Approximation by Shifts of Compositions of Dirichlet L-Functions with the Gram Function

Abstract

In this paper, a joint approximation of analytic functions by shifts of Dirichlet L-functions $L(s+i{a}_1{t}_{\tau },{\chi }_1),\dots ,L(s+i{a}_r{t}_{\tau },{\chi }_r)$, where $a_1,\dots ,a_r$ are non-zero real algebraic numbers linearly independent over the field $\mathbb{Q}$ and $t_{\tau }$ is the Gram function, is considered. It is proved that the set of their shifts has a positive lower density.

1. Introduction

Let $\chi :\mathbb{N}\to \mathbb{C}$ be a Dirichlet character modulo $q\in \mathbb{N}$. Note that $\chi \left(m\right)$ is periodic with period q, completely multiplicative (i.e., $\chi \left(mn\right)=\chi \left(m\right)\chi \left(n\right)$ for all $m,n\in \mathbb{N}$ and $\chi \left(1\right)=1$), $\chi \left(m\right)=0$ for $(m,q)=1$ and $\chi \left(m\right)\ne 0$ for $(m,q)=1$. Let $s=\sigma +it$. In [1], L. Dirichlet introduced a function

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

(1)

which is now called the Dirichlet L-function. In virtue of the complete multiplicativity of $\chi \left(m\right)$, the function (1) can be written as an Euler product

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

where $\mathbb{P}$ is the set of all prime numbers and has a meromorphic continuation to the whole complex plane with a unique simple pole at the point $s=1$ (if $\chi$ is the principal character modulo q) with residue ${\prod }_{p\mid q}(1-1/p)$. Since then, the function (1) has become a subject of intensive investigation. See, for instance, References [2,3,4] for some very recent papers on its zeros and moments. For $q=1$, the function $L(s,\chi )$ becomes the Riemann zeta-function $\zeta \left(s\right)$.

In Reference [5], S. M. Voronin established the universality of Dirichlet L-functions. He proved that if $f\left(s\right)$ is a continuous non-vanishing function on the disc $\left|s\right|\le r$ with any fixed r, $0<r<1/4$, and analytic in the interior of that disc, then, for every $\epsilon >0$, there exists a real number $\tau =\tau \left(\epsilon \right)$ such that

$$\underset{\left|s\right|\le r}{max}|L(s+3/4+i\tau ,\chi )-f\left(s\right)|<\epsilon .$$

The Voronin theorem was extended to more general compact sets independently in References [6,7,8]. Denote by $\mathcal{K}$ the class of compact subsets of the strip $D=\{s\in \mathbb{C}:1/2<\sigma <1\}$ with connected complements, and by $H_0\left(K\right)$, where $K\in \mathcal{K}$, the class of continuous non-vanishing functions on K that are analytic in the interior of K. Then the modern version of the Voronin theorem asserts that if $K\in \mathcal{K}$ and $f\left(s\right)\in H_0\left(K\right)$, then, for every $\epsilon >0$,

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

where $\mathrm{meas}A$ stands for the Lebesgue measure of a measurable set $A\subset \mathbb{R}$ (see, for example, Reference [9]). The latter inequality shows that there are infinitely many shifts $L(s+i\tau ,\chi )$ approximating a given function from the class $H_0\left(K\right)$.

In Reference [10], Voronin considered the joint functional independence of Dirichlet L-functions using the joint universality. We recall that two Dirichlet characters are called non-equivalent if they are not generated by the same primitive character. Thus, the following statement is valid [10,11]; see also References [9,12,13].

Theorem 1.

Let ${\chi }_1,\dots ,{\chi }_r$ be pairwise non-equivalent Dirichlet characters. For $j=1,\dots ,r$, let $K_j\in \mathcal{K}$, and $f_j\left(s\right)\in H_0\left(K_j\right)$. Then, for every $\epsilon >0$,

$$\underset{T\to \infty }{lim\; inf}\frac{1}{T}\mathrm{meas}\left\{\tau \in [0,T]:\underset{1\le j\le r}{sup}\underset{s\in K_j}{sup}|L(s+i\tau ,{\chi }_j)-f_j\left(s\right)|<\epsilon \right\}>0.$$

The non-equivalence of the characters ${\chi }_1,\dots ,{\chi }_r$ ensures a certain independence of the functions $L(s,{\chi }_1),\dots ,L(s,{\chi }_r)$ which is necessary for a simultaneous approximation of the collection $f_1\left(s\right),\dots ,f_r\left(s\right)$. Later, it turned out that, in place of non-equivalent characters, different shifts can be used. This was observed by Nakamura [14]. More precisely, he proved the following theorem.

Theorem 2.

Let $a_1=1,a_2,\dots ,a_r$ be real algebraic numbers linearly independent over the field of rational numbers $\mathbb{Q}$ and ${\chi }_1,\dots ,{\chi }_r$ be arbitrary Dirichlet characters. For $j=1,\dots ,r$, let $K_j\in \mathcal{K}$, and let $f_j\left(s\right)\in H_0\left(K_j\right)$. Then, for every $\epsilon >0$ and $a\in \mathbb{R}\backslash \left\{0\right\}$,

$$\underset{T\to \infty }{lim\; inf}\frac{1}{T}\mathrm{meas}\left\{\tau \in [0,T]:\underset{1\le j\le r}{sup}\underset{s\in K_j}{sup}|L(s+ia{a}_j\tau ,{\chi }_j)-f_j\left(s\right)|<\epsilon \right\}>0.$$

In Reference [15], Pańkowski obtained the joint universality of Dirichlet L-functions using the shifts $L(s+i{\alpha }_j{\tau }^{a_j}{log}^{b_j}\tau ,{\chi }_j)$, $j=1,\dots ,r$, where ${\alpha }_1,\dots ,{\alpha }_r\in \mathbb{R}$, $a_1,\dots ,a_r\in {\mathbb{R}}^{+}$ are distinct, $b_1,\dots ,b_r$ are distinct and satisfy

$$b_j\in \left\{\begin{array}{cc}\mathbb{R}\hfill & \mathrm{if}{a}_j\notin \mathbb{N},\hfill \\ (-\infty ,0]\cup (1+\infty )\hfill & \mathrm{if}{a}_j\in \mathbb{N}.\hfill \end{array}\right.$$

The aim of this paper is to introduce new shifts of Dirichlet L-functions that approximate collections of analytic functions from the class $H_0\left(K\right)$. Let, as usual, $\mathsf{\Gamma }\left(s\right)$ be the Euler gamma-function. For $t>0$, denote the increment $\theta \left(t\right)$ of the argument of the function ${\pi }^{-s/2}\mathsf{\Gamma }\left(s/2\right)$ along the segment connecting the points $s=1/2$ and $s=1/2+it$. Then it is known (see, for example, Reference [16] [Lemma 1.1]) that, for $\tau \ge 0$, the equation

$$\theta \left(t\right)=(\tau -1)\pi$$

has the unique solution $t_{\tau }$ satisfying ${\theta }^{\prime }\left(t_{\tau }\right)>0$. For $n\in \mathbb{N}$, the numbers $t_n$ are called the Gram points. They were introduced and studied in Reference [17]. Therefore, we call $t_{\tau }$ the Gram function. A very interesting property of the Gram points is the relation $t_n\sim {\gamma }_n$ as $n\to \infty$, where ${\gamma }_n>0$ are imaginary parts of non-trivial zeros of the Riemann zeta-function. In the paper, we will consider the joint approximation of analytic functions by shifts of Dirichlet L-functions involving the Gram function. More precisely, we will prove the following joint universality theorem.

Theorem 3.

Suppose that $a_1,\dots ,a_r$ are real non-zero algebraic numbers linearly independent over $\mathbb{Q}$, and ${\chi }_1,\dots ,{\chi }_r$ are arbitrary Dirichlet characters. For $j=1,\dots ,r$, let $K_j\in \mathcal{K}$ and $f_j\left(s\right)\in H_0\left(K_j\right)$. Then, for every $\epsilon >0$,

$$\underset{T\to \infty }{lim\; inf}\frac{1}{T-2}\mathrm{meas}\left\{\tau \in [2,T]:\underset{1\le j\le r}{sup}\underset{s\in K_j}{sup}|L(s+i{a}_j{t}_{\tau },{\chi }_j)-f_j\left(s\right)|<\epsilon \right\}>0.$$

Moreover, the limit

$$\underset{T\to \infty }{lim}\frac{1}{T-2}\mathrm{meas}\left\{\tau \in [2,T]:\underset{1\le j\le r}{sup}\underset{s\in K_j}{sup}|L(s+i{a}_j{t}_{\tau },{\chi }_j)-f_j\left(s\right)|<\epsilon \right\}>0$$

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

For the proof of Theorem 3, we will use the probabilistic approach based on weakly convergent probability measures in the space of analytic functions.

2. Lemmas

We start with a lemma on the functional properties of the function $t_{\tau }$. (Its proof can be found in Reference [16] [Lemma 1.1].)

Lemma 1.

Suppose that $\tau \to \infty$. Then

$$\begin{array}{c}{t}_{\tau }=\frac{2\pi \tau }{log\tau }\left(1+\frac{loglog\tau }{log\tau }(1+o\left(1\right))\right),\hfill \\ t_{\tau }^{\prime }=\frac{2\pi }{log\tau }\left(1+\frac{loglog\tau }{log\tau }(1+o\left(1\right))\right)\hfill \end{array}$$

and

$$t_{\tau }^{″}=-\frac{\pi }{\tau {(log\tau )}^2}\left(1+\frac{loglog\tau }{log\tau }(2+o\left(1\right))\right).$$

The next lemma provides an estimate for certain trigonometric integral.

Lemma 2.

Suppose that $F\left(x\right)$ is a real differentiable function, the derivative $F^{\prime }\left(x\right)$ is monotonic and $F^{\prime }\left(x\right)\ge \lambda >0$ or $F^{\prime }\left(x\right)\le -\lambda <0$ on the interval $(a,b)$. Then

$$\left|{\int }_a^{b}exp\left\{iF\left(x\right)\right\}\mathrm{d}x\right|\le \frac{4}{\lambda }.$$

The proof of the lemma is given, for example, in Reference [11].

We will also use Baker’s theorem on linear forms in logarithms of algebraic numbers (see, for example, Reference [18]).

Lemma 3.

Suppose that ${\lambda }_1,\dots ,{\lambda }_r\in \overline{\mathbb{Q}}$ are such that their logarithms $log{\lambda }_1,\dots ,log{\lambda }_r$ are linearly independent over the field of rational numbers $\mathbb{Q}$. Then, for any algebraic numbers ${\beta }_0,\dots ,{\beta }_r$, not all zero, we have

$$|{\beta }_0+{\beta }_{1}log{\lambda }_1+\cdots +{\beta }_{r}log{\lambda }_r|>H^{-C},$$

where H is the maximum of the heights of ${\beta }_0,{\beta }_1,\dots ,{\beta }_r$, and C is an effectively computable constant depending on r, ${\lambda }_1,\dots ,{\lambda }_r$ and the maximum of the degrees of ${\beta }_0,{\beta }_1,\dots ,{\beta }_r$.

Let $\gamma =\{s\in \mathbb{C}:|s|=1\}$, and

$$\mathsf{\Omega }=\prod _{p\in \mathbb{P}}{\gamma }_p,$$

where ${\gamma }_p=\gamma$ for all $p\in \mathbb{P}$. With the product topology and pointwise multiplication, the infinite-dimensional torus $\mathsf{\Omega }$ is a compact topological Abelian group. Define

$${\mathsf{\Omega }}^r={\mathsf{\Omega }}_1\times \dots \times {\mathsf{\Omega }}_r,$$

where ${\mathsf{\Omega }}_j=\mathsf{\Omega }$ for $j=1,\dots ,r$. Then ${\mathsf{\Omega }}^r$ is also a compact topological Abelian group. Therefore, denoting by $\mathcal{B}\left(\mathbb{X}\right)$ the Borel $\sigma$-field of the space $\mathbb{X}$, we see that, on $({\mathsf{\Omega }}^r,\mathcal{B}\left({\mathsf{\Omega }}^r\right))$, the probability Haar measure $m_H^r$ exists. This gives the probability space $({\mathsf{\Omega }}^r,\mathcal{B}\left({\mathsf{\Omega }}^r\right),m_H^r)$.

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

$$Q_T\left(A\right)=\frac{1}{T-2}\mathrm{meas}\left\{\tau \in [2,T]:\left(\left(p^{-i{a}_1{t}_{\tau }}:p\in \mathbb{P}\right),\dots ,\left(p^{-i{a}_r{t}_{\tau }}:p\in \mathbb{P}\right)\right)\in A\right\}.$$

Then the following limit theorem holds.

Lemma 4.

Under hypotheses of Theorem 2 on the numbers $a_1,\dots ,a_r$, $Q_T$ converges weakly to the Haar measure $m_H^r$ as $T\to \infty$.

Proof. 

We apply the Fourier transform method. It is well known that the dual group of ${\mathsf{\Omega }}^r$ is isomorphic to the group

$$\underset{j=1}{\overset{r}{⨁}}\underset{p\in \mathbb{P}}{⨁}{\mathbb{Z}}_{jp},$$

where ${\mathbb{Z}}_{jp}=\mathbb{Z}$ for all $j=1,\dots ,r$, $p\in \mathbb{P}$. Hence it follows that characters of the group ${\mathsf{\Omega }}^r$ are of the form

$$\prod _{j=1}^r\underset{p\in \mathbb{P}}{{\prod }^{*}}{\omega }_j^{k_{jp}}\left(p\right),$$

where ${\omega }_j\left(p\right)$ is the pth component of an element ${\omega }_j\in {\mathsf{\Omega }}_j$, $j=1,\dots ,r$, and the sign “$*$” means that only a finite number of integers $k_{jp}$ are distinct from zero. Therefore

$$\begin{array}{c}\hfill {\int }_{{\mathsf{\Omega }}^r}\left(\prod _{j=1}^r\underset{p\in \mathbb{P}}{{\prod }^{*}}{\omega }_j^{k_{jp}}\left(p\right)\right)\mathrm{d}\mu \end{array}$$

(2)

is the Fourier transform of a measure $\mu$ on $({\mathsf{\Omega }}^r,\mathcal{B}\left({\mathsf{\Omega }}^r\right))$.

Let $g_{Q_T}\left(\underline{k}\right)$, $\underline{k}=({\underline{k}}_1,\dots ,{\underline{k}}_r)$, ${\underline{k}}_j=(k_{jp}:k_{jp}\in \mathbb{Z},p\in \mathbb{P})$, $j=1,\dots ,r$, be the Fourier transform of $Q_T$. In view of (2) we have

$$g_{Q_T}\left(\underline{k}\right))={\int }_{{\mathsf{\Omega }}^r}\left(\prod _{j=1}^r\underset{p\in \mathbb{P}}{{\prod }^{*}}{\omega }_j^{k_{jp}}\left(p\right)\right)\mathrm{d}{Q}_T.$$

Thus, by the definition of $Q_T$,

$$\begin{array}{cc}\hfill g_{Q_T}\left(\underline{k}\right)& =\frac{1}{T-2}{\int }_2^T\prod _{j=1}^r\underset{p\in \mathbb{P}}{{\prod }^{*}}{p}^{-i{k}_{jp}{a}_j{t}_{\tau }}\mathrm{d}\tau \hfill \\ \hfill & =\frac{1}{T-2}{\int }_2^{T}exp\left\{-i{t}_{\tau }\sum _{j=1}^r\underset{p\in \mathbb{P}}{{\sum }^{*}}{a}_j{k}_{jp}logp\right\}\mathrm{d}\tau .\hfill \end{array}$$

(3)

Obviously, if $\underline{k}=(\underline{0},\dots ,\underline{0})$, then

$$g_{Q_T}\left(\underline{k}\right)=1.$$

(4)

Now suppose that $\underline{k}=({\underline{k}}_1,\dots ,{\underline{k}}_r)\ne (\underline{0},\dots ,\underline{0})$. Note that

$$\begin{array}{c}\hfill A_{\underline{k}}\stackrel{def}{=}\sum _{j=1}^r\underset{p\in \mathbb{P}}{{\sum }^{*}}{a}_j{k}_{jp}logp=\underset{p\in \mathbb{P}}{{\sum }^{*}}logp\sum _{j=1}^r{a}_j{k}_{jp}.\end{array}$$

Since ${\underline{k}}_j\ne \underline{0}$ for some $j\in \{1,2,\dots ,r\}$, there is a prime number p such that $k_{jp}\ne 0$. For this p, the sum ${\beta }_p\stackrel{def}{=}{\sum }_{j=1}^r{a}_j{k}_{jp}$ is non-zero, because the numbers $a_1,\dots ,a_r$ are linearly independent over $\mathbb{Q}$. It is well known that the set $\{logp:p\in \mathbb{P}\}$ is linearly independent over $\mathbb{Q}$. Therefore, in view of Lemma 3,

$$\begin{array}{c}\hfill A_{\underline{k}}=\underset{p\in \mathbb{P}}{{\sum }^{*}}{\beta }_{p}logp\ne 0.\end{array}$$

(5)

Now, (3) and Lemmas 1 and 2 show that, in the case $\underline{k}\ne (\underline{0},\dots ,\underline{0})$,

$$g_{Q_T}\left(\underline{k}\right)\ll \frac{logT}{T{A}_{\underline{k}}}.$$

This together with (4) and (5) give

$$\underset{T\to \infty }{lim}{g}_{Q_T}\left(\underline{k}\right)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\underline{k}=(\underline{0},\dots ,\underline{0}),\hfill \\ 0\hfill & \mathrm{if}\underline{k}\ne (\underline{0},\dots ,\underline{0}).\hfill \end{array}\right.$$

Since the right-hand side of the latter equality is the Fourier transform of the Haar measure $m_H^r$, the lemma follows by a continuity theorem for probability measures on compact groups. □

$H\left(D\right)$ denotes the space of analytic functions on the strip D endowed with the topology of uniform convergence on compacta. Lemma 4 implies a limit theorem for probability measures on $\left(H\right(D),\mathcal{B}(H\left(D\right)\left)\right)$ defined by means of absolutely convergent Dirichlet series.

For a fixed number $\theta >1/2$ and $m,n\in \mathbb{N}$, set

$$\begin{array}{c}\hfill v_n\left(m\right)=exp\left\{-{\left(\frac{m}{n}\right)}^{\theta }\right\}.\end{array}$$

(6)

Then we define the series

$$L_n(s,{\chi }_j)=\sum _{m=1}^{\infty }\frac{{\chi }_j\left(m\right)v_n\left(m\right)}{m^s}$$

and

$$L_n(s,{\omega }_j,{\chi }_j)=\sum _{m=1}^{\infty }\frac{{\chi }_j\left(m\right){\omega }_j\left(m\right)v_n\left(m\right)}{m^s},$$

$j=1,\dots ,r$, where the functions ${\omega }_j\left(p\right)$ are extended to the set $\mathbb{N}$ by the formula

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

Denote the elements of ${\mathsf{\Omega }}^r$ by $\omega =({\omega }_1,\dots ,{\omega }_r)$. Put $\underline{\chi }=({\chi }_1,\dots ,{\chi }_r)$, and set

$$\begin{array}{c}\hfill {\underline{L}}_n(s,\underline{\chi })=\left(L_n(s,{\chi }_1),\dots ,L_n(s,{\chi }_r)\right)\end{array}$$

(7)

and

$${\underline{L}}_n(s,\omega ,\underline{\chi })=\left(L_n(s,{\omega }_1,{\chi }_1),\dots ,L_n(s,{\omega }_r,{\chi }_r)\right).$$

Moreover, let $u_n:{\mathsf{\Omega }}^r\to H^r\left(D\right)$ be given by the formula

$$u_n\left(\omega \right)={\underline{L}}_n(s,\omega ,\underline{\chi }).$$

The absolute convergence of the series for $L_n(s,{\omega }_j,{\chi }_j)$ implies the continuity of the mapping $u_n$. Let $V_n=m_H^r{u}_n^{-1}$, where, for $A\in \mathcal{B}\left(H^r\left(D\right)\right)$,

$$\begin{array}{c}\hfill V_n\left(A\right)=m_H^r{u}_n^{-1}\left(A\right)=m_H^r\left(u_n^{-1}A\right).\end{array}$$

(8)

In view of (7) and (8) we conclude that Lemma 4, the continuity of $u_n$ and the well-known property on preservation of weak convergence under mapping lead to the following statement.

Lemma 5.

Under hypothesis of Theorem 3 on the numbers $\underline{a}=(a_1,\dots ,a_r)$, we have

$$P_{T,n}\left(A\right)\stackrel{def}{=}\frac{1}{T-2}\mathrm{meas}\left\{\tau \in [2,T]:{\underline{L}}_n(s+i\underline{a}{t}_{\tau },\underline{\chi })\in A\right\},A\in \mathcal{B}\left(H^r\left(D\right)\right),$$

converges weakly to the measure $V_n$ as $T\to \infty$.

The probability measure $V_n$ is very important for the proof of Theorem 3. Let

$$\underline{L}(s,\omega ,\underline{\chi })=\left(L(s,{\omega }_1,{\chi }_1),\dots ,L(s,{\omega }_r,{\chi }_r)\right),$$

where

$$\begin{array}{c}\hfill \underline{L}(s,\omega ,\underline{\chi })=\prod _{p\in \mathbb{P}}{\left(1-\frac{{\omega }_j\left(p\right){\chi }_j\left(p\right)}{p^s}\right)}^{-1},j=1,\dots ,r.\end{array}$$

(9)

Note that the latter products are uniformly convergent on compact subsets of the strip D for almost all ${\omega }_j\in {\mathsf{\Omega }}_j$, and define the $H\left(D\right)$-valued random elements on the probability space $({\mathsf{\Omega }}_j,\mathcal{B}\left({\mathsf{\Omega }}_j\right),m_{jH})$, where $m_{jH}$ is the probability Haar measure on $({\mathsf{\Omega }}_j,\mathcal{B}\left({\mathsf{\Omega }}_j\right))$. Therefore, $\underline{L}(s,\omega ,\underline{\chi })$ is the $H^r\left(D\right)$-valued random element on $({\mathsf{\Omega }}^r,\mathcal{B}\left({\mathsf{\Omega }}^r\right),m_H^r)$. Denote by $P_{\underline{L}}$ the distribution of the random element $\underline{L}(s,\omega ,\underline{\chi })$, that is,

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

We recall that the support of a probability measure P on $(\mathbb{X},\mathcal{B}(\mathbb{X}\left)\right)$, where the space $\mathbb{X}$ is separable, is a minimal closed set $S_P\subset \mathbb{X}$ such that $P\left(S_P\right)=1$. The set $S_P$ consists of all elements $x\in \mathbb{X}$ such that, for every open neighbourhood G of x, the inequality $P\left(G\right)>0$ is satisfied.

The measure $V_n$ is independent on any hypothesis. Therefore, from Reference [19] it follows that:

Lemma 6.

The measure $V_n$ converges weakly to $P_{\underline{L}}$ as $n\to \infty$. Moreover, the support of $P_{\underline{L}}$ is the set $S^r$, where

$$S=\{g\in H(D):g(s)\ne 0org(s)\equiv 0\}.$$

Proof. 

To be precise, in Reference [19] it was proved that a certain measure $P_N$ converges weakly to a certain probability measure P on $(H^r\left(D\right),\mathcal{B}\left(H^r\left(D\right)\right))$ (as $N\to \infty$), and the measure P is the limit measure of $V_n$ as $n\to \infty$. Moreover, it was proved that $P=P_{\underline{L}}$.

It remains to prove that the support of $P_{\underline{L}}$ is the set $S^r$. It is well known that the support of the random element

$$\prod _{p\in \mathbb{P}}{\left(1-\frac{\omega \left(p\right)\chi \left(p\right)}{p^s}\right)}^{-1},\omega \in \mathsf{\Omega },$$

(10)

is the set S for every Dirichlet character $\chi$. Since the space $H^r\left(D\right)$ is separable, we have

$$\mathcal{B}\left(H^r\left(D\right)\right)=\underset{r}{\underbrace{\mathcal{B}\left(H\right(D\left)\right)\times \dots \times \mathcal{B}\left(H\right(D\left)\right)}}$$

(see [20]). Therefore, it suffices to consider the measure $P_{\underline{L}}$ on the sets

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

Since the Haar measure $m_H^r$ is the product of the Haar measures $m_{jH}$ on $({\mathsf{\Omega }}_j,\mathcal{B}\left({\mathsf{\Omega }}_j\right))$, $j=1,\dots ,r$, we deduce that

$$m_H^r\{\omega \in {\mathsf{\Omega }}^r:\underline{L}(s,\omega ,\underline{\chi })\in A\}=\prod _{j=1}^r{m}_{jH}\{{\omega }_j\in {\mathsf{\Omega }}_j:L(s,{\omega }_j,{\chi }_j)\in A_j\}.$$

This equality and the minimality of the support together with remark on the support of the element (10) show that the support of $P_{\underline{L}}$ is the set $S^r$. □

3. Mean Square Estimates

Define

$$\begin{array}{c}\hfill \underline{L}(s,\underline{\chi })=\left(L(s,{\chi }_1),\dots ,L(s,{\chi }_r)\right).\end{array}$$

(11)

To pass from ${\underline{L}}_n(s+i\underline{a}{t}_{\tau },\underline{\chi })$ (defined by (7)) to $\underline{L}(s+i\underline{a}{t}_{\tau },\underline{\chi })$, certain mean square estimates for Dirichlet L-functions are necessary. Let $\chi$ be an arbitrary character modulo q.

Lemma 7.

Suppose that σ, $1/2<\sigma <1$, and $a\in \mathbb{R}\backslash \left\{0\right\}$ are fixed. Then, for $t\in \mathbb{R}$,

$${\int }_2^T|L(\sigma +it+ia{t}_{\tau },\chi )|{}^2\mathrm{d}\tau \ll T(1+|t\left|\right).$$

Proof. 

It is well known that, for fixed $\sigma >1/2$,

$${\int }_2^T{\left|L(\sigma +it,\chi )\right|}^2\mathrm{d}t{\ll }_{\sigma }T.$$

Therefore, in view of Lemma 1, for $1/2<\sigma <1$,

$$\begin{array}{cc}\hfill {\int }_2^T{\left|L(\sigma +it+ia{t}_{\tau },\chi )\right|}^2\mathrm{d}\tau & =\frac{1}{a}{\int }_2^T\frac{1}{t_{\tau }^{\prime }}{\left|L(\sigma +it+ia{t}_{\tau },\chi )\right|}^2\mathrm{d}\left(a{t}_{\tau }\right)\hfill \\ & =\frac{1}{a}{\int }_2^T\frac{1}{t_{\tau }^{\prime }}\mathrm{d}\left({\int }_2^{t+a{t}_{\tau }}{\left|L(\sigma +iu,\chi )\right|}^2\mathrm{d}u\right)\hfill \\ & \ll \frac{logT}{a}{\int }_2^{\left|t\right|+\left|a\right|t_T}{\left|L(\sigma +iu,\chi )\right|}^2\mathrm{d}u\hfill \\ & {\ll }_{\sigma ,a}logT\left(\left|t\right|+\left|a\right|\frac{T}{logT}\right){\ll }_{\sigma ,a}T(1+|t\left|\right),\hfill \end{array}$$

which is the required estimate. □

For $g_1,g_2\in H\left(D\right)$, define

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

(12)

where $\left\{K_l\right\}\subset D$ is a sequence of compact subsets such that

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

$K_l\subset K_{l+1}$ for all $l\in \mathbb{N}$, and if $K\subset D$ is a compact set, then $K\subset K_l$ for some $l\in \mathbb{N}$. Then $\rho$ is a metric in the space $H\left(D\right)$ inducing the topology of uniform convergence on compacta. Now, putting, for ${\underline{g}}_1=(g_{11},\dots ,g_{1r}),{\underline{g}}_2=(g_{21},\dots ,g_{2r})\in H^r\left(D\right)$,

$$\begin{array}{c}\hfill \underline{\rho }({\underline{g}}_1,{\underline{g}}_2)=\underset{1\le j\le r}{max}\rho (g_{1j},g_{2j})\end{array}$$

(13)

gives a metric in $H^r\left(D\right)$ inducing the product topology. The next lemma provides a certain approximation of $\underline{L}(s,\underline{\chi })$ (see definition (11)) by ${\underline{L}}_n(s,\underline{\chi })$.

Lemma 8.

Suppose that $\underline{a}\ne (0,\dots ,0)$. Then

$$\underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\frac{1}{T-2}{\int }_2^T\underline{\rho }\left(\underline{L}(s+i\underline{a}{t}_{\tau },\underline{\chi }),{\underline{L}}_n(s+i\underline{a}{t}_{\tau },\underline{\chi })\right)\mathrm{d}\tau =0.$$

Proof. 

From the definition (13) of the metric $\underline{\rho }$, it follows that it suffices to prove that, for $a\ne 0$,

$$\underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\frac{1}{T-2}{\int }_2^T\rho \left(L(s+ia{t}_{\tau },{\chi }_j),L_n(s+ia{t}_{\tau },\chi )\right)\mathrm{d}\tau =0$$

(14)

for every $j=1,\dots ,r$. We will prove the above equality for the character $\chi$ modulo q.

Let $\theta$ be from the definition (6) of $v_n\left(m\right)$, and

$$\begin{array}{c}\hfill l_n\left(s\right)=\frac{s}{\theta }\mathsf{\Gamma }\left(\frac{s}{\theta }\right)n^s.\end{array}$$

(15)

Then the representation

$$L_n(s,\chi )=\frac{1}{2\pi i}{\int }_{\theta -i\infty }^{\theta +i\infty }L(s+z,\chi )l_n\left(z\right)\frac{\mathrm{d}z}{z},$$

is true. Its proof is the same as in Section 5.4 of [21] for the Riemann zeta-function. Hence, taking ${\theta }_1>0$, by the residue theorem, we obtain

$$L_n(s,\chi )-L(s,\chi )=\frac{1}{2\pi i}{\int }_{-{\theta }_1-i\infty }^{-{\theta }_1+i\infty }L(s+z,\chi )l_n\left(z\right)\frac{\mathrm{d}z}{z}+R_n(s,\chi ),$$

(16)

where

$$R_n(s,\chi )=\left\{\begin{array}{cc}0\hfill & \mathrm{if}\chi \mathrm{is}\mathrm{a}\mathrm{non}-\mathrm{principal}\mathrm{character},\hfill \\ {\displaystyle \prod _{p\mid q}}\left(1-\frac{1}{p}\right)\frac{l_n(1-s)}{1-s}\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Let $K\subset D$ be an arbitrary compact set. Denote by $s=\sigma +iv$ the points of K, and suppose that $1/2+2\epsilon \le \sigma \le 1-\epsilon$ with fixed $\epsilon >0$ for $s\in K$. More precisely, we select ${\theta }_1=\sigma -\epsilon -1/2\ge \epsilon >0$. Then, in view of (16),

$$\begin{array}{c}|L_n(s+ia{t}_{\tau },\chi )-L(s+ia{t}_{\tau },\chi )|\hfill \\ \ll {\int }_{-\infty }^{\infty }|L(s+ia{t}_{\tau }-{\theta }_1+it,\chi )|\frac{|l_n(-{\theta }_1+it)|}{|-{\theta }_1+it|}\mathrm{d}t+\left|R_n(s+ia{t}_{\tau },\chi )\right|.\hfill \end{array}$$

Now, taking t in place of $t+v$, we get that, for $s\in K$,

$$\begin{array}{c}|L_n(s+ia{t}_{\tau },\chi )-L(s+ia{t}_{\tau },\chi )|\hfill \\ \ll {\int }_{-\infty }^{\infty }\left|L(1/2+\epsilon +i(t+a{t}_{\tau }),\chi )\right|\frac{|l_n(1/2+\epsilon -s+it)|}{|1/2+\epsilon -s+it|}\mathrm{d}t\hfill \\ +|R_n(s+ia{t}_{\tau },\chi )|.\hfill \end{array}$$

This implies the estimate

$$\begin{array}{c}\frac{1}{T-2}{\int }_2^T\underset{s\in K}{sup}|L(s+ia{t}_{\tau },\chi )-L_n(s+ia{t}_{\tau },\chi )|\mathrm{d}\tau \hfill \\ \ll \frac{1}{T-2}{\int }_2^T{\int }_{-\infty }^{\infty }\left|L(1/2+\epsilon +i(t+a{t}_{\tau }),\chi )\right|\underset{s\in K}{sup}\frac{|l_n(1/2+\epsilon -s+it)|}{|1/2+\epsilon -s+it|}\mathrm{d}t\mathrm{d}\tau \hfill \\ \phantom{\ll }+\frac{1}{T-2}{\int }_2^T\underset{s\in K}{sup}\left|R_n(s+ia{t}_{\tau },\chi )\right|\mathrm{d}\tau \hfill \\ \ll J_1+J_2,\hfill \end{array}$$

(17)

where

$$\begin{array}{c}\hfill J_1={\int }_{-\infty }^{\infty }\frac{1}{T-2}{\int }_2^T\left(\left|L\right(1/2+\epsilon +i(t+a{t}_{\tau }),\chi \left)\right|\mathrm{d}\tau \right)\underset{s\in K}{sup}\frac{|l_n(1/2+\epsilon -s+it)|}{|1/2+\epsilon -s+it|}\mathrm{d}t\end{array}$$

and

$$\begin{array}{c}\hfill J_2=\frac{1}{T-2}{\int }_2^T\underset{s\in K}{sup}\left|R_n(s+ia{t}_{\tau },\chi )\right|\mathrm{d}\tau .\end{array}$$

(18)

It is well known that uniformly in $\sigma$, ${\sigma }_1\le \sigma \le {\sigma }_2$, with arbitrary ${\sigma }_1<{\sigma }_2$,

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

Therefore, by the definition (15) of the function $l_n\left(s\right)$, we find that, for $s\in K$,

$$\begin{array}{cc}\hfill \left|\frac{l_n(1/2+\epsilon -s+it)}{1/2+\epsilon -s+it}\right|& =\frac{n^{1/2+\epsilon -\sigma }}{\theta }\left|\mathsf{\Gamma }\left(\frac{1/2+\epsilon -\sigma }{\theta }+\frac{i(t-v)}{\theta }\right)\right|\hfill \\ \hfill & {\ll }_{\theta ,K}{n}^{-\epsilon }exp\left\{-\frac{c_1}{\theta }\left|t\right|\right\},c_1>0.\hfill \end{array}$$

(19)

In the same way, for $s\in K$, we obtain

$$R_n(s+ia{t}_{\tau },\chi ){\ll }_{\theta ,q,K}{n}^{1-\sigma }exp\left\{-\frac{c_2}{\theta }\left|a\right|t_{\tau }\right\}.$$

(20)

Suppose that $\theta =1/2+\epsilon$. Then (17), (19) and Lemma 7 lead to the bound

$$J_1{\ll }_{\epsilon ,K}{n}^{-\epsilon }{\int }_{-\infty }^{\infty }\left(1+\left|t\right|\right)exp\{-c_3\left|t\right|\}\mathrm{d}t{\ll }_{\epsilon ,K,a}{n}^{-\epsilon },c_3>0.$$

(21)

Moreover, by (18), Lemma 1 and (20),

$$\begin{array}{cc}\hfill J_2& {\ll }_{\epsilon ,K,q}{n}^{1/2-2\epsilon }\frac{1}{T-2}{\int }_2^{T}exp\left\{-c_4\left|a\right|\frac{\tau }{log\tau }\right\}\mathrm{d}\tau \hfill \\ & {\ll }_{\epsilon ,K,q}{n}^{1/2-2\epsilon }\frac{logT}{T-2}+\frac{n^{1/2-2\epsilon }}{T-2}{\int }_{logT}^{T}exp\left\{-c_4\left|a\right|\frac{\tau }{log\tau }\right\}\mathrm{d}\tau \hfill \\ & {\ll }_{\epsilon ,K,q,a}{n}^{1/2-2\epsilon }\frac{logT}{T-2}.\hfill \end{array}$$

Thus, in view of (17) and (21),

$$\frac{1}{T-2}{\int }_2^T\underset{s\in K}{sup}|L(s+ia{t}_{\tau },\chi )-L_n(s+ia{t}_{\tau },\chi )|\mathrm{d}\tau {\ll }_{\epsilon ,K,q,a}{n}^{-\epsilon }+n^{1/2-2\epsilon }\frac{logT}{T-2}.$$

From this, it follows that

$$\underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\frac{1}{T-2}{\int }_2^T\underset{s\in K}{sup}|L(s+ia{t}_{\tau },\chi )-L_n(s+ia{t}_{\tau },\chi )|\mathrm{d}\tau =0.$$

(22)

Now, the definition (12) of the metric $\rho$ implies (14), which completes the proof of Lemma 8. □

4. A Limit Theorem

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

$$\begin{array}{c}\hfill P_T\left(A\right)=\frac{1}{T-2}\mathrm{meas}\left\{\tau \in [2,T]:\underline{L}(s+i\underline{a}{t}_{\tau },\underline{\chi })\in A\right\}.\end{array}$$

(23)

In this section, we will prove the following statement.

Theorem 4.

Suppose that $a_1,\dots ,a_r$ are non-zero real algebraic numbers linearly independent over $\mathbb{Q}$, and ${\chi }_1,\dots ,{\chi }_r$ are arbitrary Dirichlet characters. Then $P_T$ converges weakly to $P_{\underline{L}}$ as $T\to \infty$. The support of $P_{\underline{L}}$ is the set $S^r$.

First we recall a useful property of convergence in distribution ($\underset{}{\overset{\mathcal{D}}{\to }}$) (see Theorem 4.2 in Reference [20]).

Lemma 9.

Suppose that the space $(\mathbb{X},d)$ is separable, the random elements $X_{kn}$ and $Y_n$, $k\in \mathbb{N}$, $n\in \mathbb{N}$, are defined on the same probability space with measure μ,

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

for every $k\in \mathbb{N}$,

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

and, for every $\epsilon >0$,

$$\underset{k\to \infty }{lim}\underset{n\to \infty }{lim\; sup}\mu \left\{d(X_{kn},Y_n)\ge \epsilon \right\}=0.$$

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

In the theory of weak convergence of probability measures, the notions of relative compactness and tightness of families of probability measures are very useful. We recall that the family $\left\{P\right\}$ of probability measures on $(\mathbb{X},\mathcal{B}(\mathbb{X}\left)\right)$ is called relatively compact if every sequence $\left\{P_n\right\}\subset \left\{P\right\}$ contains a weakly convergent subsequence to a certain measure on $(\mathbb{X},\mathcal{B}(\mathbb{X}\left)\right)$, and this family is called tight, if for every $\epsilon >0$, there exists a compact set $K=K\left(\epsilon \right)\subset \mathbb{X}$ such that

$$P\left(K\right)>1-\epsilon$$

for all $P\in \left\{P\right\}$. By the direct Prokhorov theorem (see Theorem 5.1 in Billingsley [20]), every tight family $\left\{P\right\}$ is relatively compact. We apply the above remarks to the sequence $\{V_n:n\in \mathbb{N}\}$, where $V_n$ (defined by (8)) is the limit measure in Lemma 5.

Lemma 10.

The sequence $\left\{V_n\right\}$ is relatively compact.

Proof. 

By the above mentioned Prokhorov theorem, it suffices to prove that the sequence $\left\{V_n\right\}$ is tight.

Suppose ${\theta }_T$ is a random variable defined on a certain probability space with measure $\mu$ and uniformly distributed on $[2,T]$. Define the $H^r\left(D\right)$-valued random element

$${\underline{X}}_{T,n}={\underline{X}}_{T,n}\left(s\right)=\left(X_{T,n,1}\left(s\right),\dots ,X_{T,n,r}\left(s\right)\right)={\underline{L}}_n(s+i\underline{a}{t}_{{\theta }_T},\underline{\chi }).$$

Moreover, let

$$\begin{array}{c}\hfill {\underline{X}}_n={\underline{X}}_n\left(s\right)=\left(X_{n1}\left(s\right),\dots ,X_{nr}\left(s\right)\right)\end{array}$$

(24)

be the $H^r\left(D\right)$-valued random element with the distribution $V_n$. Then Lemma 5 implies the relation

$${\underline{X}}_{T,n}\underset{T\to \infty }{\overset{\mathcal{D}}{\to }}{\underline{X}}_n.$$

(25)

By Lemma 7 with $t=0$, we have, for $1/2<\sigma <1$,

$${\int }_2^T{\left|L(\sigma +i{a}_j{t}_{\tau },{\chi }_j)\right|}^2\mathrm{d}\tau {\ll }_{\sigma ,a_j}T,j=1,\dots ,r.$$

(26)

Let $K_l$ be a compact set from the definition of the metric $\rho$. Then (26) together with the Cauchy integral formula show that

$${\int }_2^T\underset{s\in K_l}{sup}\left|L(s+i{a}_j{t}_{\tau },{\chi }_j)\right|\mathrm{d}\tau {\ll }_{l,a_j}T,j=1,\dots ,r.$$

This combined with (22) implies the inequality

$$\underset{n\in \mathbb{N}}{sup}\underset{T\to \infty }{lim\; sup}\frac{1}{T-2}{\int }_2^T\underset{s\in K_l}{sup}\left|L_n(s+i{a}_j{t}_{\tau },{\chi }_j)\right|\mathrm{d}\tau \ll R_{lj},j=1,\dots ,r.$$

(27)

Fix $\epsilon >0$, and define $M_{lj}=M_{lj}\left(s\right)=2^{l}r{R}_{lj}{\epsilon }^{-1}$. Then, in view of (27), we find that, for each $n\in \mathbb{N}$,

$$\begin{array}{c}\underset{T\to \infty }{lim\; sup}\mu \left\{\exists j:\underset{s\in K_l}{sup}\left|X_{T,n,j}\left(s\right)\right|>M_{lj}\right\}\hfill \\ \le \sum _{j=1}^r\underset{T\to \infty }{lim\; sup}\mu \left\{\underset{s\in K_l}{sup}\left|X_{T,n,j}\left(s\right)\right|>M_{lj}\right\}\hfill \\ \le \sum _{j=1}^r\underset{T\to \infty }{lim\; sup}\frac{1}{(T-2)M_{lj}}{\int }_2^T\underset{s\in K_l}{sup}\left|L_n(s+i{a}_j{t}_{\tau },{\chi }_j)\right|\mathrm{d}\tau \le \sum _{j=1}^r\frac{R_{lj}}{M_{lj}}=\frac{\epsilon }{2^r}.\hfill \end{array}$$

This together with (25) shows that, for all $l,n\in \mathbb{N}$,

$$\mu \left\{\exists j:\underset{s\in K_l}{sup}\left|X_{n,j}\left(s\right)\right|>M_{lj}\right\}\le \frac{\epsilon }{2^l}.$$

(28)

Define the set

$$K_j=K_j\left(s\right)=\left\{g\in H\left(D\right):\underset{s\in K_l}{sup}\left|g\left(s\right)\right|\le M_{lj},l\in \mathbb{N}\right\}.$$

Then $K_j$ is a compact set in $H\left(D\right)$, and, in virtue of (24) and (28),

$$\mu \{{\underline{X}}_n\in K\}\ge 1-\epsilon$$

for all $n\in \mathbb{N}$. In other words, we have

$$V_n\left(K\right)\ge 1-\epsilon$$

for all $n\in \mathbb{N}$. Thus, the sequence $\{V_n:n\in \mathbb{N}\}$ is tight. □

Proof of Theorem 4.

By Lemma 10, there exists a subsequence $\left\{V_{n_k}\right\}$ of the sequence $\left\{V_n\right\}$ that is weakly convergent to a certain probability measure P on $(H^r\left(D\right),\mathcal{B}\left(H^r\left(D\right)\right))$ as $k\to \infty$. This can be written as

$${\underline{X}}_{n_k}\underset{k\to \infty }{\overset{\mathcal{D}}{\to }}P.$$

(29)

Define one more $H^r\left(D\right)$-valued random element

$${\underline{X}}_T={\underline{X}}_T\left(s\right)=\underline{L}(s+i\underline{a}{t}_{{\theta }_T},\underline{\chi }).$$

Then Lemma 8 implies that, for every $\epsilon >0$,

$$\begin{array}{c}\underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\mu \left\{\underline{\rho }(X_T,X_{T,n})\ge \epsilon \right\}\hfill \\ \le \underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\frac{1}{(T-2)\epsilon }{\int }_2^T\underline{\rho }\left(\underline{L}(s+i\underline{a}{t}_{\tau },\underline{\chi }),{\underline{L}}_n(s+i\underline{a}{t}_{\tau },\underline{\chi })\right)\mathrm{d}\tau =0.\hfill \end{array}$$

The latter equality together with (25), (29), and Lemma 9 shows that

$${\underline{X}}_T\underset{T\to \infty }{\overset{\mathcal{D}}{\to }}P,$$

(30)

or, in other words, $P_T$ converges weakly to P as $T\to \infty$. Moreover, by the relation (30), the measure P is independent of the subsequence $\left\{V_{n_k}\right\}$. Thus, we deduce that

$${\underline{X}}_n\underset{n\to \infty }{\overset{\mathcal{D}}{\to }}P,$$

or $V_n$ converges weakly to P as $n\to \infty$. Therefore, the theorem follows by Lemma 6. □

5. Proof of Universality

The proof of Theorem 3 is based on Mergelyan’s theorem on the approximation of analytic functions by polynomials [22], Theorem 4, and the properties of weak convergence. For convenience, we state them as lemmas.

Lemma 11

(Mergelyan theorem). Suppose that $K\subset \mathbb{C}$ is a compact set with connected complement, and $f\left(s\right)$ be a continuous function on K and analytic in the interior of K. Then, for every $\epsilon >0$, there exists a polynomial $p\left(s\right)$ such that

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

We recall that $A\in \mathcal{B}\left(\mathbb{X}\right)$ is called a continuity set of the measure P on $(\mathbb{X},\mathcal{B}(\mathbb{X}\left)\right)$ if $P(\partial A)=0$, where $\partial A$ is a boundary of A.

Lemma 12.

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

1°

$P_n$ converges weakly to P as $n\to \infty$;

2°

For every open set $G\subset \mathbb{X}$,

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

3°

For every continuity set A of P,

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

The above lemma is a part of Theorem 2.1 from Reference [20]. Now, we can give the proof of Theorem 3.

Proof of Theorem 3.

First part. In view of Lemma 11, there exist polynomials $p_1\left(s\right),\dots ,p_r\left(s\right)$ such that

$$\underset{1\le j\le r}{sup}\underset{s\in K_j}{sup}\left|f_j\left(s\right)-{\mathrm{e}}^{p_j\left(s\right)}\right|<\frac{\epsilon }{2}.$$

(31)

The set

$$\begin{array}{c}\hfill G_{\epsilon }^r=\left\{(g_1,\dots ,g_r)\in H^r\left(D\right):\underset{1\le j\le r}{sup}\underset{s\in K_j}{sup}\left|g_j\left(s\right)-{\mathrm{e}}^{p_j\left(s\right)}\right|<\frac{\epsilon }{2}\right\}\end{array}$$

(32)

is an open neighbourhood of the element $\left({\mathrm{e}}^{p_1\left(s\right)},\dots ,{\mathrm{e}}^{p_r\left(s\right)}\right)\in S^r$. Thus, by Theorem 4, $P_{\underline{L}}\left(G_{\epsilon }^r\right)>0$, where the distribution $P_{\underline{L}}$ is defined by (9). Hence, from Theorem 4 again and Lemma 12,

$$\underset{T\to \infty }{lim\; inf}{P}_T\left(G_{\epsilon }^r\right)\ge P_{\underline{L}}\left(G_{\epsilon }^r\right)>0,$$

and the definitions (23) and (32) of $P_T$ and $G_{\epsilon }^r$ together with (31) prove the first part of the theorem.

Second part. Introduce one more set

$$\begin{array}{c}\hfill A_{\epsilon }=\left\{(g_1,\dots ,g_r)\in H^r\left(D\right):\underset{1\le j\le r}{sup}\underset{s\in K_j}{sup}\left|g_j\left(s\right)-f_j\left(s\right)\right|<\epsilon \right\}.\end{array}$$

(33)

Then the boundary of $A_{\epsilon }$ lies in the set

$$\left\{(g_1,\dots ,g_r)\in H^r\left(D\right):\underset{1\le j\le r}{sup}\underset{s\in K_j}{sup}\left|g_j\left(s\right)-f_j\left(s\right)\right|=\epsilon \right\},$$

thus, $\partial A_{{\epsilon }_1}\cap \partial A_{{\epsilon }_2}=⌀$ for different ${\epsilon }_1>0$ and ${\epsilon }_2>0$. This shows that the set $A_{\epsilon }$ is a continuity set of the measure $P_{\underline{L}}$ for all but at most countably many $\epsilon >0$. Therefore, by Lemma 12,

$$\underset{T\to \infty }{lim}{P}_T\left(A_{\epsilon }\right)=P_{\underline{L}}\left(A_{\epsilon }\right)$$

(34)

for all but at most countably many $\epsilon >0$. Moreover, (31) shows the inclusion $G_{\epsilon }^r\subset A_{\epsilon }$. This, (34) and the definitions (23) and (33) of $P_T$ and $A_{\epsilon }$ prove the second assertion of the theorem. □