Discrete Limit Bohr–Jessen Type Theorem for the Epstein Zeta-Function in Short Intervals

Abstract

We prove a probabilistic limit theorem for the Epstein zeta-function $\zeta (s;Q)$ in the interval $[N,N+M]$ as $N\to \infty$, using discrete shifts $\zeta (\sigma +ikh;Q)$, where $h>0$ and $\sigma >{\displaystyle \frac{n-1}{2}}$ are fixed. Here, Q is a positive-definite $n\times n$ matrix, and the interval length M satisfies $h^{-1}{\left(Nh\right)}^{27/82}⩽M⩽h^{-1}{\left(Nh\right)}^{1/2}$. The limit measure is given explicitly. This theorem is the first result in short intervals for $\zeta (s;Q)$. The obtained theorem improves the known results established for the interval of length N. Since the considered probability measures are defined in terms of frequency, theorems in short intervals have a certain advantage in the detection of $\zeta (\sigma +ikh;Q)$ with a given property, as well as in the characterization of the asymptotic behaviour of $\zeta (s;Q)$ in general.

1. Introduction

Let $s=\sigma +it$ be a complex variable. The term “zeta-function” steams from the Riemann zeta-function

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

Thus, zeta-functions in the classical sense are analytic functions of a complex variable in some half-plane $\sigma >{\sigma }_0$ defined by Dirichlet series

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

with coefficients $a_m$ of certain arithmetical nature, and have meromorphic continuation to the region $\sigma <{\sigma }_0$. In other words, zeta-functions are various generalizations of the function $\zeta \left(s\right)$.

The function $\zeta \left(s\right)$ appears to be quite simple; however, its value distribution is especially complicated. This is clearly illustrated by the Riemann hypothesis on the location of the nontrivial zeros of $\zeta \left(s\right)$. Roughly speaking, the value distribution of zeta-functions is chaotic, and information on their majority concrete values is insufficient. On the other hand, in the context of certain problems, precise information on values of $\zeta \left(s\right)$ is not necessary: certain average data are sufficient. This situation led H. Bohr to the idea that statistical methods could be applied for investigation of $\zeta \left(s\right)$ (see ref. [1]). By examining values of $\zeta \left(s\right)$ on vertical lines $\sigma +it$ with a fixed $\sigma$, one can take measurable sets $A\subset \mathbb{C}$ and consider the frequency of t such that $\zeta (\sigma +it)\in A$. The first result of such a kind, for fixed $\sigma >1$, was obtained in [2]. Denote by J the Jordan measure on the real line, and let R be a rectangle with edges parallel to the coordinate axis. Then, the main result of [2] states that the limit

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

exists and depends only on R and $\sigma$. The case for $\sigma >\frac{1}{2}$ is more complicated than that of $\sigma >1$, as it depends on the possible zeros of $\zeta \left(s\right)$. Therefore, in [3], the existence of the limit for the frequency

$${\displaystyle \frac{1}{T}}\mathrm{J}\left\{t\in [0,T]:\sigma +it\in G,log\zeta (\sigma +it)\in R\right\}$$

as $T\to \infty$ was considered. Here,

$$G=\left\{s\in \mathbb{C}:\sigma >{\displaystyle \frac{1}{2}}\right\}\setminus \bigcup _{\varrho =\beta +i\gamma }\left\{s=\sigma +i\gamma :{\displaystyle \frac{1}{2}}<\sigma \le \beta \right\},$$

where the union is taken over all zeros $\varrho$ of $\zeta \left(s\right)$ located in the strip $\left\{s\in \mathbb{C}:\sigma \in \left(\frac{1}{2},1\right)\right\}$. The proofs of these results are based on a theory of convex curves developed by the authors themselves [4].

Bohr–Jessen theorems were further developed in [5,6], leading to probabilistic limit theorems on weakly convergent probability measures. Let $\mathbb{X}$ be a topological space with Borel $\sigma$-field $\mathcal{B}\left(\mathbb{X}\right)$, and let P and $P_n$, $n\in \mathbb{N}$, be probability measures on $(\mathbb{X},\mathcal{B}(\mathbb{X}\left)\right)$. Recall that $P_n$ converges weakly to P as $n\to \infty$, denoted $P_n\underset{n\to \infty }{\overset{\mathrm{w}}{\to }}P$, if for every bounded continuous real-valued function g on $\mathbb{X}$,

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

In this terminology, Bohr–Jessen theorems can be rephrased as follows: on the space $(\mathbb{C},\mathcal{B}(\mathbb{C}\left)\right)$, there exists a probability measure $P_{\sigma }$ such that, for fixed $\sigma >\frac{1}{2}$, and

$$P_{T,\sigma }\left(A\right)\stackrel{\mathrm{def}}{=}{\displaystyle \frac{1}{T}}\mathrm{meas}\left\{t\in [0,T]:\zeta (\sigma +it)\in A\right\},A\in \mathcal{B}\left(\mathbb{C}\right),$$

the relation $P_{T,\sigma }\underset{T\to \infty }{\overset{\mathrm{w}}{\to }}{P}_{\sigma }$ holds (see, for example, refs. [7,8]). Here and throughout, $\mathrm{meas}\{\ast \}$ denotes the Lebesgue measure on the real line.

A new stage in the development of the probabilistic theory of zeta-functions was initiated by B. Bagchi. In his thesis [9], he created a theory of probabilistic limit theorems in the space of analytic functions and applied it to prove universality property of zeta-functions on approximation of classes of analytic functions. In his theorems, Bagchi proposed a new way for identification of the limit measure. In [10], we applied Bagchi’s method to prove a limit theorem for the Epstein zeta-function.

Suppose that Q is a positive-definite quadratic matrix of order $n\in \mathbb{N}$. The Epstein zeta-function $\zeta (s;Q)$ is defined, for $\sigma >\frac{n}{2}$, by the series

$$\zeta (s;Q)=\sum _{\underline{x}\in {\mathbb{Z}}^n\setminus \left\{\underline{0}\right\}}{\left({\underline{x}}^{\mathrm{T}}Q\underline{x}\right)}^{-s},$$

where ${\underline{x}}^{\mathrm{T}}$ is the transpose of $\underline{x}$. Moreover, $\zeta (s;Q)$ has analytic continuation to the whole complex plane, except for the point $s=\frac{n}{2}$, which is a simple pole with residue ${\pi }^{n/2}{\left(\Gamma \left(\frac{n}{2}\right)\sqrt{\det Q}\right)}^{-1}$, where $\Gamma \left(s\right)$ is the Euler gamma-function. The function $\zeta (s;Q)$ was introduced and studied in [11]. Its author aimed to define the most general zeta-function with the functional equation of the Riemann type, i.e.,

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

The attempt was successful: Epstein proved the following functional equation

$${\pi }^{-s}\Gamma \left(s\right)\zeta (s;Q)=\sqrt{\det Q}{\pi }^{s-{\displaystyle \frac{n}{2}}}\Gamma \left({\displaystyle \frac{n}{2}}-s\right)\zeta \left({\displaystyle \frac{n}{2}}-s;Q^{-1}\right),s\in \mathbb{C},$$

for $\zeta (s;Q)$, where $Q^{-1}$ denotes the inverse of Q.

We observe that in a particular case, the function $\zeta (s;Q)$ can be expressed as an ordinary Dirichlet series. Actually, suppose that ${\underline{x}}^{\mathrm{T}}Q\underline{x}\in \mathbb{Z}$ for all $\underline{x}\in {\mathbb{Z}}^n\setminus \left\{\underline{0}\right\}$, and, for $m\in \mathbb{N}$, $r_Q\left(m\right)$ is the number of $\underline{x}\in {\mathbb{Z}}^n$ such that ${\underline{x}}^{\mathrm{T}}Q\underline{x}=m$. Then, we have

$$\zeta (s;Q)=\sum _{m=1}^{\infty }{\displaystyle \frac{r_Q\left(m\right)}{m^s}},\sigma >{\displaystyle \frac{n}{2}}.$$

(1)

For example, for the n-dimensional unit matrix $I_n$ and $\sigma >1$, $\zeta (s;I_1)=2\zeta \left(2s\right)$, $\zeta (s;I_2)=4\zeta \left(s\right)L(s,{\chi }_4)$, where $L(s,\chi )$ is the Dirichlet L-function, and ${\chi }_4$ is the non-principal Dirichlet character modulo 4.

Note that the Epstein zeta-function is not only an object of interest in pure mathematics, particularly in algebra and algebraic number theory, but it also has practical applications. In fact, the function $\zeta (s;Q)$ appears in crystallography [12], as well as in physics, including quantum-field theory [13], and in studies related to energy and temperature [14,15].

The function $\zeta (s;Q)$ is, in general, significantly more complicated than the Riemann zeta-function. For example, the analytic continuation and functional equation of $\zeta (s;Q)$ depend essentially on the signature and rank of the real quadratic form Q on ${\mathbb{R}}^n$: when Q is positive-definite, the function $\zeta (s;Q)$ admits a meromorphic continuation to the whole complex plane and satisfies a classical functional equation. In contrast, when Q is indefinite, the analytic behavior of $\zeta (s;Q)$ becomes more intricate due to the presence of a continuous spectrum and its connections to non-holomorphic automorphic forms, such as Maass forms. Therefore, it is difficult to expect any general results about the value distribution characteristics of $\zeta (s;Q)$ for all matrices Q.

As previously mentioned, a Bohr–Jessen type theorem for $\zeta (s;Q)$ was obtained under the following restrictions in [10]. Based on (1), it was shown in [16] that

$$\zeta (s;Q)=\zeta (s;E_Q)+\zeta (s;F_Q),$$

where $\zeta (s;E_Q)$ and $\zeta (s;F_Q)$ are zeta-functions of a certain Eisenstein series and a cusp form, respectively, related to the coefficients $r_Q\left(m\right)$. Moreover, even for $n⩾4$, it is known that the mentioned Eisenstein series is a modular form of weight $\frac{n}{2}$ and level $q\in \mathbb{N}$ such that $q{\left(2Q\right)}^{-1}$ is an integral matrix [17], and then the zeta-function $\zeta (s;E_Q)$ is a certain combination of Dirichlet L-functions [18]. This leads to the representation

$$\zeta (s;Q)=\sum _{k=1}^K\sum _{l=1}^L{\displaystyle \frac{a_{kl}}{k^s{l}^s}}L(s,{\chi }_k)L\left(s-{\displaystyle \frac{n}{2}}+1,{\psi }_l\right)+\sum _{m=1}^{\infty }{\displaystyle \frac{b_Q\left(m\right)}{m^s}},$$

(2)

where k and l run over positive divisors of the level q, and K and L are finite integers; the characters ${\chi }_k$, $1⩽k⩽K$, and ${\psi }_l$, $1⩽l⩽L$ are the pairwise nonequivalent Dirichlet characters modulo $q/k$ and $q/l$, respectively, and $L(s,{\chi }_k)$ and $L(s,{\psi }_l)$ are the corresponding Dirichlet L-functions. The coefficients $a_{kl}\in \mathbb{C}$ are certain complex numbers. Furthermore, the series with coefficients $b_Q\left(m\right)$ is absolutely convergent for $\sigma >\frac{n-1}{2}$.

The formulation of the main limit theorem relies on the following object

$$\mathsf{\Omega }=\prod _{p\in \mathbb{P}}\{s\in \mathbb{C}:|s|=1\},$$

where $\mathbb{P}$ is the set of all prime numbers. With the product topology and operation of pairwise multiplication, $\mathsf{\Omega }$ is a compact topological group, and this leads to the probability space $(\mathsf{\Omega },\mathcal{B}(\mathsf{\Omega }),\mu )$, where $\mu$ is the probability Haar measure on $(\mathsf{\Omega },\mathcal{B}(\mathsf{\Omega }\left)\right)$. Denote by $\omega =\left(\omega \right(p):p\in \mathbb{P})$ the elements of $\mathsf{\Omega }$, and extend $\omega \left(p\right)$ to the set $\mathbb{N}$ by using the formula

$$\omega \left(m\right)=\underset{p^{l+1}\nmid m}{\prod _{p^l\mid m}}{\omega }^l\left(p\right),m\in \mathbb{N}.$$

Now, on the probability space $(\mathsf{\Omega },\mathcal{B}(\mathsf{\Omega }),\mu )$, define the complex-valued random element

$$\begin{array}{ccc}\hfill \zeta \left(\sigma ,\omega ;Q\right)& =& {\displaystyle \sum _{k=1}^K}{\displaystyle \sum _{l=1}^L}{\displaystyle \frac{a_{kl}\omega \left(k\right)\omega \left(l\right)}{k^{\sigma }{l}^{\sigma }}}L\left(\sigma ,\omega ,{\chi }_k\right)L\left(\sigma -{\displaystyle \frac{n}{2}}+1,\omega ,{\psi }_l\right)\hfill \\ \hfill & +& {\displaystyle \sum _{m=1}^{\infty }}{\displaystyle \frac{b_Q\left(m\right)\omega \left(m\right)}{m^{\sigma }}},\sigma >{\displaystyle \frac{n-1}{2}},\hfill \end{array}$$

(3)

where

$$L(\sigma ,\omega ,{\chi }_k)=\prod _p{\left(1-{\displaystyle \frac{{\chi }_k\left(p\right)\omega \left(p\right)}{p^{\sigma }}}\right)}^{-1}\mathrm{and}L\left(\sigma -{\displaystyle \frac{n}{2}}+1,\omega ,{\phi }_l\right)=\prod _p{\left(1-{\displaystyle \frac{{\phi }_l\left(p\right)\omega \left(p\right)}{p^{\sigma -n/2+1}}}\right)}^{-1}.$$

Denote by $P_{\zeta ,\sigma }$ the distribution

$$P_{\zeta ,\sigma }\left(A\right)=\mu \left\{\omega \in \mathsf{\Omega }:\zeta (\sigma ,\omega ;Q)\in A\right\},A\in \mathcal{B}\left(\mathbb{C}\right),$$

of the random element $\zeta (\sigma ,\omega ;Q)$. Then, the main result of [10] is the following.

Proposition 1 

(Theorem 2 of [10]). Suppose that $\sigma >{\displaystyle \frac{n-1}{2}}$ is fixed. Then, the probability measure

$${\displaystyle \frac{1}{T}}\mathrm{meas}\left\{t\in [0,T]:\zeta (\sigma +it;Q)\in A\right\},A\in \mathcal{B}\left(\mathbb{C}\right),$$

converges weakly to the measure $P_{\zeta ,\sigma }$ as $T\to \infty$.

In [19], a discrete version of Proposition 1 is presented. To state it, some definitions are needed. A number h is called type 1 if the number $exp\left\{\frac{2\pi k}{h}\right\}$ is irrational for all $k\in \mathbb{N}$. In the opposite case, h is type 2. Then, in this case, there exists the smallest $r_0$ such that

$$exp\left\{{\displaystyle \frac{2\pi r_0}{h}}\right\}={\displaystyle \frac{a}{b}}\mathrm{with}\mathrm{cooprimes}a,b\in \mathbb{N}.$$

Let ${\mathbb{P}}_0$ be a subset of $\mathbb{P}$:

$${\mathbb{P}}_0=\left\{p\in \mathbb{P}:\prod _{p\in \mathbb{P}}{p}^{l_p}={\displaystyle \frac{a}{b}},l_p\ne 0\right\}.$$

Then, $\#{\mathbb{P}}_0<\infty$, where $\#A$ denotes the cardinality of the set A.

We now return to the group $\mathsf{\Omega }$. For $h>0$, let ${\mathsf{\Omega }}_h$ be the closed subgroup of $\mathsf{\Omega }$ generated by $\left\{p^{-ih}:p\in \mathbb{P}\right\}$. Then, ${\mathsf{\Omega }}_h$, as $\mathsf{\Omega }$, is a compact group. Therefore, on $({\mathsf{\Omega }}_h,\mathcal{B}\left({\mathsf{\Omega }}_h\right))$, the probability Haar measure ${\mu }_h$ can be defined. It is known (see Lemma 4.2.2 of [9] and Lemma 1 of [20]) that

$${\mathsf{\Omega }}_h=\left\{\begin{array}{cc}\mathsf{\Omega }\hfill & \mathrm{if}h\mathrm{is}\mathrm{of}\mathrm{type}1,\hfill \\ \{\omega \in \mathsf{\Omega }:\omega (a)=\omega (b\left)\right\}\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Let ${\zeta }_h(\sigma ,\omega ;Q)$, for $\omega \in {\mathsf{\Omega }}_h$, be the complex-valued random element on the probability space $({\mathsf{\Omega }}_h,\mathcal{B}\left({\mathsf{\Omega }}_h\right),{\mu }_h)$ given by the Representation (3), and $P_{\zeta ,\sigma }^h$ denote its distribution. Let N run over the set ${\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$. Then, in [19], the following statement has been obtained.

Proposition 2 

(Theorem 1 of [19]). Suppose that $\sigma >{\displaystyle \frac{n-1}{2}}$ and $h>0$ are fixed. Then, the probability measure

$${\displaystyle \frac{1}{N}}\#\left\{0⩽k⩽N:\zeta (\sigma +ikh;Q)\in A\right\},A\in \mathcal{B}\left(\mathbb{C}\right),$$

converges weakly to the measure $P_{\zeta ,\sigma }^h$ as $N\to \infty$.

In Propositions 1 and 2, the weak convergence of probability measures defined by frequencies in the intervals $[0,T]$ and $[0,N]$ is discussed. It is well known that frequencies in short intervals contain more information on the discussed objects. This motivates the refinement of the limit theorems in Propositions 1 and 2, leading to limit theorems in short intervals, i.e., intervals of length shorter than T and N. A result of this type, corresponding to Proposition 1, was obtained in [21].

Proposition 3 

(Theorem 2 of [21]). Suppose that $\sigma >{\displaystyle \frac{n-1}{2}}$ is fixed, and $T^{27/82}⩽H⩽T^{1/2}$. Then, the probability measure

$${\displaystyle \frac{1}{H}}\mathrm{meas}\left\{t\in [T,T+H]:\zeta (\sigma +it;Q)\in A\right\},A\in \mathcal{B}\left(\mathbb{C}\right),$$

converges weakly to the measure $P_{\zeta ,\sigma }$ as $T\to \infty$.

The purpose of this paper is to develop a more complex version of Proposition 2 in short intervals. For brevity, we use the notation

$$D_{N,M}(\dots )={\displaystyle \frac{1}{M+1}}\#\left\{N⩽k⩽N+M:\dots \right\},M\in \mathbb{N},$$

where the dots indicate a condition to be satisfied by k. The main result is the following theorem.

Theorem 1. 

Suppose that $\sigma >{\displaystyle \frac{n-1}{2}}$ and $h>0$ are fixed, and $h^{-1}{\left(Nh\right)}^{27/82}⩽M⩽h^{-1}{\left(Nh\right)}^{1/2}$. Then, the probability measure

$$P_{\zeta ,\sigma ,N,M}^h\left(A\right)=D_{N,M}\left(\zeta (\sigma +ikh;Q)\in A\right),A\in \mathcal{B}\left(\mathbb{C}\right),$$

converges weakly to the measure $P_{\zeta ,\sigma }^h$ as $N\to \infty$.

Theorem 1 is theoretical, it extends and develops Bohr–Jessen’s ideas. Recall that even the eminent mathematician Atle Selberg, who was awarded the Fields Medal, devoted much attention to probabilistic limit theorems for zeta-functions; see [22,23]. Note that results concerning short intervals are highly valued in analytic number theory, especially those related to the distribution of zeros of zeta-functions and prime numbers. Moreover, problems in short intervals are more complicated. Theorem 1 continues investigations in this direction, providing new results that confirm the chaotic behaviour of the function $\zeta (s;Q)$. The results presented here could be useful for researchers studying this function in applied mathematics.

Theorem 1 will be proved in Section 3; the cases where h is type 1 and type 2 will be considered separately. Before that, we prove limit lemmas (Lemmas 7 and 8) for probability measures defined on the one-dimensional torus ${\mathsf{\Omega }}_h$, the structure of which depends on the arithmetic nature of the number h. Using Lemmas 7 and 8, we obtain limit lemmas (Lemmas 11 and 13) for probability measures defined in terms of ${\zeta }_r(s;Q)$ involving absolutely convergent Dirichlet series. Section 2 is devoted to certain discrete mean estimates for $\zeta (s;Q)$ in short intervals. Lemma 6 occupies a central place with respect to short intervals. It shows that the functions $\zeta (s;Q)$ and ${\zeta }_r(s;Q)$ are close in the mean in such intervals.

2. Estimates in Short Intervals

We start with recalling the mean square estimate for the Hurwitz zeta-function in short intervals. Let $0<\alpha ⩽1$ be a fixed parameter. The Hurwitz zeta-function $\zeta (s,\alpha ]$, for $\sigma >1$, is defined by the Dirichlet series

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

and has meromorphic continuation to the whole complex plane with the unique simple pole at the point $s=1$ with residue 1.

Lemma 1 

(Theorem 2 of [24]). Suppose that $\alpha \in (0,1)$ and $\frac{1}{2}<\sigma ⩽\frac{7}{12}$ are fixed, and $T^{27/82}⩽H⩽T^{\sigma }$. Then, uniformly in H, the estimate

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

holds.

We recall that the notation $z{\ll }_{\theta }x$, $z\in \mathbb{C}$, $x>0$ shows that there is a constant $c=c\left(\theta \right)>0$ such that $\left|z\right|⩽cx$.

Let $\chi$ be an arbitrary Dirichlet character modulo $q\in \mathbb{N}$, and let $L(s,\chi )$ be the corresponding Dirichlet L function. For $\sigma >1$, the function $L(s,\chi )$ is given by the series

$$L(s,\chi )=\sum _{m=1}^{\infty }{\displaystyle \frac{\chi \left(m\right)}{m^s}}.$$

The function has analytic continuation to the whole complex plane if $\chi$ is a non-principal character, and has a simple pole at the point $s=1$ with residue

$$\prod _{p\mid q}\left(1-{\displaystyle \frac{1}{p}}\right),p\in \mathbb{P},$$

if $\chi$ is the principal character modulo q ($\chi$ is the principal character modulo q if $\chi \left(m\right)=1$ for all $m\in \mathbb{N}$ coprime to q).

Lemma 2. 

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

$$\underset{T-H}{\overset{T+H}{\int }}{\left|L(\sigma +it,\chi )\right|}^2\mathrm{d}t{\ll }_{\sigma ,q}H$$

holds.

Proof. 

We use the formula

$$L(s,\chi )={\displaystyle \frac{1}{q^s}}\sum _{k=1}^q\chi \left(k\right)\zeta \left(s,{\displaystyle \frac{k}{q}}\right),$$

(4)

see, for example, [25]. Since $\left|\chi \right(k\left)\right|⩽1$, we have

$$L(s,\chi ){\ll }_q\sum _{k=1}^q\left|\zeta \left(s,{\displaystyle \frac{k}{q}}\right)\right|{\ll }_q\underset{1⩽k⩽q}{max}\sum _{k=1}^q\left|\zeta \left(s,{\displaystyle \frac{k}{q}}\right)\right|.$$

Therefore,

$$\underset{T-H}{\overset{T+H}{\int }}{\left|L(\sigma +it,\chi )\right|}^2\mathrm{d}t{\ll }_q\sum _{k=1}^q\underset{T-H}{\overset{T+H}{\int }}{\left|\zeta \left(\sigma +it,{\displaystyle \frac{m}{q}}\right)\right|}^2\mathrm{d}t{\ll }_{\sigma ,q}H$$

in virtue of Lemma 1. □

For our purposes, we need a discrete version of Lemma 2. For this, we will apply the Gallagher lemma on the connection of continuous and discrete mean squares of certain functions (see refs. [26,27]).

Lemma 3 

(Lemma 1.4 of [27]). For $\delta >0$, suppose that $T_1,T_2⩾\delta$, $\mathcal{A}\ne \varnothing$ is a finite set, $\mathcal{A}\subset \left[T_1+\delta /2,T_1+T_2-\delta /2\right]$, and, for $t\in \mathcal{A}$,

$$M_{\delta }\left(t\right)=\underset{|t-\tau |<\delta }{\sum _{\tau \in \mathcal{A}}}1.$$

Let $g\left(t\right)$ be a continuous function in the interval $[T_1,T_1+T_2]$, which has a continuous derivative $g^{\prime }\left(t\right)$ in $(T_1,T_1+T_2)$. Then, the inequality

$$\sum _{t\in \mathcal{A}}{M}_{\delta }^{-1}{\left(t\right)\left|g\left(t\right)\right|}^2⩽{\displaystyle \frac{1}{\delta }}\underset{T_1}{\overset{T_1+T_2}{\int }}{\left|g\left(t\right)\right|}^2\mathrm{d}t+{\left(\underset{T_1}{\overset{T_1+T_2}{\int }}{\left|g\left(t\right)\right|}^2\mathrm{d}t\underset{T_1}{\overset{T_1+T_2}{\int }}{\left|g^{\prime }\left(t\right)\right|}^2\mathrm{d}t\right)}^{1/2}$$

is valid.

Lemma 4. 

Suppose that $\frac{1}{2}<\sigma ⩽\frac{7}{12}$ and $h>0$ are fixed: $h^{-1}{\left(Nh\right)}^{27/82}⩽M⩽h^{-1}{\left(Nh\right)}^{1/2}$ and $\left|t\right|⩽{log}^2\left(Nh\right)$. Then, uniformly in M, the estimate

$$\sum _{k=N}^{N+M}{\left|L(\sigma +ikh+it,\chi )\right|}^2\mathrm{d}t{\ll }_{\sigma ,q,h}M(1+|t\left|\right)$$

holds.

Proof. 

We will apply Lemmas 2 and 3. In Lemma 3, take $\delta =1$, $T_1=N-\frac{1}{2}$, $T_2=M+1$, $\mathcal{A}=\{k\in \mathbb{N}:k\in [N,N+M\left]\right\}$, and $g\left(\tau \right)=L(\sigma +it+i\tau ,\chi )$. Then, obviously,

$$M_{\delta }\left(k\right)=\underset{|m-k|<1}{\sum _{m\in \mathcal{A}}}1=1.$$

Therefore, in view of Lemma 3,

$$\begin{array}{ccc}\hfill & & {\displaystyle \sum _{k=N}^{N+M}}{\left|L(\sigma +ikh+it,\chi )\right|}^2\ll \underset{N-1/2}{\overset{N+M+1/2}{\int }}{\left|L(\sigma +it+ih\tau ,\chi )\right|}^2\mathrm{d}\tau \hfill \\ \hfill & & +{\left(\underset{N-1/2}{\overset{N+M+1/2}{\int }}{\left|L(\sigma +it+ih\tau ,\chi )\right|}^2\mathrm{d}t\underset{N-1/2}{\overset{N+M+1/2}{\int }}{\left|L^{\prime }(\sigma +it+ih\tau ,\chi )\right|}^2\mathrm{d}t\right)}^{1/2}.\hfill \end{array}$$

(5)

It is easily seen that

$$\begin{array}{c}\hfill \underset{N-1/2}{\overset{N+M+1/2}{\int }}{\left|L(\sigma +it+ih\tau ,\chi )\right|}^2\mathrm{d}\tau {\ll }_h\underset{(N-1/2)h-\left|t\right|}{\overset{(N+M+1/2)h+\left|t\right|}{\int }}{\left|L(\sigma +i\tau ,\chi )\right|}^2\mathrm{d}\tau .\end{array}$$

(6)

Moreover, since $(M+\frac{1}{2})h+\left|t\right|⩾Mh+\left|t\right|⩾{\left(Nh\right)}^{27/82}$ and $(M+\frac{1}{2})h+\left|t\right|⩽{\left(Nh\right)}^{1/2}+\frac{h}{2}+{log}^2\left(Nh\right)⩽{\left(Nh\right)}^{\sigma }$ because $\sigma >\frac{1}{2}$. These remarks allow us to apply Lemma 2. Thus, in view of Estimate (6),

$$\begin{array}{c}\hfill \underset{N-1/2}{\overset{N+M+1/2}{\int }}{\left|L(\sigma +it+ih\tau ,\chi )\right|}^2\mathrm{d}\tau {\ll }_{\sigma ,q,h}Mh+{\displaystyle \frac{h}{2}}+\left|t\right|{\ll }_{\sigma ,q,h}M\left(1+\left|t\right|\right).\end{array}$$

(7)

Application of the Cauchy integral formula

$$L^{\prime }(s+it+ih\tau ,\chi )={\displaystyle \frac{1}{2\pi i}}\underset{\mathcal{L}}{\int }{\displaystyle \frac{L(z+it+ih\tau ,\chi )}{{(z-s)}^2}}\mathrm{d}z,$$

where $\mathcal{L}$ is a suitable simple closed contour enclosing s, and (7) gives

$$\underset{N-1/2}{\overset{N+M+1/2}{\int }}{\left|L^{\prime }(\sigma +it+ih\tau ,\chi )\right|}^2\mathrm{d}\tau {\ll }_{\sigma ,q,h}M\left(1+\left|t\right|\right).$$

From this and estimates (5) and (7), the lemma follows. □

This result ensures that we can bound discrete means analogously to continuous ones, which is essential for our approximation step.

Now, we will utilise Lemma 4 for approximation of the function $\zeta (s;Q)$ by a certain simple function involving absolutely convergent Dirichlet series. Let $\theta >\frac{1}{2}$ be a fixed number, $r\in \mathbb{N}$, and

$$w_r\left(m\right)=exp\left\{-{\left({\displaystyle \frac{m}{r}}\right)}^{\theta }\right\},m\in \mathbb{N}.$$

Return to the representation (2), and define

$$L_r\left(s_n,{\psi }_l\right)=\sum _{m=1}^{\infty }{\displaystyle \frac{{\psi }_l\left(m\right)w_r\left(m\right)}{m^{s_n}}},s_n=s-{\displaystyle \frac{n}{2}}+1.$$

In virtue of the exponential decreasing with respect to m for $w_r\left(m\right)$, the series for $L_r(s_n,{\psi }_l)$ is absolutely convergent in any half-plane $\sigma >{\sigma }_0$ with finite ${\sigma }_0$. Using $L_r\left(s_n,{\psi }_l\right)$, we introduce

$${\zeta }_r(s;Q)=\sum _{k=1}^K\sum _{l=1}^L{\displaystyle \frac{a_{kl}}{k^s{l}^s}}L(s,{\chi }_k)L_r(s_n,{\psi }_l)+\sum _{m=1}^{\infty }{\displaystyle \frac{b_Q\left(m\right)}{m^s}},$$

and we will approximate $\zeta (s;Q)$ by ${\zeta }_r(s;Q)$. For this, we will use a certain integral representation for $L_r(s_n,{\psi }_l)$. Set

$${\kappa }_r\left(s\right)={\displaystyle \frac{1}{\theta }}\Gamma \left({\displaystyle \frac{s}{\theta }}\right)r^s,$$

where $\theta >\frac{1}{2}$ is the number from the definition of $w_r\left(m\right)$.

Lemma 5. 

The representation

$$L_r(s_n,{\psi }_l)={\displaystyle \frac{1}{2\pi i}}\underset{\theta -i\infty }{\overset{\theta +i\infty }{\int }}L(s_n+z,{\psi }_l){\kappa }_r\left(z\right)\mathrm{d}z$$

is valid.

Proof. 

By the Mellin formula

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

and definitions of $w_r\left(m\right)$ and ${\kappa }_r\left(z\right)$, we have

$$\begin{array}{c}\hfill w_r\left(m\right)={\displaystyle \frac{1}{2\pi i}}\underset{1-i\infty }{\overset{1+i\infty }{\int }}\Gamma \left(z\right){\left({\displaystyle \frac{m}{r}}\right)}^{-\theta z}\mathrm{d}z={\displaystyle \frac{1}{2\pi i}}\underset{\theta -i\infty }{\overset{\theta +i\infty }{\int }}{\displaystyle \frac{1}{\theta }}\Gamma \left({\displaystyle \frac{r}{\theta }}\right){\displaystyle \frac{r^z}{m^z}}\mathrm{d}z={\displaystyle \frac{1}{2\pi i}}\underset{\theta -i\infty }{\overset{\theta +i\infty }{\int }}{\kappa }_r\left(z\right){\displaystyle \frac{\mathrm{d}z}{m^z}}.\end{array}$$

Therefore,

$$\begin{array}{c}\hfill {\displaystyle \frac{{\psi }_l\left(m\right)w_r\left(m\right)}{m^{s_n}}}={\displaystyle \frac{1}{2\pi i}}\underset{\theta -i\infty }{\overset{\theta +i\infty }{\int }}{\displaystyle \frac{{\psi }_l\left(m\right)}{m^{s_n+z}}}{\kappa }_r\left(z\right)\mathrm{d}z.\end{array}$$

(8)

Since, for the Gamma function, the estimate

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

(9)

uniformly in $\sigma \in [{\sigma }_1,{\sigma }_2]$ with arbitrary ${\sigma }_1<{\sigma }_2$ is valid, and $\Re s_n+\theta >1$ for $\sigma >\frac{n-1}{2}$, from (8) we find that

$$L_r(s_n,{\psi }_l)={\displaystyle \frac{1}{2\pi i}}\underset{\theta -i\infty }{\overset{\theta +i\infty }{\int }}\left(\sum _{m=1}^{\infty }{\displaystyle \frac{{\psi }_l\left(m\right)}{m^{s_n+z}}}\right){\kappa }_r\left(z\right)\mathrm{d}z={\displaystyle \frac{1}{2\pi i}}\underset{\theta -i\infty }{\overset{\theta +i\infty }{\int }}L(s_n+z,{\psi }_l){\kappa }_r\left(z\right)\mathrm{d}z.$$

□

Lemma 6. 

Suppose that $\sigma >\frac{n-1}{2}$ is fixed, and $h^{-1}{\left(Nh\right)}^{27/82}⩽M⩽h^{-1}{\left(Nh\right)}^{1/2}$. Then,

$$\underset{r\to \infty }{lim}\underset{N\to \infty }{lim\; sup}{\displaystyle \frac{1}{N+1}}\sum _{k=N}^{N+M}\left|\zeta \left(\sigma +ikh;Q\right)-{\zeta }_r\left(\sigma +ikh;Q\right)\right|=0.$$

Proof. 

We begin by proving that

$$\underset{r\to \infty }{lim}\underset{N\to \infty }{lim\; sup}{\displaystyle \frac{1}{N+1}}\sum _{k=N}^{N+M}\left|L\left({\sigma }_n+ikh,{\psi }_l\right)-L_r\left({\sigma }_n+ikh,{\psi }_l\right)\right|=0,$$

(10)

where ${\sigma }_n=\sigma -{\displaystyle \frac{n}{2}}+1$. First, we consider the case $\frac{n-1}{2}<\sigma ⩽\frac{n}{2}+\frac{1}{4}$. To proceed, we will apply Lemma 5 and the residue theorem. Let $\epsilon >0$ be a small fixed number, take $\theta ={\displaystyle \frac{n-1}{2}}+\epsilon$, and $\frac{n-1}{2}+2\epsilon ⩽\sigma ⩽\frac{n}{2}+\frac{1}{4}$. Moreover, let ${\theta }_1={\displaystyle \frac{n-1}{2}}+\epsilon -\sigma$. Clearly, ${\theta }_1<0$ and ${\theta }_1⩾\epsilon -\frac{3}{4}$. Therefore, the function

$$L(s_n+z,{\psi }_l){\kappa }_r\left(z\right)$$

in the strip ${\theta }_1⩽\Re z⩽\theta$ has a simple pole at the point $z=0$, which is the pole of $\Gamma \left(\frac{z}{\theta }\right)$, and a simple pole at the point $z=1-s_n$ of $L(s_n+z,{\psi }_l)$ if ${\psi }_l$ is the principal character modulo $q/l$. These observations, Lemma 5, and the residue theorem show that

$$L_r(s_n,{\psi }_l)-L(s_n,{\psi }_l)={\displaystyle \frac{1}{2\pi i}}\underset{{\theta }_1-i\infty }{\overset{{\theta }_1+i\infty }{\int }}L(s_n+z,{\psi }_l){\kappa }_r\left(z\right)\mathrm{d}z+R_r\left(s_n\right),$$

where

$$R_r\left(s_n\right)=\underset{z=1-s_n}{\mathrm{Res}}L(s_n+z,{\psi }_l){\kappa }_r\left(z\right)=\left\{\begin{array}{c}{\kappa }_r(1-s_n)\mathrm{if}{\psi }_l\mathrm{is}\mathrm{the}\mathrm{principal}\mathrm{character},\hfill \\ 0\mathrm{otherwise}.\hfill \end{array}\right.$$

From the latter equality, we derive

$$\begin{array}{ccc}\hfill L_r(s_n,{\psi }_l)-L(s_n,{\psi }_l)& =& {\displaystyle \frac{1}{2\pi }}\underset{-\infty }{\overset{\infty }{\int }}L\left(s_n+{\displaystyle \frac{n-1}{2}}+\epsilon -\sigma +iv,{\psi }_l\right)\hfill \\ & & \times {\kappa }_r\left({\displaystyle \frac{n-1}{2}}+\epsilon -\sigma +iv\right)\mathrm{d}v\hfill \\ & =& {\displaystyle \frac{1}{2\pi }}\underset{-\infty }{\overset{\infty }{\int }}L\left({\displaystyle \frac{1}{2}}+\epsilon +it+iv,{\psi }_l\right){\kappa }_r\left({\displaystyle \frac{n-1}{2}}+\epsilon -\sigma +iv\right)\mathrm{d}v\hfill \\ & & +{\kappa }_r(1-s_n).\hfill \end{array}$$

Hence,

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{1}{M+1}}{\displaystyle \sum _{k=N}^{N+M}}\left|L_r\left(\sigma +ikh-{\displaystyle \frac{n}{2}}+1,{\psi }_l\right)-L\left(\sigma +ikh-{\displaystyle \frac{n}{2}}+1,{\psi }_l\right)\right|\hfill \\ \hfill & & \ll \underset{-\infty }{\overset{\infty }{\int }}\left({\displaystyle \frac{1}{M+1}}{\displaystyle \sum _{k=N}^{N+M}}\left|L\left({\displaystyle \frac{1}{2}}+\epsilon +ikh+iv,{\psi }_l\right)\right|\right)\left|{\kappa }_r\left({\displaystyle \frac{n-1}{2}}+\epsilon -\sigma +iv\right)\right|\mathrm{d}v\hfill \\ & & +{\displaystyle \frac{1}{M+1}}{\displaystyle \sum _{k=N}^{N+M}}\left|{\kappa }_r\left(1-\sigma -ikh+{\displaystyle \frac{n-1}{2}}-1\right)\right|\hfill \\ \hfill & =& \left(\underset{-\infty }{\overset{-{log}^{2}Nh}{\int }}+\underset{{log}^{2}Nh}{\overset{\infty }{\int }}+\underset{-{log}^{2}Nh}{\overset{{log}^{2}Nh}{\int }}\right){\displaystyle \frac{1}{M+1}}{\displaystyle \sum _{k=N}^{N+M}}\left|L\left({\displaystyle \frac{1}{2}}+\epsilon +ikh+iv,{\psi }_l\right)\right|\hfill \\ \hfill & & \times \left|{\kappa }_r\left({\displaystyle \frac{n-1}{2}}+\epsilon -\sigma +iv\right)\right|\mathrm{d}v+{\displaystyle \frac{1}{M+1}}{\displaystyle \sum _{k=N}^{N+M}}\left|{\kappa }_r\left(1-\sigma -ikh+{\displaystyle \frac{n}{2}}-1\right)\right|.\hfill \end{array}$$

(11)

By the definition of ${\kappa }_r\left(s\right)$ and Estimate (9), we obtain

$${\kappa }_r\left({\displaystyle \frac{n-1}{2}}+\epsilon -\sigma +iv\right)\ll r^{(n-1)/2+\epsilon -\sigma }exp\left\{-{\displaystyle \frac{c}{\theta }}\left|v\right|\right\}{\ll }_{\epsilon }{r}^{-\epsilon }exp\{-c_1\left|v\right|\},c_1>0.$$

(12)

Moreover, it is known (see, for example, ref. [28]) that, for $\sigma >\frac{1}{2}$,

$$\zeta (\sigma +it,\alpha ){\ll }_{\sigma ,\alpha }{\left|t\right|}^{{\displaystyle \frac{1}{2}}},\sigma ⩾{\displaystyle \frac{1}{2}}.$$

Hence, in view of Representation (4),

$$L\left({\displaystyle \frac{1}{2}}+\epsilon +ikh+iv,{\psi }_l\right){\ll }_{\epsilon ,q}{\left(kh\right)}^{{\displaystyle \frac{1}{2}}}+{\left|v\right|}^{{\displaystyle \frac{1}{2}}}.$$

(13)

Therefore, by Equation (12),

$$\begin{array}{ccc}& & \left(\underset{-\infty }{\overset{{log}^{2}Nh}{\int }}+\underset{{log}^{2}Nh}{\overset{\infty }{\int }}\right)\left({\displaystyle \frac{1}{M+1}}\sum _{k=N}^{N+M}\left|L\left({\displaystyle \frac{1}{2}}+\epsilon +ikh+iv,{\psi }_l\right)\right|\right)\hfill \\ & & \times \left|{\kappa }_r\left({\displaystyle \frac{n-1}{2}}+\epsilon -\sigma +iv\right)\right|\mathrm{d}v\hfill \\ & {\ll }_{\epsilon ,q}& r^{-\epsilon }\left(\underset{-\infty }{\overset{{log}^{2}Nh}{\int }}+\underset{{log}^{2}Nh}{\overset{\infty }{\int }}\right)\left({\left(2N\right)}^{{\displaystyle \frac{1}{2}}}+{\left|v\right|}^{{\displaystyle \frac{1}{2}}}\right)exp\{-c_1\left|v\right|\}\mathrm{d}v\hfill \\ & {\ll }_{\epsilon ,q,h}& r^{-\epsilon }{N}^{{\displaystyle \frac{1}{2}}}exp\{-c_2{log}^{2}Nh\},c_2>0.\hfill \end{array}$$

For $v\in [-{log}^{2}Nh,{log}^{2}Nh]$, we apply Lemma 4. Thus,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{M+1}}\sum _{k=N}^{N+M}\left|L\left({\displaystyle \frac{1}{2}}+\epsilon +ikh+iv,{\psi }_l\right)\right|& \ll & {\left({\displaystyle \frac{1}{M+1}}\sum _{k=N}^{N+M}{\left|L\left({\displaystyle \frac{1}{2}}+\epsilon +ikh+iv,{\psi }_l\right)\right|}^2\right)}^{1/2}\hfill \\ & {\ll }_{\epsilon ,q,h}& {\left(1+\left|v\right|\right)}^{{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

Hence,

$$\begin{array}{ccc}& & \underset{-{log}^{2}Nh}{\overset{{log}^{2}Nh}{\int }}\left({\displaystyle \frac{1}{M+1}}{\displaystyle \sum _{k=N}^{N+M}}\left|L\left({\displaystyle \frac{1}{2}}+\epsilon +ikh+iv,{\psi }_l\right)\right|\right)\left|{\kappa }_r\left({\displaystyle \frac{n-1}{2}}+\epsilon -\sigma +iv\right)\right|\mathrm{d}v\hfill \\ & {\ll }_{\epsilon ,q,h}& r^{-\epsilon }\underset{-{log}^{2}Nh}{\overset{{log}^{2}Nh}{\int }}{\left(1+\left|v\right|\right)}^{{\displaystyle \frac{1}{2}}}exp\{-c_1\left|v\right|\}\mathrm{d}v\hfill \\ & {\ll }_{\epsilon ,q,h}& r^{-\epsilon }\underset{-\infty }{\overset{\infty }{\int }}{\left(1+\left|v\right|\right)}^{{\displaystyle \frac{1}{2}}}exp\{-c_1\left|v\right|\}\mathrm{d}v{\ll }_{\epsilon ,q}{r}^{-\epsilon }.\hfill \end{array}$$

(14)

Using Estimate (9) again yields

$${\kappa }_r\left(1-\sigma -ikh+{\displaystyle \frac{n}{2}}-1\right)\ll r^{-\sigma +n/2-1}exp\left\{-c_{3}kh\right\}\ll r^{1/2-2\epsilon }exp\{-c_{3}kh\},c_3>0.$$

Hence,

$${\displaystyle \frac{1}{M+1}}\sum _{k=N}^{N+M}\left|{\kappa }_r\left(-\sigma -ikh+{\displaystyle \frac{n}{2}}\right)\right|\ll r^{1/2-2\epsilon }{\displaystyle \frac{1}{M+1}}\sum _{k=N}^{N+M}exp\{-c_{3}kh\}\ll r^{1/2-2\epsilon }exp\{-c_{3}Nh\}.$$

This, together with (11), (13), and (14), shows that the left-hand side of (10) is estimated as

$$\begin{array}{c}\hfill {\ll }_{\epsilon ,q,h}{r}^{-\epsilon }{N}^{{\displaystyle \frac{1}{2}}}exp\{-c_2{log}^{2}Nh\}+r^{-\epsilon }+r^{1/2-2\epsilon }exp\{-c_{3}Nh\}.\end{array}$$

Thus, letting $N\to \infty$ and subsequently $r\to \infty$, we obtain the Equality (10).

The case of $\sigma >\frac{n}{2}+\frac{1}{4}$ is simpler due to the absolute convergence of the series for $L\left({\sigma }_n,{\psi }_l\right)$. Thus, with small $\epsilon >0$, we have

$$L\left(\sigma -\epsilon +ikh+iv,{\psi }_l\right){\ll }_{\sigma ,\epsilon ,q}1,$$

for $\sigma -\epsilon ⩾\frac{n}{2}+\frac{1}{4}$. Hence,

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{M+1}}\sum _{k=N}^{N+M}\left|L_r\left(\sigma +ikh-{\displaystyle \frac{n}{2}}+1,{\psi }_l\right)-L\left(\sigma +ikh-{\displaystyle \frac{n}{2}}+1,{\psi }_l\right)\right|\hfill \\ & {\ll }_{\sigma ,\epsilon ,q}& \underset{-\infty }{\overset{\infty }{\int }}\left|{\kappa }_r\left(-\epsilon +iv\right)\right|+{\displaystyle \frac{1}{M+1}}\sum _{k=N}^{N+M}\left|{\kappa }_r\left(1-\sigma -ikh+{\displaystyle \frac{n}{2}}-1\right)\right|,\hfill \end{array}$$

and this together with (9) proves the lemma in this case. □

3. Proof of the Theorem 1

For the proof of Theorem 1, we will apply the method of Fourier transforms, the preservation of weak convergence of probability measures under certain mappings, as well as a connection of weak convergence and convergence in distribution. For the identification of the limit measure, we will apply the results of [10,19].

We start with a limit lemma for probability measures on $(\mathsf{\Omega },\mathcal{B}(\mathsf{\Omega }\left)\right)$. For $A\in \mathcal{B}\left(\mathsf{\Omega }\right)$, set

$$Q_{N,M}^h\left(A\right)=D_{N,M}\left(\left(p^{-ikh}:p\in \mathbb{P}\right)\in A\right).$$

Lemma 7. 

Suppose that $h>0$ is type 1, and $M\to \infty$ as $N\to \infty$. Then, $Q_{N,M}^h$ converges weakly to the Haar measure μ as $N\to \infty$.

Proof. 

The group $\mathsf{\Omega }$ is compact, as the product of compact sets. Therefore, it provides a reason to study the Fourier transform of the measure $Q_{N,M}^h$, and, from its convergence, derive the weak convergence for $Q_{N,M}^h$. Denote by $\widehat{\mathsf{\Omega }}$ the dual group, or character group, of $\mathsf{\Omega }$. It is a well-known and widely used fact that $\widehat{\mathsf{\Omega }}$ is isomorphic to the group

$$G\stackrel{\mathrm{def}}{=}\underset{p\in \mathbb{P}}{⨁}{\mathbb{Z}}_p,$$

where ${\mathbb{Z}}_p=\mathbb{Z}$ for all $p\in \mathbb{P}$, and an element $\underline{l}=\left(l_p:l_p\in \mathbb{N}\right)\in G$, where only a finite number of integers $l_p$ are non-zero, acts on $\widehat{\mathsf{\Omega }}$ by the formula

$$\omega ⟼{\omega }^{\underline{l}}=\prod _{p\in \mathbb{P}}{\omega }^{l_p},\omega =\left(\omega \left(p\right):p\in \mathbb{P}\right)\in \mathsf{\Omega }.$$

From this, it follows that the characters of $\mathsf{\Omega }$ are given by the product

$$\underset{p\in \mathbb{P}}{{\prod }^{*}}{\omega }^{l_p}\left(p\right),$$

where the sign “*” shows that only a finite number of $l_p\ne 0$. This implies that the Fourier transform $f_{N,M}\left(\underline{l}\right)$ of the measure $Q_{N,M}^h$ is

$$f_{N,M}\left(\underline{l}\right)=\underset{\mathsf{\Omega }}{\int }\left(\underset{p\in \mathbb{P}}{{\prod }^{*}}{\omega }^{l_p}\left(p\right)\right)\mathrm{d}{Q}_{N,M}^h.$$

Hence, the definition of the measure $Q_{N,M}^h$ yields

$$f_{N,M}\left(\underline{l}\right)={\displaystyle \frac{1}{M+1}}\sum _{k=N}^{N+M}\left(\underset{p\in \mathbb{P}}{{\prod }^{*}}{p}^{-ikh{l}_p}\right)={\displaystyle \frac{1}{M+1}}\sum _{k=N}^{N+M}exp\left\{-ikh\underset{p\in \mathbb{P}}{{\sum }^{*}}{l}_{p}logp\right\}.$$

(15)

For brevity, let

$$L_h\left(\underline{l}\right)\stackrel{\mathrm{def}}{=}h\underset{p\in \mathbb{P}}{{\sum }^{*}}{l}_{p}logp.$$

We observe that the requirement that h is type 1 is equivalent to the linear independence over the field of rational numbers $\mathbb{Q}$ for the multi-set $\left\{(logp:p\in \mathbb{P}),\frac{2\pi }{h}\right\}\stackrel{\mathrm{def}}{=}{A}_h$. Actually, if the set $A_h$ is linearly dependent over $\mathbb{Q}$, then there exist the numbers $k_1,\dots ,k_r\in \mathbb{Z}$, and $m\in \mathbb{Z}\setminus \left\{0\right\}$ that

$$k_{1}log{p}_1+\dots +k_{r}log{p}_r+{\displaystyle \frac{2\pi m}{h}}=0,$$

and

$$exp\left\{{\displaystyle \frac{2\pi m}{h}}\right\}=p_1^{k_1}\dots p_r^{k_r},$$

and this contradicts the definition of type 1. Thus, h of type 1 implies the linear independence of $A_h$. Similarly, if h is not type 1, then there exists $m_0\in \mathbb{Z}\setminus \left\{0\right\}$ such that

$$exp\left\{{\displaystyle \frac{2\pi m_0}{h}}\right\}=p_1^{k_1}\dots p_r^{k_r}$$

is a rational number. Hence, the set $A_h$ is linearly dependent over $\mathbb{Q}$. This shows that linear independence of $A_h$ implies the type 1 for h.

The above remarks imply that

$$h{L}_h\left(\underline{l}\right)=2\pi m$$

with $m\in \mathbb{Z}$ if and only if $\underline{l}=\underline{0}$ and $m=0$. Hence, in view of Expression (15),

$$f_{N,M}\left(\underline{l}\right)=1$$

(16)

if and only if $\underline{l}=\underline{0}$. If $\underline{l}\ne \underline{0}$, then $L_h\left(\underline{l}\right)\ne 0$, and (15) gives

$$f_{N,M}\left(\underline{l}\right)={\displaystyle \frac{exp\{-iN{L}_h\left(\underline{l}\right)\}-exp\{-i(N+M+1)L_h\left(\underline{l}\right)\}}{(M+1)(1-exp\{-i{L}_h\left(\underline{l}\right)\})}}.$$

Hence, for $\underline{l}\ne \underline{0}$,

$$\underset{N\to \infty }{lim}{f}_{N,M}\left(\underline{l}\right)=0.$$

(17)

Let

$$f\left(\underline{l}\right)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\underline{l}=\underline{0},\hfill \\ 0\hfill & \mathrm{if}\underline{l}\ne \underline{0}.\hfill \end{array}\right.$$

Then, Equalities (16) and (17) show that

$$\underset{N\to \infty }{lim}{f}_{N,M}\left(\underline{l}\right)=f\left(\underline{l}\right).$$

(18)

Since $f\left(\underline{l}\right)$ is the Fourier transform of the Haar measure $\mu$ on $(\mathsf{\Omega },\mathcal{B}(\mathsf{\Omega }\left)\right)$, and $\mathsf{\Omega }$, as a compact group, is the Lévy group, in view of Theorem 1.4.2 of [29] and Equation (18), we determine that the measure $Q_{N,M}^h$ converges weakly to $\mu$ as $N\to \infty$. □

Now consider the case with h of type 2.

Lemma 8. 

Suppose that $h>0$ is type 2, and $M\to \infty$ as $N\to \infty$. Then, $Q_{N,M}^h$ converges weakly to the Haar measure ${\mu }^h$ as $N\to \infty$.

Proof. 

It is known, see the proof of Lemma 4 of [19], that in this case, characters of ${\mathsf{\Omega }}_h$ are of the form

$$\prod _{p\in \mathbb{P}\setminus {\mathbb{P}}_0}{\omega }^{l_p}\left(p\right)\prod _{p\in {\mathbb{P}}_0}{\omega }^{l_p+l{\beta }_p}\left(p\right),l\in \mathbb{Z},$$

and only a finite number of integers $l_p$ are not zero. Here,

$${\mathbb{P}}_0=\left\{p\in \mathbb{P}:\prod _{p\in \mathbb{P}}{p}^{{\alpha }_p}={\displaystyle \frac{a}{b}},{\alpha }_p\ne 0\right\}$$

is a finite subset of $\mathbb{P}$. Hence, the Fourier transform $f_{N,M}\left(\underline{l}\right)$ of $Q_{N,M}^h$ is

$$\begin{array}{ccc}\hfill f_{N,M}\left(\underline{l}\right)& =& {\displaystyle \frac{1}{M+1}}{\displaystyle \sum _{k=N}^{N+M}}\prod _{p\in \mathbb{P}\setminus {\mathbb{P}}_0}{p}^{-ikh{l}_p}\prod _{p\in {\mathbb{P}}_0}{p}^{-ikh(l_p+l{\beta }_p)}\hfill \\ \hfill & =& {\displaystyle \frac{1}{M+1}}{\displaystyle \sum _{k=N}^{N+M}}\prod _{p\in \mathbb{P}\setminus {\mathbb{P}}_0}{p}^{-ikh{l}_p}\prod _{p\in {\mathbb{P}}_0}{p}^{-ikh{l}_p}\hfill \end{array}$$

(19)

due to the equality

$$\begin{array}{c}\hfill \prod _{p\in {\mathbb{P}}_0}{p}^{-ikhl{\beta }_p}={\displaystyle \frac{a^{-ikhl}}{b^{-ikhl}}}=1,l\in \mathbb{Z},\end{array}$$

(20)

since $\omega \left(a\right)=\omega \left(b\right)$ for $\omega \in {\mathsf{\Omega }}_h$. Taking into account (19) and (20), we find that

$$\begin{array}{c}\hfill f_{N,M}\left(\underline{l}\right)=1\end{array}$$

(21)

for $l_p=0$, $p\in \mathbb{P}\setminus {\mathbb{P}}_0$, and $l_p=l{\beta }_p$, $p\in {\mathbb{P}}_0$.

Now, suppose that $l_p\ne 0$ for some $p\in \mathbb{P}\setminus {\mathbb{P}}_0$, or there is $l\in \mathbb{Z}$ such that $l_p\ne l{\beta }_p$ for all $p\in {\mathbb{P}}_0$. Then, we have

$$exp\left\{-ih\left(\sum _{p\in \mathbb{P}\setminus {\mathbb{P}}_0}{l}_{p}logp+\sum _{p\in {\mathbb{P}}_0}(l_p+l{\beta }_p)logp\right)\right\}\ne 1.$$

(22)

Actually, if (22) is not true, then

$$exp\left\{\sum _{p\in \mathbb{P}\setminus {\mathbb{P}}_0}{l}_{p}logp+\sum _{p\in {\mathbb{P}}_0}(l_p+l{\beta }_p)logp\right\}=exp\left\{{\displaystyle \frac{2\pi l_0}{h}}\right\}$$

(23)

with some integer $l_0$. If $l_0$ is a multiple of $r_0$ ($r_0$ is from the definition of type 2), then

$$exp\left\{{\displaystyle \frac{2\pi l_0}{h}}\right\}=\prod _{p\in {\mathbb{P}}_0}{p}^{l_1{\beta }_p}$$

with $l_1\in \mathbb{Z}$. Hence, by (23),

$$\sum _{p\in \mathbb{P}\setminus {\mathbb{P}}_0}{l}_{p}logp+\sum _{p\in {\mathbb{P}}_0}\left(l_p+l_2{\beta }_p\right)logp=0$$

with some integer $l_2$, and this contradicts the linear independence over Q of logarithms of prime numbers. If $l_0$ is not a multiple of $r_0$, then the number

$$exp\left\{{\displaystyle \frac{2\pi l_0}{h}}\right\}$$

is irrational, and this contradicts (23) since

$$exp\left\{\sum _{p\in \mathbb{P}\setminus {\mathbb{P}}_0}{l}_{p}logp+\sum _{p\in {\mathbb{P}}_0}(l_p+l{\beta }_p)logp\right\}=\prod _{p\in \mathbb{P}\setminus {\mathbb{P}}_0}{p}^{l_p}\prod _{p\in {\mathbb{P}}_0}{p}^{l_p+l{\beta }_p},$$

is a rational number. These contradictions show that inequality (22) is true. Let, for brevity,

$${\widehat{L}}_h(\underline{l},l)=\sum _{p\in \mathbb{P}\setminus {\mathbb{P}}_0}{l}_{p}logp+\sum _{p\in {\mathbb{P}}_0}(l_p+l{\beta }_p)logp.$$

Then, (19) and (22) yield that

$$\begin{array}{ccc}\hfill f_{N,M}\left(\underline{l}\right)& =& {\displaystyle \frac{1}{M+1}}\sum _{k=N}^{N+M}exp\left\{-ikh\left(\sum _{p\in \mathbb{P}\setminus {\mathbb{P}}_0}{l}_{p}logp+\sum _{p\in {\mathbb{P}}_0}(l_p+l{\beta }_p)logp\right)\right\}\hfill \\ & =& {\displaystyle \frac{exp\{-ihN{\widehat{L}}_h(l_p,l)\}-exp\{-ih(N+M+1){\widehat{L}}_h(l_p,l)\}}{(M+1)(1-exp\{-ih{\widehat{L}}_h(l_p,l)\})}},l\in \mathbb{Z}.\hfill \end{array}$$

This and Equality (21) show that

$$\underset{N\to \infty }{lim}f\left(\underline{l}\right)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}{l}_p=0\mathrm{for}p\in \mathbb{P}\setminus {\mathbb{P}}_0,\mathrm{and}{l}_p=l{\beta }_p\mathrm{for}\mathrm{all}p\in {\mathbb{P}}_0,\hfill \\ 0\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

and the proof of the lemma is complete because the right-hand side of the latter equality is the Fourier transform of ${\mu }^h$. □

Lemma 7 serves as an important component for the proof of a limit lemma for the function ${\zeta }_r(s;Q)$. For this, we recall one simple but useful property of weak convergence of probability measures.

Let ${\mathbb{X}}_1$ and ${\mathbb{X}}_2$ be two topological spaces, and $g:{\mathbb{X}}_1⟼{\mathbb{X}}_2$ a continuous mapping. Then, g is $\left(\mathcal{B}\left({\mathbb{X}}_1\right),\mathcal{B}\left({\mathbb{X}}_2\right)\right)$-measurable, i.e.,

$${\mathcal{B}}^{-1}\left({\mathbb{X}}_2\right)\subset \mathcal{B}\left({\mathbb{X}}_1\right).$$

Then, every probability measure P on $\left({\mathbb{X}}_1,\mathcal{B}\left({\mathbb{X}}_1\right)\right)$ induces the unique probability measure $P{g}^{-1}$ on $\left({\mathbb{X}}_2,\mathcal{B}\left({\mathbb{X}}_2\right)\right)$ defined by

$$P{g}^{-1}\left(A\right)=P\left(g^{-1}A\right),A\in \mathcal{B}\left({\mathbb{X}}_2\right),$$

where $g^{-1}\left(A\right)$ is the preimage of A.

Lemma 9 

(Section 5 of [30]). Let $P_n$, $n\in \mathbb{N}$, and P be probability measures on $\left({\mathbb{X}}_1,\mathcal{B}\left({\mathbb{X}}_1\right)\right)$, and $g:{\mathbb{X}}_1⟼{\mathbb{X}}_2$ be a continuous mapping. Suppose that $P_n\underset{n\to \infty }{\overset{\mathrm{w}}{\to }}P$. Then, $P_n{g}^{-1}\underset{n\to \infty }{\overset{\mathrm{w}}{\to }}P{g}^{-1}$.

For $\omega \in \mathsf{\Omega }$, define

$$\begin{array}{ccc}\hfill {\zeta }_r(s,\omega ;Q)& =& \sum _{k=1}^K\sum _{l=1}^L{\displaystyle \frac{a_{kl}\omega \left(k\right)\omega \left(l\right)}{k^s{l}^s}}L(s,\omega ,{\chi }_k)L\left(s-{\displaystyle \frac{n}{2}}+1,\omega ,{\psi }_l\right)\hfill \\ & +& \sum _{m=1}^{\infty }{\displaystyle \frac{b_Q\left(m\right)\omega \left(m\right)}{m^s}},\hfill \end{array}$$

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

$$P_{N,M,r,\sigma }^h\left(A\right)=D_{N,M}\left({\zeta }_r(\sigma +ikh;Q)\in A\right).$$

Lemma 10. 

Suppose that $h>0$ is type 1, $\sigma >{\displaystyle \frac{n-1}{2}}$, and $M\to \infty$ as $N\to \infty$. Then, on $(\mathbb{C},\mathcal{B}(\mathbb{C}\left)\right)$, there exists a probability measure $P_{r,\sigma }$ such that $P_{N,M,r,\sigma }^h\underset{N\to \infty }{\overset{\mathrm{w}}{\to }}{P}_{r,\sigma }$.

Proof. 

Consider the mapping $g_{r,\sigma }:\mathsf{\Omega }⟼\mathbb{C}$ given by the formula

$$g_{r,\sigma }\left(\omega \right)={\zeta }_r(\sigma ,\omega ;Q),\omega \in \mathsf{\Omega }.$$

All functions involved in the definition of ${\zeta }_r(\sigma ,\omega ;Q)$ are absolutely convergent, hence, uniform in $\omega$. Therefore, $g_{r,\sigma }$ is a continuous mapping. Moreover, we have

$$g_{r,\sigma }\left(p^{-ikh}:p\in \mathbb{P}\right)={\zeta }_r(\sigma +ikh;Q).$$

Therefore,

$$P_{N,M,r,\sigma }^h\left(A\right)=D_{N,M}\left(\left(p^{-ikh}:p\in \mathbb{P}\right)\in g_{r,\sigma }^{-1}A\right)=Q_{N,M}^h\left(g^{-1}A\right)$$

for all $A\in \mathcal{B}\left(\mathbb{C}\right)$. Thus, we have $P_{N,M,r,\sigma }^h=Q_{N,M}^h{g}_{r,\sigma }^{-1}$. This, continuity of $g_{r,\sigma }$ and Lemmas 7 and 9 proves that $P_{N,M,r,\sigma }^h\underset{N\to \infty }{\overset{\mathrm{w}}{\to }}\mu g_{r,\sigma }^{-1}\stackrel{\mathrm{def}}{=}{P}_{r,\sigma }$. □

A separate analysis of weak convergence for the measure $P_{r,\sigma }$ as $n\to \infty$ is unnecessary, as it has already been investigated in [19].

Lemma 11. 

Suppose that $\sigma >{\displaystyle \frac{n-1}{2}}$ is fixed and $h>0$ is type 1. Then, $P_{r,\sigma }\underset{r\to \infty }{\overset{\mathrm{w}}{\to }}{P}_{\zeta ,\sigma }$.

We notice that the statement of the lemma has been obtained in the proof of Theorem 2 of [10]. For identification of the limit measure of $P_{r,\sigma }$ as $r\to \infty$, elements of the ergodic theory have been involved; more precisely, the classical Birkhoff–Khintchine ergodic theorem (see, ref. [31]) has been applied.

Now, we resume the analysis of the case where h is type 2. Repeating the proof of Lemma 10 and using Lemma 8 leads to the following statement.

Lemma 12. 

Suppose that $h>0$ is type 2. Then, on $(\mathbb{C},\mathcal{B}(\mathbb{C}\left)\right)$, there exists a probability measure $P_{r,\sigma }^h$ such that $P_{N,M,r,\sigma }^h\underset{N\to \infty }{\overset{\mathrm{w}}{\to }}{P}_{r,\sigma }^h$.

Proof. 

We notice that, in the case of type 2, the mapping $g_{r,\sigma }$ depends on h. Thus, $P_{r,\sigma }^h={\mu }^h{g}_{r,\sigma }^{-1}$. □

The measure $P_{r,\sigma }^h$ does not depend on M. Thus, we may use the results of [19].

Lemma 13 

([19], pp. 12–14). Suppose that $\sigma >{\displaystyle \frac{n-1}{2}}$ is fixed and $h>0$ is type 2. Then $P_{r,\sigma }^h\underset{r\to \infty }{\overset{\mathrm{w}}{\to }}{P}_{\zeta ,\sigma }^h$.

To complete the proof of Theorem 1, one statement on convergence in distribution is needed. Let $X_n$, $n\in \mathbb{N}$, and X be $\mathbb{X}$-valued random elements on a certain probability space $(\widehat{\mathsf{\Omega }},\mathcal{B},\mathcal{Q})$, and $P_n\left(A\right)=\mathcal{Q}\left\{\widehat{\omega }\in \widehat{\mathsf{\Omega }}:X_n\in A\right\}$ and $P\left(A\right)=\mathcal{Q}\left\{\widehat{\omega }\in \widehat{\mathsf{\Omega }}:X\in A\right\}$, $A\in \mathcal{B}\left(\mathbb{C}\right)$ be the corresponding distributions. We say that $X_n$ converges to X in distribution $\left(X_n\underset{n\to \infty }{\overset{\mathcal{D}}{\to }}X\right)$ if $P_n\underset{n\to \infty }{\overset{\mathrm{w}}{\to }}P$.

Lemma 14 

(Theorem 4.2 of [30]).Let $Y_n$ and $X_{ln}$, $n,l\in \mathbb{N}$, be $(\mathbb{X},\varrho )$-valued random elements on the probability space $(\widehat{\mathsf{\Omega }},\mathcal{B},\mathcal{Q})$, and the space $\mathbb{X}$ is separable. Suppose that, for every $l\in \mathbb{N}$,

$$X_{ln}\underset{n\to \infty }{\overset{\mathcal{D}}{\to }}{X}_{l}and{X}_l\underset{l\to \infty }{\overset{\mathcal{D}}{\to }}X.$$

Moreover, let, for every $\epsilon >0$,

$$\underset{l\to \infty }{lim}\underset{n\to \infty }{lim\; sup}\mathcal{Q}\left\{\varrho \left(X_{ln},Y_n\right)⩾\epsilon \right\}=0.$$

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

Proof of Theorem 1. 

Case for h of type 1. On a certain probability space $(\widehat{\mathsf{\Omega }},\mathcal{B},\mathcal{Q})$, define a random variable ${\xi }_{N,M}^h$ having the distribution

$$\mathcal{Q}\left\{{\xi }_{N,M}^h=kh\right\}={\displaystyle \frac{1}{M+1}},k=N,N+1,\dots ,M.$$

For $\sigma >{\displaystyle \frac{n-1}{2}}$, set

$$X_{N,M,r}^h=X_{N,M,r}^h\left(\sigma \right)={\zeta }_r(\sigma +i{\xi }_{N,M}^h;Q),$$

and

$$Y_{N,M}^h=Y_{N,M}^h\left(\sigma \right)=\zeta (\sigma +i{\xi }_{N,M}^h;Q).$$

In virtue of Lemma 10, it follows that

$$X_{N,M,r}^h\underset{N\to \infty }{\overset{\mathcal{D}}{\to }}{X}_r,$$

(24)

where $X_r=X_r\left(\sigma \right)$ is the $\mathbb{C}$-valued random element with the distribution $P_{r,\sigma }$. Moreover, in view of Lemma 11,

$$X_r\underset{r\to \infty }{\overset{\mathcal{D}}{\to }}{P}_{\zeta ,\sigma }.$$

(25)

Now, we apply Lemma 6 and find that, for fixed $\sigma >{\displaystyle \frac{n-1}{2}}$ and $\epsilon >0$,

$$\begin{array}{ccc}& & \underset{r\to \infty }{lim}\underset{N\to \infty }{lim\; sup}\mathcal{Q}\left\{\left|Y_{N,M}^h\left(\sigma \right)-X_{N,M,r}^h\left(\sigma \right)\right|⩾\epsilon \right\}\hfill \\ \hfill =& & \underset{r\to \infty }{lim}\underset{N\to \infty }{lim\; sup}{\displaystyle \frac{1}{(M+1)\epsilon }}\sum _{k=N}^{N+M}\left|\zeta (\sigma +ikh;Q)-{\zeta }_r(\sigma +ikh;Q)\right|=0.\hfill \end{array}$$

The later equality, relations (24) and (25) together with Lemma 14 show that

$$Y_{N,M}\underset{N\to \infty }{\overset{\mathcal{D}}{\to }}{P}_{\zeta ,\sigma },$$

i.e., $P_{N,M,\sigma }\underset{N\to \infty }{\overset{\mathrm{w}}{\to }}{P}_{\zeta ,\sigma }$.

Case for h of type 2. We repeat the proof of Theorem 1 in the case of h of type preserving the notations with minor changes.

By Lemma 12, we have

$$X_{N,M,r}^h\left(\sigma \right)\underset{N\to \infty }{\overset{\mathcal{D}}{\to }}{X}_r^h\left(\sigma \right),$$

(26)

where $X_r^h\left(\sigma \right)$ is the $\mathbb{C}$-valued random element with the distribution $P_{r,\sigma }^h$, and from Lemma 13 it follows that

$$X_r^h\left(\sigma \right)\underset{r\to \infty }{\overset{\mathcal{D}}{\to }}{P}_{\zeta ,\sigma }^h.$$

(27)

Using Lemma 6 yields, for $\sigma >{\displaystyle \frac{n-1}{2}}$ and $\epsilon >0$,

$$\begin{array}{c}\hfill \underset{r\to \infty }{lim}\underset{N\to \infty }{lim\; sup}\mathcal{Q}\left\{\left|Y_{N,M}^h\left(\sigma \right)-X_{N,M,r}^h\left(\sigma \right)\right|⩾\epsilon \right\}=0.\end{array}$$

(28)

Now, (26)–(28) and Lemma 14 prove the theorem in the case of h of type 2, i.e.,

$$P_{N,M,\sigma }^h\underset{N\to \infty }{\overset{\mathrm{w}}{\to }}{P}_{\zeta ,\sigma }^h.$$

The proof of Theorem 1 is complete. □

4. Conclusions

Assuming that Q is a positive-definite quadratic matrix of order $n\in 2\mathbb{N}$, $n⩾4$, and that ${\underline{x}}^{T}Q\underline{x}\in \mathbb{Z}$ for all $\underline{x}\in {\mathbb{Z}}^n\setminus \left\{\underline{0}\right\}$, we considered asymptotic properties of the frequency of the values $\zeta (\sigma +ikh;Q)$ of the Epstein zeta-function in the interval $[N,N+M]$, where $h^{-1}{\left(Nh\right)}^{27/82}⩽M⩽h^{-1}{\left(Nh\right)}^{1/2}$, as $N\to \infty$, and $h>0$ and $\sigma >{\displaystyle \frac{n-1}{2}}$ are fixed. Applying a probabilistic technique, we obtained that the above frequency converges weakly to the distribution of a certain explicitly given complex-valued random element. This random element depends on the arithmetic of the number h. The result obtained improves our earlier result, proved for the interval $[0,N]$, and is closely connected to the mean square estimate

$$\underset{T-H}{\overset{T+H}{\int }}{\left|L(\sigma +it,\chi )\right|}^2\mathrm{d}t{\ll }_{\sigma }H,\sigma >{\displaystyle \frac{1}{2}},$$

(29)

which holds in short intervals ($H=o\left(T\right)$ as $T\to \infty$) for Dirichlet L-functions $L(s,\chi )$. We believe that the lower bound for M can be decreased by using a more precise estimate for H in (29).

In the future, we plan to extend the results of this paper to the space $H\left(D\right)$, i.e., to obtain weak convergence of probability measures

$${\displaystyle \frac{1}{H}}\mathrm{meas}\left\{t\in [T,T+H]:\zeta (s+i\tau ;Q)\in A\right\}\mathrm{and}{D}_{N,M}\left(\zeta (s+i\tau ;Q)\in A\right)$$

with $A\in \mathcal{B}\left(H\right(D\left)\right)$ in short intervals ($H=o\left(T\right)$ as $T\to \infty$, and $H=o\left(N\right)$ as $N\to \infty$), where $D=\{s\in \mathbb{C}:\frac{n-1}{2}<\sigma <\frac{n}{2}\}$, and $H\left(D\right)$ denotes the space of analytic functions on D endowed with the topology of uniform convergence on compact sets. Such limit theorems and equivalents of weak convergence lead to the universality of $\zeta (s;Q)$ in short intervals. Note that the universality theorem in short intervals is an important step towards the effectivization of universality, i.e., the detection of approximating shifts $\zeta (s+i\tau ;Q)$ for a given analytic function.