On the Mishou Theorem for Zeta-Functions with Periodic Coefficients

Abstract

Let $\mathfrak{a}=\left\{a_m\right\}$ and $\mathfrak{b}=\left\{b_m\right\}$ be two periodic sequences of complex numbers, and, additionally, $\mathfrak{a}$ is multiplicative. In this paper, the joint approximation of a pair of analytic functions by shifts $({\zeta }_{n_T}(s+i\tau ;\mathfrak{a}),{\zeta }_{n_T}(s+i\tau ,\alpha ;\mathfrak{b}))$ of absolutely convergent Dirichlet series ${\zeta }_{n_T}(s;\mathfrak{a})$ and ${\zeta }_{n_T}(s,\alpha ;\mathfrak{b})$ involving the sequences $\mathfrak{a}$ and $\mathfrak{b}$ is considered. Here, $n_T\to \infty$ and $n_T\ll T^2$ as $T\to \infty$. The coefficients of these series tend to $a_m$ and $b_m$, respectively. It is proved that the set of the above shifts in the interval $[0,T]$ has a positive density. This generalizes and extends the Mishou joint universality theorem for the Riemann and Hurwitz zeta-functions.

1. Introduction

Let $\mathfrak{a}=\{a_m:m\in \mathbb{N}\}$ and $\mathfrak{b}=\{b_m:m\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}\}$ be two periodic sequences of complex numbers with minimal periods $q_1\in \mathbb{N}$ and $q_2\in {\mathbb{N}}_0$, respectively, $0<\alpha ⩽1$ a fixed parameter, and $s=\sigma +it$ a complex variable. The periodic $\zeta (s;\mathfrak{a})$ and periodic Hurwitz $\zeta (s,\alpha ;\mathfrak{b})$ zeta-functions are defined, for $\sigma >1$, by the Dirichlet series

$$\zeta (s;\mathfrak{a})=\sum _{m=1}^{\infty }\frac{a_m}{m^s}\mathrm{and}\zeta (s,\alpha ;\mathfrak{b})=\sum _{m=0}^{\infty }\frac{b_m}{{(m+\alpha )}^s}.$$

When $a_m\equiv 1$ and $b_m\equiv 1$, the functions $\zeta (s;\mathfrak{a})$ and $\zeta (s,\alpha ;\mathfrak{b})$ reduce to the classical Riemann zeta-function $\zeta \left(s\right)$ and Hurwitz zeta-function $\zeta (s,\alpha )$, respectively. In view of the periodicity of the sequences $\mathfrak{a}$ and $\mathfrak{b}$, for $\sigma >1$, it follows that

$$\zeta (s;\mathfrak{a})=\frac{1}{q_1^s}\sum _{l=1}^{q_1}{a}_l\zeta \left(s,\frac{l}{q_1}\right)\mathrm{and}\zeta (s,\alpha ;\mathfrak{b})=\frac{1}{q_2^s}\sum _{l=0}^{q_2-1}{b}_l\zeta \left(s,\frac{l+\alpha }{q_2}\right).$$

Thus, the properties of the function $\zeta (s,\alpha )$ imply the analytic continuation for the functions $\zeta (s;\mathfrak{a})$ and $\zeta (s,\alpha ;\mathfrak{b})$ to the whole complex plane, except the point $s=1$ which is a simple pole with residues

$$a\stackrel{\mathrm{def}}{=}\frac{1}{q_1}\sum _{l=1}^{q_1}{a}_l\mathrm{and}b\stackrel{\mathrm{def}}{=}\frac{1}{q_2}\sum _{l=0}^{q_2-1}{b}_l,$$

respectively. If $a=0$, then the function $\zeta (s;\mathfrak{a})$ is entire, and if $b=0$, then the function $\zeta (s,\alpha ;\mathfrak{b})$ is entire.

Examples of the function $\zeta (s;\mathfrak{a})$ are Dirichlet L-functions

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

and of the function $\zeta (s,\alpha ;\mathfrak{b})$ they are Lerch zeta-functions

$$L(\lambda ,\alpha ,s)=\sum _{m=0}^{\infty }\frac{{\mathrm{e}}^{2\pi i\lambda m}}{{(m+\alpha )}^s},\sigma >1,$$

with rational parameter $\lambda$.

The analytic properties of the functions $\zeta (s;\mathfrak{a})$ and $\zeta (s,\alpha ;\mathfrak{b})$, including the universality property of the approximation of the analytic functions by shifts $\zeta (s+i\tau ;\mathfrak{a})$ and $\zeta (s+i\tau ,\alpha ;\mathfrak{b})$, $\tau \in \mathbb{R}$, are closely connected to the sequence $\mathfrak{a}$, the sequence $\mathfrak{b}$, and parameter $\alpha$, respectively.

Let $\Delta =\{s\in \mathbb{C}:1/2<\sigma <1\}$. Denote by $\mathfrak{K}$ the class of compact sets of the strip $\Delta$ with connected complements, by $\mathcal{H}\left(K\right)$ with $K\in \mathfrak{K}$ the class of continuous functions on K that are analytic in the interior of K, and by ${\mathcal{H}}_0\left(K\right)$ the subclass of $\mathcal{H}\left(K\right)$ of non-vanishing on K functions. Let $\mathfrak{M}A$ stand for the Lebesgue measure of a measurable set $A\subset \mathbb{R}$.

The first universality results for the function $\zeta (s;\mathfrak{a})$ were obtained by J. Steuding. In [1], he proved that if $\mathfrak{a}$ is not a multiple of the Dirichlet character modulo $q_1$, and $a_m=0$ for $(m,q_1)>1$, then for $K\in \mathfrak{K}$, $f\left(s\right)\in \mathcal{H}\left(K\right)$ and all $\epsilon >0$,

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

(1)

Under the above conditions on the sequence $\mathfrak{a}$, this sequence is not multiplicative. We recall that the sequence $\mathfrak{a}$ is multiplicative, if $a_1=1$ and $a_{mn}=a_m{a}_n$ for all $m,n\in \mathbb{N}$, $(m,n)=1$. The universality of the function $\zeta (s;\mathfrak{a})$ with multiplicative sequence $\mathfrak{a}$ was proved in [2]. In [3], it was obtained that there exists a constant $c_0=c_0\left(\mathfrak{a}\right)$ such that, for $K\in \mathfrak{K}$, ${max}_{s\in K}\mathrm{Im}s-{min}_{s\in K}\mathrm{Im}s⩽c_0$, $f\left(s\right)\in H_0\left(K\right)$ and $\epsilon >0$, equality (1) holds.

The universality properties of the function $\zeta (s,\alpha )$ are included in the following theorem [4,5,6]. Suppose that α is transcendental or rational, not equal to 1 or $1/2$. Let $K\in \mathfrak{K}$ and $f\left(s\right)\in \mathcal{H}\left(K\right)$. Then, for all $\epsilon >0$,

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

The universality of $\zeta (s,\alpha )$ with algebraic irrational $\alpha$ remains an open problem up to our days. A certain approximation to this problem is given in [7], and see also [8]. The best result in this direction was obtained in [9]. The universality property of the function $\zeta (s,\alpha ;\mathfrak{b})$ was first studied in [10], and similar theorems to those for $\zeta (s,\alpha )$ with transcendental and algebraic irrational $\alpha$ were obtained in [11,12]. The case of rational $\alpha$ is studied in [13]. In this case, some hypotheses for the sequence $\mathfrak{b}$ are also involved.

The aim of this paper is the joint universality of certain Dirichlet series connected to the functions $\zeta (s;\mathfrak{a})$ and $\zeta (s,\alpha ;\mathfrak{b})$. Recall that the first joint universality theorem for the functions $\zeta \left(s\right)$ and $\zeta (s,\alpha )$ with transcendental $\alpha$ was obtained by H. Mishou in [14]. Suppose that $K_1,K_2\in \mathfrak{K}$ and $f_1\left(s\right)\in {\mathcal{H}}_0\left(K_1\right)$, $f_2\left(s\right)\in \mathcal{H}\left(K_2\right)$. Then, he proved that, for all $\epsilon >0$,

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

A similar result for the functions $\zeta (s;\mathfrak{a})$ and $\zeta (s,\alpha ;\mathfrak{b})$ was given in [15]. The approximation problem of a pair of analytic functions by shifts $\left(\zeta \right(s+i\tau ;\mathfrak{a}),\zeta (s+i\tau ,\alpha ;\mathfrak{b}\left)\right)$ with algebraic irrational $\alpha$ was considered in [16]. More general joint universality results for periodic and periodic Hurwitz zeta-functions can be found in [17,18,19,20]. A weighted generalization of the Mishou theorem was obtained in [21].

The abovementioned universality results are of a continuous type because $\tau$ in shifts takes arbitrary real values. Moreover, there are results of a discrete type when $\tau$ takes values in a certain discrete set, see, for example, [22,23,24,25,26,27,28,29,30].

Let $\theta >1/2$ be a fixed number, $u>0$, and

$$v_u(m;\theta )=exp\left\{-{\left(\frac{m}{u}\right)}^{\theta }\right\},v_u(m,\alpha ;\theta )=exp\left\{-{\left(\frac{m+\alpha }{u}\right)}^{\theta }\right\}.$$

Define the series

$${\zeta }_u(s;\mathfrak{a})=\sum _{m=1}^{\infty }\frac{a_m{v}_u(m;\theta )}{m^s},{\zeta }_u(s,\alpha ;\mathfrak{b})=\sum _{m=0}^{\infty }\frac{b_m{v}_u(m,\alpha ;\theta )}{{(m+\alpha )}^s}.$$

Then, the latter series are absolutely convergent for $\sigma >1/2$. Really, in view of the exponential decreasing of $v_u(m;\theta )$ and $v_u(m,\alpha ;\theta )$, these series are absolutely convergent for $\sigma >{\sigma }_0$ for all finite ${\sigma }_0$. We will consider the approximation of pairs of analytic functions by shifts $({\zeta }_{n_T}(s+i\tau ;\mathfrak{a}),{\zeta }_{n_T}(s+i\tau ,\alpha ;\mathfrak{b}))$, where $n_T\to \infty$ as $T\to \infty$. For the statement of a theorem, we need some definitions. Denote by $\eta$ the unit circle on the complex plane, and by $\mathcal{B}\left(\mathcal{X}\right)$ the Borel $\sigma$-field of the space $\mathcal{X}$. Define two tori

$${\mathrm{\Omega }}_1=\prod _{p\in \mathbb{P}}{\eta }_p\mathrm{and}{\mathrm{\Omega }}_2=\prod _{m\in {\mathbb{N}}_0}{\eta }_m,$$

where ${\eta }_p=\eta$ for all $p\in \mathbb{P}$ ($\mathbb{P}$ is the set of all prime numbers), and ${\eta }_m=\eta$ for all $m\in {\mathbb{N}}_0$. With the product topology and pointwise multiplication, the tori ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$ are compact topological Abelian groups. Therefore, by the Tikhonov theorem [31],

$$\mathrm{\Omega }={\mathrm{\Omega }}_1\times {\mathrm{\Omega }}_2$$

also is a compact topological group. Thus, on $(\mathrm{\Omega },\mathcal{B}(\mathrm{\Omega }\left)\right)$, we can define the probability Haar measure ${\mu }_H$, and we have the probability space $(\mathrm{\Omega },\mathcal{B}\left(\mathrm{\Omega }\right),{\mu }_H)$. Denote by $\omega \left(p\right)$ the pth component of an element ${\omega }_1\in {\mathrm{\Omega }}_1$, $p\in \mathbb{P}$, and by ${\omega }_2\left(m\right)$ the mth component of an element ${\omega }_2\in {\mathrm{\Omega }}_2$, $m\in {\mathbb{N}}_0$. Extend the functions ${\omega }_1\left(p\right)$ to the set $\mathbb{N}$ by the formula

$${\omega }_1\left(m\right)=\underset{p^{l+1}\nmid m}{\prod _{p^l\mid m}}{\omega }_1^l\left(p\right),m\in \mathbb{N}.$$

Denote by $\mathcal{H}(\Delta )$ the space of analytic functions on $\Delta$ equipped with the topology of uniform convergence on compact sets, let ${\mathcal{H}}^2(\Delta )=\mathcal{H}(\Delta )\times \mathcal{H}(\Delta )$, and, on the probability space $(\mathrm{\Omega },\mathcal{B}\left(\mathrm{\Omega }\right),{\mu }_H)$, define the ${\mathcal{H}}^2(\Delta )$-valued random element

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

where

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

Note that the latter series are uniformly convergent on compact subsets of the strip $\Delta$ for almost all ${\omega }_1$ and ${\omega }_2$ with respect to the Haar measures ${\mu }_{1H}$ on $({\mathrm{\Omega }}_1,\mathcal{B}\left({\mathrm{\Omega }}_1\right))$ and ${\mu }_{2H}$ on $({\mathrm{\Omega }}_2,\mathcal{B}\left({\mathrm{\Omega }}_2\right))$, respectively. The notation $x{\ll }_{\xi }y$, $y>0$, means that there exists a constant $c=c\left(\xi \right)>0$ such that $\left|x\right|⩽cy$.

Theorem 1.

Suppose that the sequence $\mathfrak{a}$ is multiplicative, α is transcendental, and $n_T\to \infty$ and $n_T\ll T^2$ as $T\to \infty$. Let $K_1,K_2\in \mathfrak{K}$, and $f_1\left(s\right)\in {\mathcal{H}}_0\left(K_1\right)$, $f_2\left(s\right)\in \mathcal{H}\left(K_2\right)$. Then, the limit

$$\begin{array}{c}\hfill \underset{T\to \infty }{lim}\frac{1}{T}\mathfrak{M}\{\tau \in [0,T]:\underset{s\in K_1}{sup}|{\zeta }_{n_T}(s+i\tau ;\mathfrak{a})-f_1\left(s\right)|<{\epsilon }_1,\\ \hfill \underset{s\in K_2}{sup}|{\zeta }_{n_T}(s+i\tau ,\alpha ;\mathfrak{b})-f_2\left(s\right)|<{\epsilon }_2\}\\ \hfill ={\mu }_H\{({\omega }_1,{\omega }_2)\in \mathrm{\Omega }:\underset{s\in K_1}{sup}|\zeta (s,{\omega }_1;\mathfrak{a})-f_1\left(s\right)|<{\epsilon }_1,\\ \hfill \underset{s\in K_2}{sup}|\zeta (s,\alpha ,{\omega }_2;\mathfrak{b})-f_2\left(s\right)|<{\epsilon }_2\}\end{array}$$

exists and is positive for all but at most countably many ${\epsilon }_1>0$ and ${\epsilon }_2>0$.

The first result on the approximation of the analytic functions by shifts of the absolutely convergent Dirichlet series was obtained in [32] and generalized in [33]. Discrete versions of the latter results are given in [34,35]

Theorem 1 extends the previous results on the universality of the Dirichlet series involving periodic sequences in two directions. Firstly, Theorem 1 is a joint universality on the simultaneous approximation of a pair of analytic functions. Secondly, the analytic functions are approximated by shifts of absolutely convergent series. This moment is a certain advantage in the estimation of approximated functions.

2. Approximation in the Mean

Recall the metric in the space $H^2(\Delta )$. There exists a sequence of compact sets $\{K_l:l\in \mathbb{N}\}\subset \Delta$ satisfying the requirements:

• $\Delta$ is the union of the sets $K_l$;
• $K_l\subset K_{l+1}$ for all $l\in \mathbb{N}$;
• For every compact set $K\subset \Delta$, there exists $K_l$ such that $K\subset K_l$.

Then,

$$\rho (F_1,F_2)=\sum _{l=1}^{\infty }{2}^{-l}\frac{{sup}_{s\in K_l}|F_1\left(s\right)-F_2\left(s\right)|}{1+{sup}_{s\in K_l}|F_1\left(s\right)-F_2\left(s\right)|},F_1,F_2\in \mathcal{H}(\Delta ),$$

is a metric in $\mathcal{H}(\Delta )$ inducing its topology of uniform convergence on compacta. Putting, for ${\underline{F}}_1=(F_{11},F_{12}),{\underline{F}}_2=(F_{21},F_{22})\in {\mathcal{H}}^2(\Delta )$,

$${\rho }_2({\underline{F}}_1,{\underline{F}}_2)=\underset{j=1,2}{max}\rho (F_{1j},F_{2j})$$

gives a metric in ${\mathcal{H}}^2(\Delta )$ inducing the product topology.

Lemma 1.

Suppose that $n_T\to \infty$ and $n_T\ll T^2$ as $T\to \infty$. Let

$$\underline{\zeta }(s,\alpha ;\mathfrak{a},\mathfrak{b})=\left(\zeta (s;\mathfrak{a}),\zeta (s,\alpha ;\mathfrak{b})\right)$$

and

$${\underline{\zeta }}_{n_T}(s,\alpha ;\mathfrak{a},\mathfrak{b})=\left({\zeta }_{n_T}(s;\mathfrak{a}),{\zeta }_{n_T}(s,\alpha ;\mathfrak{b})\right).$$

Then,

$$\underset{T\to \infty }{lim}\frac{1}{T}{\int }_0^T{\rho }_2\left(\underline{\zeta }(s+i\tau ,\alpha ;\mathfrak{a},\mathfrak{b}),{\underline{\zeta }}_{n_T}(s+i\tau ,\alpha ;\mathfrak{a},\mathfrak{b})\right)\mathrm{d}\tau =0.$$

Proof. 

By the definition of the metric ${\rho }_2$, it suffices to show that

$$\underset{T\to \infty }{lim}\frac{1}{T}{\int }_0^T\rho \left(\zeta (s+i\tau ;\mathfrak{a}),{\zeta }_{n_T}(s+i\tau ;\mathfrak{a})\right)\mathrm{d}\tau =0$$

and

$$\underset{T\to \infty }{lim}\frac{1}{T}{\int }_0^T\rho \left(\zeta (s+i\tau ,\alpha ;\mathfrak{b}),{\zeta }_{n_T}(s+i\tau ,\alpha ;\mathfrak{b})\right)\mathrm{d}\tau =0.$$

The first of these equalities follows from Lemma 2 of [33] which states that, for every compact set $K\subset \Delta$,

$$\underset{T\to \infty }{lim}\frac{1}{T}{\int }_0^T\underset{s\in K}{sup}\left|\zeta (s+i\tau ;\mathfrak{a})-{\zeta }_{n_T}(s+i\tau ;\mathfrak{a})\right|\mathrm{d}\tau =0,$$

and from the definition of the metric $\rho$. The second equality is obtained similarly using the representation

$${\zeta }_{n_T}(s,\alpha ;\mathfrak{b})=\frac{1}{2\pi i}{\int }_{\theta -i\infty }^{\theta +i\infty }\zeta (s+z,\alpha ;\mathfrak{b})l_{n_T}(z;\theta )\mathrm{d}z,$$

where $s\in \Delta$, $\mathrm{\Gamma }\left(s\right)$ is the Euler gamma-function, and

$$l_{n_T}(s;\theta )=\frac{1}{\theta }\mathrm{\Gamma }\left(\frac{s}{\theta }\right)n_T^s.$$

□

3. Limit Theorem

We will apply a limit theorem in the space ${\mathcal{H}}^2(\Delta )$ obtained in [15]. For $A\in \mathcal{B}\left({\mathcal{H}}^2(\Delta )\right)$, define

$$P_{T,\alpha ,\mathfrak{a},\mathfrak{b}}\left(A\right)=\frac{1}{T}\mathfrak{M}\left\{\tau \in [0,T]:\underline{\zeta }(s+i\tau ,\alpha ;\mathfrak{a},\mathfrak{b})\in A\right\}.$$

Moreover, let $P_{\underline{\zeta },\alpha ,\mathfrak{a},\mathfrak{b}}$ be the distribution of the random element $\underline{\zeta }(s+i\tau ,\alpha ,{\omega }_1,{\omega }_2;\mathfrak{a},\mathfrak{b})$, i.e.,

$$P_{\underline{\zeta },\alpha ,\mathfrak{a},\mathfrak{b}}\left(A\right)={\mu }_H\{({\omega }_1,{\omega }_2)\in \mathrm{\Omega }:\underline{\zeta }(s+i\tau ,\alpha ,{\omega }_1,{\omega }_2;\mathfrak{a},\mathfrak{b})\in A\}.$$

Lemma 2.

Suppose that the sequence $\mathfrak{a}$ is multiplicative and the parameter α is transcendental. Then, $P_{T,\alpha ,\mathfrak{a},\mathfrak{b}}$ converges weakly to $P_{\underline{\zeta },\alpha ,\mathfrak{a},\mathfrak{b}}$ as $T\to \infty$. Moreover, the support of the measure $P_{\underline{\zeta },\alpha ,\mathfrak{a},\mathfrak{b}}$ is the set

$$\{g\in \mathcal{H}(\Delta ):\mathrm{either}g(s)\ne 0\mathrm{on}\Delta ,\mathrm{or}g(s)\equiv 0\}\times \mathcal{H}(\Delta ).$$

Proof. 

The lemma is the union of Theorem 6 and Lemma 12 from [15]. □

Now, we consider a limit theorem for ${\underline{\zeta }}_{n_T}(s+i\tau ,\alpha ;\mathfrak{a},\mathfrak{b})$. For $A\in \mathcal{B}\left({\mathcal{H}}^2(\Delta )\right)$, define

$${\widehat{P}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}}\left(A\right)=\frac{1}{T}\mathfrak{M}\left\{\tau \in [0,T]:{\underline{\zeta }}_{n_T}(s+i\tau ,\alpha ;\mathfrak{a},\mathfrak{b})\in A\right\}.$$

Theorem 2.

Suppose that the sequence $\mathfrak{a}$ is multiplicative, the parameter α is transcendental, and $n_T\to \infty$ and $n_T\ll T^2$ as $T\to \infty$. Then, ${\widehat{P}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}}$ converges weakly to $P_{\underline{\zeta },\alpha ,\mathfrak{a},\mathfrak{b}}$ as $T\to \infty$.

Proof. 

Let ${\theta }_T$ be a random variable defined on a certain probability space $(\widehat{\mathrm{\Omega }},\mathcal{A},P)$ and uniformly distributed on the segment $[0,T]$. Define the ${\mathcal{H}}^2(\Delta )$-valued random elements

$${\underline{X}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}}={\underline{X}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}}\left(s\right)=\left(X_{T,\mathfrak{a}}\left(s\right),X_{T,\alpha ,\mathfrak{b}}\left(s\right)\right),$$

where

$$X_{T,\mathfrak{a}}\left(s\right)=\zeta (s+i{\theta }_T;\mathfrak{a}),X_{T,\alpha ,\mathfrak{b}}\left(s\right)=\zeta (s+i{\theta }_T,\alpha ;\mathfrak{b}),$$

and

$${\underline{\widehat{X}}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}}={\underline{\widehat{X}}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}}\left(s\right)=\left({\widehat{X}}_{T,\mathfrak{a}}\left(s\right),{\widehat{X}}_{T,\alpha ,\mathfrak{b}}\left(s\right)\right),$$

where

$${\widehat{X}}_{T,\mathfrak{a}}\left(s\right)={\zeta }_{n_T}(s+i{\theta }_T;\mathfrak{a}),{\widehat{X}}_{T,\alpha ,\mathfrak{b}}\left(s\right)={\zeta }_{n_T}(s+i{\theta }_T,\alpha ;\mathfrak{b}).$$

By the definitions of ${\theta }_T$, ${\underline{X}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}}$ and ${\underline{\widehat{X}}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}}$, for $A\in \mathcal{B}\left({\mathcal{H}}^2(\Delta )\right)$, we have

$$P\left\{{\underline{X}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}}\in A\right\}=P_{T,\alpha ,\mathfrak{a},\mathfrak{b}}\left(A\right)$$

(2)

and

$$P\left\{{\underline{\widehat{X}}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}}\in A\right\}={\widehat{P}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}}\left(A\right).$$

(3)

Fix $\epsilon >0$, a closed set $F\subset {\mathcal{H}}^2(\Delta )$, and define

$$F_{\epsilon }=\left\{\underline{F}\in {\mathcal{H}}^2(\Delta ):{\rho }_2(\underline{F},F)⩽\epsilon \right\},$$

where ${\rho }_2(\underline{F},F)={inf}_{\underline{\widehat{F}}\in F}{\rho }_2(\underline{F},\underline{\widehat{F}})$. Then, Lemma 2, equality (2), and the equivalent of weak convergence in terms of closed sets [36] show that

$$\underset{T\to \infty }{lim\; sup}{P}_{T,\alpha ,\mathfrak{a},\mathfrak{b}}\left(F_{\epsilon }\right)=\underset{T\to \infty }{lim\; sup}P\{X_{T,\alpha ,\mathfrak{a},\mathfrak{b}}\in F_{\epsilon }\}⩽P_{\underline{\zeta },\alpha ,\mathfrak{a},\mathfrak{b}}\left(F_{\epsilon }\right).$$

(4)

It is easily seen that

$$\left\{{\underline{\widehat{X}}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}}\in F\right\}\subset \left\{{\underline{X}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}}\in F_{\epsilon }\right\}\cup \left\{{\rho }_2({\underline{X}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}},{\underline{\widehat{X}}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}})⩾\epsilon \right\}.$$

Note that ${\rho }_2({\underline{X}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}},{\underline{\widehat{X}}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}})$ is a random variable, and, by the definition of ${\theta }_T$, its expectation is

$$\frac{1}{T}{\int }_0^T{\rho }_2\left(\underline{\zeta }(s+i\tau ,\alpha ;\mathfrak{a},\mathfrak{b}),{\underline{\zeta }}_{n_T}(s+i\tau ,\alpha ;\mathfrak{a},\mathfrak{b})\right)\mathrm{d}\tau .$$

Thus,

$$P\left\{{\underline{\widehat{X}}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}}\in F\right\}⩽P\left\{{\underline{X}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}}\in F_{\epsilon }\right\}+P\left\{{\rho }_2({\underline{X}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}},{\underline{\widehat{X}}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}})⩾\epsilon \right\},$$

(5)

and Lemma 1 together with Chebyshev’s type inequality

$$\begin{array}{cc}\hfill \mathfrak{M}& \left\{\tau \in [0,T]:{\rho }_2\left(\underline{\zeta }(s+i\tau ,\alpha ;\mathfrak{a},\mathfrak{b}),{\underline{\zeta }}_{n_T}(s+i\tau ,\alpha ;\mathfrak{a},\mathfrak{b})\right)⩾\epsilon \right\}\hfill \\ & ⩽\frac{1}{\epsilon }{\int }_0^T{\rho }_2\left(\underline{\zeta }(s+i\tau ,\alpha ;\mathfrak{a},\mathfrak{b}),{\underline{\zeta }}_{n_T}(s+i\tau ,\alpha ;\mathfrak{a},\mathfrak{b})\right)\mathrm{d}\tau \hfill \end{array}$$

implies that

$$\begin{array}{cc}\hfill P\left\{{\rho }_2\left({\underline{X}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}},{\underline{\widehat{X}}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}}\right)⩾\epsilon \right\}& ⩽\frac{1}{\epsilon T}{\int }_0^T{\rho }_2\left(\underline{\zeta }(s+i\tau ,\alpha ;\mathfrak{a},\mathfrak{b}),{\underline{\zeta }}_{n_T}(s+i\tau ,\alpha ;\mathfrak{a},\mathfrak{b})\right)\mathrm{d}\tau \hfill \\ & =0.\hfill \end{array}$$

(6)

Therefore, in view of (5) and (6),

$$\underset{T\to \infty }{lim\; sup}P\left\{{\underline{\widehat{X}}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}}\in F\right\}⩽\underset{T\to \infty }{lim\; sup}P\left\{{\underline{X}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}}\in F_{\epsilon }\right\},$$

and, by (2), (3), and (4),

$$\underset{T\to \infty }{lim\; sup}{\widehat{P}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}}\left(F\right)⩽P_{\underline{\zeta },\alpha ,\mathfrak{a},\mathfrak{b}}\left(F_{\epsilon }\right).$$

Because $F_{\epsilon }\to F$ as $\epsilon \to +0$, this gives

$$\underset{T\to \infty }{lim\; sup}{\widehat{P}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}}\left(F\right)⩽P_{\underline{\zeta },\alpha ,\mathfrak{a},\mathfrak{b}}\left(F\right),$$

and the equivalent of weak convergence in terms of closed sets proves the theorem. □

Let $K_1$, $K_2$, and $f_1\left(s\right)$, $f_2\left(s\right)$ be as in Theorem 1. For $A\in \mathcal{B}\left({\mathbb{R}}^2\right)$, define

$$\begin{array}{c}\hfill Q_{T,\alpha ,\mathfrak{a},\mathfrak{b}}\left(A\right)=\frac{1}{T}\mathfrak{M}\{\tau \in [0,T]:(\underset{s\in K_1}{sup}|{\zeta }_{n_T}(s+i\tau ;\mathfrak{a})-f_1\left(s\right)|\\ \hfill \underset{s\in K_2}{sup}|{\zeta }_{n_T}(s+i\tau ,\alpha ;\mathfrak{b})-f_2\left(s\right)|)\in A\}.\end{array}$$

Corollary 1.

Under hypotheses of Theorem 2, $Q_{T,\alpha ,\mathfrak{a},\mathfrak{b}}$ converges weakly to the measure

$$\begin{array}{cc}\hfill {\mu }_H\{({\omega }_1,{\omega }_2)\in \mathrm{\Omega }:& (\underset{s\in K_1}{sup}|{\zeta }_{n_T}(s,{\omega }_1;\mathfrak{a})-f_1\left(s\right)|,\hfill \\ & \underset{s\in K_2}{sup}|{\zeta }_{n_T}(s,\alpha ,{\omega }_2;\mathfrak{a})-f_2\left(s\right)|)\in A\},A\in \mathcal{B}\left({\mathbb{R}}^2\right),\hfill \end{array}$$

as $T\to \infty$.

Proof. 

Define the function $h:{\mathcal{H}}^2(\Delta )\to {\mathbb{R}}^2$ by the formula

$$h(F_1,F_2)=\left(\underset{s\in K_1}{sup}|F_1\left(s\right)-f_1\left(s\right)|,\underset{s\in K_2}{sup}|F_2\left(s\right)-f_2\left(s\right)|\right).$$

Because the space $\mathcal{H}(\Delta )$ is equipped with the topology of the uniform convergence on compacta, the function h is continuous. Therefore, using a property of weak convergence preservation under continuous mappings [36], by Theorem 2, we have that ${\widehat{P}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}}{h}^{-1}$ converges weakly to $P_{\underline{\zeta },\alpha ,\mathfrak{a},\mathfrak{b}}{h}^{-1}$ as $T\to \infty$. However,

$$\begin{array}{cc}\hfill {\widehat{P}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}}{h}^{-1}\left(A\right)& ={\widehat{P}}_{T,\alpha ,\mathfrak{a},\mathfrak{b}}\left(h^{-1}A\right)=\frac{1}{T}\mathfrak{M}\{\tau \in [0,T]:{\underline{\zeta }}_{n_T}(s+i\tau ,\alpha ;\mathfrak{a},\mathfrak{b})\in h^{-1}A\}\hfill \\ & =Q_{T,\alpha ,\mathfrak{a},\mathfrak{b}}\left(A\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & P_{\underline{\zeta },\alpha ,\mathfrak{a},\mathfrak{b}}{h}^{-1}\left(A\right)=P_{\underline{\zeta },\alpha ,\mathfrak{a},\mathfrak{b}}\left(h^{-1}A\right)\hfill \\ & ={\mu }_H\left\{({\omega }_1,{\omega }_2)\in \mathrm{\Omega }:\left(\underset{s\in K_1}{sup}|\zeta (s,{\omega }_1;\mathfrak{a})-f_1\left(s\right)|,\underset{s\in K_2}{sup}|\zeta (s,\alpha ,{\omega }_2;\mathfrak{a})-f_2\left(s\right)|\right)\in A\right\}.\hfill \end{array}$$

This proves the corollary. □

Taking $A=(-\infty ,{\epsilon }_1)\times (-\infty ,{\epsilon }_2)$ in the definition of $Q_{T,\alpha ,\mathfrak{a},\mathfrak{b}}$ and its limit measure, we obtain the distribution functions

$$\begin{array}{c}\hfill F_{T,\alpha ,\mathfrak{a},\mathfrak{b}}({\epsilon }_1,{\epsilon }_2)=\frac{1}{T}\mathfrak{M}\{\tau \in [0,T]:\underset{s\in K_1}{sup}|{\zeta }_{n_T}(s+i\tau ;\mathfrak{a})-f_1\left(s\right)|<{\epsilon }_1,\\ \hfill \underset{s\in K_2}{sup}|{\zeta }_{n_T}(s+i\tau ,\alpha ;\mathfrak{b})-f_2\left(s\right)|<{\epsilon }_2\}\end{array}$$

and

$$\begin{array}{c}\hfill F_{\underline{\zeta },\alpha ,\mathfrak{a},\mathfrak{b}}({\epsilon }_1,{\epsilon }_2)={\mu }_H\{({\omega }_1,{\omega }_2)\in \mathrm{\Omega }:\underset{s\in K_1}{sup}|\zeta (s,{\omega }_1;\mathfrak{a})-f_1\left(s\right)|<{\epsilon }_1,\\ \hfill \underset{s\in K_2}{sup}|\zeta (s,\alpha ,{\omega }_2;\mathfrak{b})-f_2\left(s\right)|<{\epsilon }_2\}.\end{array}$$

It is well-known that the weak convergence of probability measures on $({\mathbb{R}}^2,\mathcal{B}\left({\mathbb{R}}^2\right))$ is equivalent to that of the corresponding distribution functions. Recall that $F_{T,\alpha ,\mathfrak{a},\mathfrak{b}}({\epsilon }_1,{\epsilon }_2)$ converges weakly to $F_{\underline{\zeta },\alpha ,\mathfrak{a},\mathfrak{b}}({\epsilon }_1,{\epsilon }_2)$ if

$$\underset{T\to \infty }{lim}{F}_{T,\alpha ,\mathfrak{a},\mathfrak{b}}({\epsilon }_1,{\epsilon }_2)=F_{\underline{\zeta },\alpha ,\mathfrak{a},\mathfrak{b}}({\epsilon }_1,{\epsilon }_2)$$

for all $({\epsilon }_1,{\epsilon }_2)$ such that ${\epsilon }_1$ and ${\epsilon }_2$ are continuity points of the functions $F_{\underline{\zeta },\alpha ,\mathfrak{a},\mathfrak{b}}({\epsilon }_1,+\infty )$ and $F_{\underline{\zeta },\alpha ,\mathfrak{a},\mathfrak{b}}(+\infty ,{\epsilon }_2)$, respectively. Thus, Corollary 1 implies the following:

Corollary 2.

Under hypotheses of Theorem 2, the distribution function $F_{T,\alpha ,\mathfrak{a},\mathfrak{b}}({\epsilon }_1,{\epsilon }_2)$ converges weakly to the distribution function $F_{\underline{\zeta },\alpha ,\mathfrak{a},\mathfrak{b}}({\epsilon }_1,{\epsilon }_2)$ as $T\to \infty$.

4. Proof of Theorem 1

Proof of Theorem 1.

Because the set of the discontinuity points of the distribution function is at most countable, by Corollary 2, the limit

$$\underset{T\to \infty }{lim}{F}_{T,\alpha ,\mathfrak{a},\mathfrak{b}}({\epsilon }_1,{\epsilon }_2)=F_{\underline{\zeta },\alpha ,\mathfrak{a},\mathfrak{b}}({\epsilon }_1,{\epsilon }_2)$$

exists for all but at most countably many ${\epsilon }_1>0$ and ${\epsilon }_2>0$. Thus, it remains to prove the positivity of $F_{\underline{\zeta },\alpha ,\mathfrak{a},\mathfrak{b}}({\epsilon }_1,{\epsilon }_2)$.

In view of the Mergelyan theorem on the approximation of analytic functions by polynomials [37], there exist polynomials $p_1\left(s\right)$ and $p_2\left(s\right)$ such that

$$\underset{s\in K_1}{sup}\left|f_1\left(s\right)-{\mathrm{e}}^{p_1\left(s\right)}\right|<\frac{{\epsilon }_1}{2}\mathrm{and}\underset{s\in K_2}{sup}|f_2\left(s\right)-p_2\left(s\right)|<\frac{{\epsilon }_2}{2}.$$

(7)

By Lemma 2, the support S of the measure $P_{\underline{\zeta },\alpha ,\mathfrak{a},\mathfrak{b}}$ is the set $\{g\in \mathcal{H}(\Delta ):\mathrm{either}g(s)\ne 0\mathrm{on}D,\mathrm{or}g(s)\equiv 0\}$. Therefore, $\left({\mathrm{e}}^{p_1\left(s\right)},p_2\left(s\right)\right)$ is an element of S. Hence,

$$P_{\underline{\zeta },\alpha ,\mathfrak{a},\mathfrak{b}}\left(G_{{\epsilon }_1,{\epsilon }_2}\right)>0,$$

(8)

where

$$G_{{\epsilon }_1,{\epsilon }_2}=\left\{(F_1,F_2)\in {\mathcal{H}}^2(\Delta ):\underset{s\in K_1}{sup}\left|F_1\left(s\right)-{\mathrm{e}}^{p_1\left(s\right)}\right|<\frac{{\epsilon }_1}{2},\underset{s\in K_2}{sup}|F_2\left(s\right)-p_2\left(s\right)|<\frac{{\epsilon }_2}{2}\right\}.$$

Define one more set

$${\widehat{G}}_{{\epsilon }_1,{\epsilon }_2}=\left\{(F_1,F_2)\in {\mathcal{H}}^2(\Delta )\underset{s\in K_1}{sup}\left|F_1\left(s\right)-f_1\left(s\right)\right|<{\epsilon }_1,\underset{s\in K_2}{sup}|F_2\left(s\right)-f_2\left(s\right)|<{\epsilon }_2\right\}.$$

The inequalities (7) show that if $(F_1,F_2)\in G_{{\epsilon }_1,{\epsilon }_2}$, then $(F_1,F_2)\in {\widehat{G}}_{{\epsilon }_1,{\epsilon }_2}$. Thus, $G_{{\epsilon }_1,{\epsilon }_2}\subset {\widehat{G}}_{{\epsilon }_1,{\epsilon }_2}$. Therefore, in virtue of (8), $P_{\underline{\zeta },\alpha ,\mathfrak{a},\mathfrak{b}}\left({\widehat{G}}_{{\epsilon }_1,{\epsilon }_2}\right)>0$, i.e., $F_{\underline{\zeta },\alpha ,\mathfrak{a},\mathfrak{b}}({\epsilon }_1,{\epsilon }_2)>0$. The theorem is proved. □

5. Conclusions

In this paper, the joint approximation of a pair of analytic functions by shifts of absolutely convergent Dirichlet series

$${\zeta }_{n_T}(s;\mathfrak{a})=\sum _{m=1}^{\infty }\frac{a_m{v}_{n_T}(m;\theta )}{m^s}\mathrm{and}{\zeta }_{n_T}(s,\alpha ;\mathfrak{b})=\sum _{m=0}^{\infty }\frac{b_m{v}_{n_T}(m,\alpha ;\theta )}{{(m+\alpha )}^s}$$

with periodic sequences $\left\{a_m\right\}$ and $\left\{b_m\right\}$, and exponentially decreasing sequences $\left\{v_{n_T}(m;\theta )\right\}$ and $\left\{v_{n_T}(m,\alpha ;\theta )\right\}$, is obtained. It is proved that if $n_T\to \infty$ and $n_T\ll T^2$ as $T\to \infty$, then the set of approximating shifts $({\zeta }_{n_T}(s+i\tau ;\mathfrak{a}),{\zeta }_{n_T}(s+i\tau ,\alpha ;\mathfrak{b}))$ has an explicitly given density on the interval $[0,T]$.

A possible improvement to the main theorem is an extension of the class of functions $n_T$. Moreover, we are planning to invite experts in numerical methods and IT into our group to obtain some numerical calculations of concrete examples. This is a very difficult problem closely connected to the effectivization of universality theorems for zeta-functions.