}_1,{\underline{k}}_2)\stackrel{\mathrm{def}}{=}\underset{p\in \mathbb{P}}{{\sum }^{*}}{k}_{1p}logp+\underset{m\in {\mathbb{N}}_0}{{\sum }^{*}}{k}_{2m}log(m+\alpha )\ne 0.$$

Hence, in view of (9),

$$F_{T,H,\alpha }({\underline{k}}_1,{\underline{k}}_2)={\displaystyle \frac{exp\{-iTA({\underline{k}}_1,{\underline{k}}_2)\}-exp\{-i(T+H)A({\underline{k}}_1,{\underline{k}}_2)\}}{iHA({\underline{k}}_1,{\underline{k}}_2)}}.$$

Thus, for $({\underline{k}}_1,{\underline{k}}_2)\ne (\underline{0},\underline{0})$,

$$\underset{T\to \infty }{lim}{F}_{T,H,\alpha }({\underline{k}}_1,{\underline{k}}_2)=0,$$

and with (11), we obtain (10). The lemma is proven to be true. □

Now, we are in position to consider the weak convergence for

$$P_{T,H,n,\alpha }\left(A\right)\stackrel{\mathrm{def}}{=}{L}_{T,H}\left({\underline{\zeta }}_n(s+i\tau ,\alpha )\in A\right),A\in \mathcal{B}\left(H^2\left(D\right)\right).$$

For this, define $u_{n,\alpha }:\mathsf{\Omega }\in H^2\left(D\right)$ by

$$u_{n,\alpha }\left(\omega \right)={\underline{\zeta }}_n(s,\omega ,\alpha ),\omega \in \mathsf{\Omega },$$

where

$${\underline{\zeta }}_n(s,\omega ,\alpha )=\left({\zeta }_n(s,{\omega }_1),{\zeta }_n(s,{\omega }_2,\alpha )\right),$$

$${\zeta }_n(s,{\omega }_1)=\sum _{m=1}^{\infty }{\displaystyle \frac{{\omega }_1\left(m\right),v_n\left(m\right)}{m^s}},{\omega }_1\left(m\right)=\underset{p^{l+1}\nmid m}{\prod _{p^l\mid m}}{\omega }_1^l\left(p\right),$$

and

$${\zeta }_n(s,{\omega }_2,\alpha )=\sum _{m=0}^{\infty }{\displaystyle \frac{{\omega }_2\left(m\right)v_n(m,\alpha )}{{(m+\alpha )}^s}}.$$

Since the series for ${\zeta }_n(s,{\omega }_1)$ and ${\zeta }_n(s,{\omega }_2,\alpha )$ are absolutely convergent in every half-plane $\sigma >{\sigma }_0$, the mapping $u_{n,\alpha }$ is continuous; hence, $(\mathcal{B}\left(\mathsf{\Omega }\right),\mathcal{B}\left(H^2\left(D\right)\right))$-measurable. Therefore, the measure $m_H$ defines, on $(H^2\left(D\right),\mathcal{B}\left(H^2\left(D\right)\right))$, the probability measure $m_H{u}_{n,\alpha }^{-1}$ by

$$m_H{u}_{n,\alpha }^{-1}\left(A\right)=m_H\left(u_{n,\alpha }^{-1}A\right),A\in \mathcal{B}\left(H^2\left(D\right)\right).$$

For brevity, let $U_{n,\alpha }=m_H{u}_{n,\alpha }^{-1}$.

Lemma 9.

Suppose that $T^{27/82}⩽H⩽T^{1/2}$, and α is transcendental. Then, $P_{T,H,n,\alpha }$ converges weakly to $U_{n,\alpha }$ as $T\to \infty$.

Proof.  

The definition of $u_{n,\alpha }$ yields

$$u_{n,\alpha }\left(\left(p^{-i\tau }:p\in \mathbb{P}\right),\left({(m+\alpha )}^{-i\tau }:m\in {\mathbb{N}}_0\right)\right)={\underline{\zeta }}_n(s+i\tau ,\alpha ).$$

Therefore, by the definitions of $P_{T,H,\alpha }^{\mathsf{\Omega }}$ and $P_{T,H,n,\alpha }$, we have

$$P_{T,H,n,\alpha }\left(A\right)=L_{T,H}\left(\left(\left(p^{-i\tau }:p\in \mathbb{P}\right),\left({(m+\alpha )}^{-i\tau }:m\in {\mathbb{N}}_0\right)\right)\in u_{n,\alpha }^{-1}A\right)$$

for all $A\in \mathcal{B}\left(H^2\left(D\right)\right)$. Hence,

$$P_{T,H,n,\alpha }=P_{T,H,\alpha }^{\mathsf{\Omega }}{u}_{n,\alpha }^{-1}.$$

This, the continuity of $u_{n,\alpha }$, Lemma 8, and Theorem 5.1 of [38] prove that $P_{T,H,n,\alpha }$ converges weakly to $U_{n,\alpha }$. □

On $(\mathsf{\Omega },\mathcal{B}\left(\mathsf{\Omega }\right),m_H)$, define the $H^2\left(D\right)$-valued random element $\underline{\zeta }(s,\omega ,\alpha )$ by

$$\underline{\zeta }(s,\omega ,\alpha )=\left(\zeta (s,{\omega }_1),\zeta (s,{\omega }_2,\alpha )\right),$$

where

$$\zeta (s,{\omega }_1)=\sum _{m=1}^{\infty }{\displaystyle \frac{{\omega }_1\left(m\right)}{m^s}}\mathrm{and}\zeta (s,{\omega }_2,\alpha )=\sum _{m=0}^{\infty }{\displaystyle \frac{{\omega }_2\left(m\right)}{{(m+\alpha )}^s}}.$$

We observe that the latter series are uniformly convergent on compact subsets of strip D for almost all ${\omega }_1$ and ${\omega }_2$, respectively (see, for example, [10,15]). Let $P_{\underline{\zeta },\alpha }$ be the distribution of the random element $\underline{\zeta }(s,\omega ,\alpha )$, i. e.,

$$P_{\underline{\zeta },\alpha }\left(A\right)=m_H\left\{\omega \in \mathsf{\Omega }:\underline{\zeta }(s,\omega ,\alpha )\in A\right\},A\in \mathcal{B}\left(H^2\left(D\right)\right).$$

In [23], for the proof of Theorem 5, a limit theorem for the functions $\zeta \left(s\right)$ and $\zeta (s,\alpha )$ with transcendental $\alpha$ was obtained. For $A\in \mathcal{B}\left(H^2\left(D\right)\right)$, let

$$P_{T,\alpha }\left(A\right)=L_T\left(\underline{\zeta }(s+i\tau ,\alpha )\in A\right).$$

Then, in [23], it was proved that $P_{T,\alpha }$, as $T\to \infty$, and $U_{n,\alpha }$, as $n\to \infty$ converges weakly to the same probability measure on $(H^2\left(D\right),\mathcal{B}\left(H^2\left(D\right)\right))$, and this measure is $P_{\underline{\zeta },\alpha }$. Thus, we have the following statement.

Lemma 10.

Suppose that α is transcendental. Then, $U_{n,\alpha }$ converges weakly to $P_{\underline{\zeta },\alpha }$ as $n\to \infty$.

Now, we are ready to prove a limit theorem for $P_{T,H,\alpha }$.

Theorem 9.

Suppose that $T^{27/82}⩽H⩽T^{1/2}$, and α is transcendental. Then $P_{T,H,\alpha }$ converges weakly to $P_{\underline{\zeta },\alpha }$ as $T\to \infty$.

Proof.  

Introduce a random variable ${\xi }_{T,H}$ defined on a certain probability space $(\widehat{\mathsf{\Omega }},\mathcal{A},\mu )$ and uniformly distributed on $[T,T+H]$. Define the $H^2\left(D\right)$-valued random elements as follows:

$${\underline{\zeta }}_{T,H,n,\alpha }={\underline{\zeta }}_{T,H,n,\alpha }\left(s\right)=\left({\zeta }_n(s+i{\xi }_{T,H}),{\zeta }_n(s+i{\xi }_{T,H},\alpha )\right)$$

and

$${\underline{\zeta }}_{T,H,\alpha }={\underline{\zeta }}_{T,H,\alpha }\left(s\right)=\left(\zeta (s+i{\xi }_{T,H}),\zeta (s+i{\xi }_{T,H},\alpha )\right).$$

Moreover, let ${\underline{\zeta }}_{n,\alpha }$ denote the $H^2\left(D\right)$-valued random element with distribution $U_{n,\alpha }$. Further on, we will use the language of convergence in distribution ($\stackrel{\mathcal{D}}{\to }$), i. e., we say that a random element ${\eta }_n$, as $n\to \infty$, converges in distribution to $\eta$ if the distribution of ${\eta }_n$, as $n\to \infty$, converges weakly to that of $\eta$.

In virtue of Lemma 10, we have

$${\underline{\zeta }}_{T,H,n,\alpha }\underset{T\to \infty }{\overset{\mathcal{D}}{\to }}{\underline{\zeta }}_{n,\alpha }.$$

(12)

By Lemma 10,

$${\underline{\zeta }}_{n,\alpha }\underset{n\to \infty }{\overset{\mathcal{D}}{\to }}\underline{\zeta }(s,\alpha ).$$

(13)

The definitions of ${\underline{\zeta }}_{T,H,n,\alpha }$, ${\underline{\zeta }}_{T,H,\alpha }$, and ${\xi }_{T,H}$ show that, for every $\epsilon >0$,

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}& \mu \left\{{\rho }_2\left({\underline{\zeta }}_{T,H,\alpha },{\underline{\zeta }}_{T,H,n,\alpha }\right)⩾\epsilon \right\}\hfill \\ & =\underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}{L}_{T,H}\left({\rho }_2\left(\underline{\zeta }(s+i\tau ,\alpha ),{\underline{\zeta }}_n(s+i\tau ,\alpha )\right)⩾\epsilon \right)\hfill \\ & ⩽\underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}{\displaystyle \frac{1}{\epsilon H}}{\int }_T^{T+H}{\rho }_2\left(\underline{\zeta }(s+i\tau ,\alpha ),{\underline{\zeta }}_n(s+i\tau ,\alpha )\right)\mathrm{d}\tau .\hfill \end{array}$$

Therefore, Lemma 7 implies that

$$\underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\mu \left\{{\rho }_2\left({\underline{\zeta }}_{T,H,\alpha },{\underline{\zeta }}_{T,H,n,\alpha }\right)⩾\epsilon \right\}=0.$$

This equality and relations (12) and (13) show that all hypotheses of Theorem 4.2 of [38] are fulfilled because the space $H^2\left(D\right)$ is separable. In consequence,

$${\underline{\zeta }}_{T,H,\alpha }\underset{T\to \infty }{\overset{\mathcal{D}}{\to }}\underline{\zeta }(s,\alpha ),$$

and this relation is equivalent to the weak convergence of $P_{T,H,\alpha }$ to $P_{\underline{\zeta },\alpha }$ as $T\to \infty$. □

4. Proof of the Main Theorem

Theorem 9 is the main ingredient of the proof of Theorem 8. However, the support of the limit measure $P_{\underline{\zeta },\alpha }$ is also needed. We recall that the support of $P_{\underline{\zeta },\alpha }$ is a minimal closed set $S_{\underline{\zeta },\alpha }\subset H^2\left(D\right)$ such that $P_{\underline{\zeta },\alpha }\left(S_{\underline{\zeta },\alpha }\right)=1$. The elements of $S_{\underline{\zeta },\alpha }$ have a property that, for every open neighborhood G of $\underline{g}$, the inequality $P_{\underline{\zeta },\alpha }\left(G\right)>0$ is satisfied.

Since the space $H^2\left(D\right)$ is separable, we have [38]

$$\mathcal{B}\left(H^2\left(D\right)\right)=\mathcal{B}\left(H\left(D\right)\right)\times \mathcal{B}\left(H\left(D\right)\right).$$

Therefore, it suffices to deal with sets of the form

$$A=A_1\times A_2,A_1,A_2\in \mathcal{B}\left(H\left(D\right)\right).$$

It is well known [10] that

$$L_T\left(\zeta (s+i\tau )\in A\right),A\in \mathcal{B}\left(H\left(D\right)\right),$$

converges weakly to the measure $P_{\zeta }$ as $T\to \infty$, where $P_{\zeta }$ is the distribution of the random element $\zeta (s,{\omega }_1)$.

$$L_T\left(\zeta (s+i\tau ,\alpha )\in A\right),A\in \mathcal{B}\left(H\left(D\right)\right),$$

with transcendental $\alpha$ converges weakly to the measure $P_{\zeta ,\alpha }$ as $T\to \infty$, where $P_{\zeta ,\alpha }$ is the distribution of the random element $\zeta (s,{\omega }_2,\alpha )$ [15]. Moreover, the support of $P_{\zeta }$ is the set

$$S\stackrel{\mathrm{def}}{=}\left\{g\in H\left(D\right):g\left(s\right)\ne 0\mathrm{or}g\left(s\right)\equiv 0\right\},$$

while the support of $P_{\zeta ,\alpha }$ is the whole $H\left(D\right)$ [10,15].

Lemma 11.

The support of the measure $P_{\underline{\zeta },\alpha }$ is the set $S\times H\left(D\right)$.

Proof.  

By a property of the Haar measures $m_{1H}$, $m_{2H}$, and $m_H$, and the above remark, we have

$$m_H(S\times H\left(D\right))=m_{1H}\left(S\right)·m_{2H}\left(H\left(D\right)\right).$$

This and the minimality of the sets S and $H\left(D\right)$ such that $m_{1H}\left(S\right)=1$ and $m_{2H}\left(H\left(D\right)\right)=1$ show that $S\times H\left(D\right)$ is a minimal set satisfying $m_H(S\times H\left(D\right))=1$. □

Proof of Theorem 8.  

By the Mergelyan theorem on the approximation of analytic functions by polynomials [39] (see also [40]), we have the existence of polynomials $p\left(s\right)$ and $q\left(s\right)$ such that

$$\underset{s\in K_1}{sup}\left|f_1\left(s\right)-{\mathrm{e}}^{p\left(s\right)}\right|<{\displaystyle \frac{\epsilon }{2}}$$

(14)

and

$$\underset{s\in K_2}{sup}\left|f_2\left(s\right)-q\left(s\right)\right|<{\displaystyle \frac{\epsilon }{2}}.$$

(15)

We stress that the Mergelyan theorem can be applied because $K_1,K_2\in \mathcal{K}$.

Define the set

$$G_{\epsilon }=\left\{(g_1,g_2)\in H^2\left(D\right):\underset{s\in K_1}{sup}\left|g_1\left(s\right)-{\mathrm{e}}^{p\left(s\right)}\right|<{\displaystyle \frac{\epsilon }{2}},\underset{s\in K_2}{sup}\left|g_2\left(s\right)-q\left(s\right)\right|<{\displaystyle \frac{\epsilon }{2}}\right\}.$$

Then, $G_{\epsilon }$ is an open neighborhood of an element $({\mathrm{e}}^{p\left(s\right)},q\left(s\right))\in S\times H\left(D\right)$. By Lemma 11 and properties of the support, we have

$$P_{\underline{\zeta },\alpha }\left(G_{\epsilon }\right)>0.$$

(16)

Let

$${\mathcal{G}}_{\epsilon }=\left\{(g_1,g_2)\in H^2\left(D\right):\underset{s\in K_1}{sup}|g_1\left(s\right)-f_1\left(s\right)|<\epsilon ,\underset{s\in K_2}{sup}|g_2\left(s\right)-f_2\left(s\right)|<\epsilon \right\}.$$

The inequalities (14)–(15) imply the inclusion of $G_{\epsilon }\subset {\mathcal{G}}_{\epsilon }$. Therefore, in view of (16),

$$P_{\underline{\zeta },\alpha }\left({\mathcal{G}}_{\epsilon }\right)>0.$$

The set ${\mathcal{G}}_{\epsilon }$ is open in $H^2\left(D\right)$. Therefore, Theorem 9 with the equivalent of weak convergence in terms of open sets (see Theorem 2.1 of [38]) gives

$$\underset{T\to \infty }{lim\; inf}{P}_{T,H,\alpha }\left({\mathcal{G}}_{\epsilon }\right)⩾P_{\underline{\zeta },\alpha }\left({\mathcal{G}}_{\epsilon }\right)>0.$$

This and the definitions of ${\mathcal{G}}_{\epsilon }$ and $P_{T,H,\alpha }$ imply the first assertion of the theorem.

The boundary of the set ${\mathcal{G}}_{\epsilon }$ is denoted by $𝜕{\mathcal{G}}_{\epsilon }$. Then, we have that $𝜕{\mathcal{G}}_{{\epsilon }_1}\cap 𝜕{\mathcal{G}}_{{\epsilon }_2}=⌀$ for different positives ${\epsilon }_1$ and ${\epsilon }_2$. The set ${\mathcal{G}}_{\epsilon }$ is a continuity set of $P_{\underline{\zeta },\alpha }$ if $P_{\underline{\zeta },\alpha }(𝜕{\mathcal{G}}_{\epsilon })=0$. From the above remark, it follows $P_{\underline{\zeta },\alpha }(𝜕{\mathcal{G}}_{\epsilon })\ne 0$ for at most countably many $\epsilon >0$. Applying Theorem 9 again in terms of continuity sets (see Theorem 2.1 of [38]), we obtain that

$$\underset{T\to \infty }{lim}{P}_{T,H,\alpha }\left({\mathcal{G}}_{\epsilon }\right)=P_{\underline{\zeta },\alpha }\left({\mathcal{G}}_{\epsilon }\right)>0$$

for all but at most countably many $\epsilon >0$. This proves the second statement of the theorem. □

5. Conclusions

Let $\zeta \left(s\right)$ and $\zeta (s,\alpha )$ denote the Riemann and Hurwitz zeta-functions, respectively, and the parameter $\alpha$ is transcendental. We obtained the set of shifts $\left(\zeta \right(s+i\tau ),\zeta (s+i\tau ,\alpha \left)\right)$, $\tau \in \mathbb{R}$, that approximate a given pair of analytic functions defined on the strip $D=\{s\in \mathbb{C}:1/2<\sigma <1\}$, has a positive lower density in the interval $[T,T+H]$, $T\to \infty$. Here, $T^{27/82}⩽H⩽T^{1/2}$. More precisely, the following result is proven. Let $K_1$ and $K_2$ be compact subsets of the strip D with connected complements, and $f_1\left(s\right)\ne 0$ and $f_2\left(s\right)$ continuous functions on $K_1$ and $K_2$ that are analytic inside of $K_1$ and $K_2$, respectively. Then, for every $\epsilon >0$,

$$\begin{array}{cc}\hfill \underset{T\to \infty }{lim\; inf}{\displaystyle \frac{1}{H}}\mathrm{meas}\{\tau \in [T,T+H]:& \underset{s\in K_1}{sup}|f_1\left(s\right)-\zeta (s+i\tau )|<\epsilon \hfill \\ & \underset{s\in K_2}{sup}|f_2\left(s\right)-\zeta (s+i\tau ,\alpha )|<\epsilon \}>0.\hfill \end{array}$$

Moreover, except for at most countably many values of $\epsilon >0$, “lim inf” can be replaced by “lim”. This result extends that of H. Mishou [23].

We are planing to consider similar problems for discrete shifts and generalized shifts $\left(\zeta \right(s+i\phi \left(\tau \right)),\zeta (s+i\phi \left(\tau \right),\alpha \left)\right)$ with a certain function $\phi \left(\tau \right)$.