Joint Universality of the Zeta-Functions of Cusp Forms

Abstract

Let F be the normalized Hecke-eigen cusp form for the full modular group and $\zeta (s,F)$ be the corresponding zeta-function. In the paper, the joint universality theorem on the approximation of a collection of analytic functions by shifts $(\zeta (s+i{h}_1\tau ,F),\dots ,\zeta (s+i{h}_r\tau ,F))$ is proved. Here, $h_1,\dots ,h_r$ are algebraic numbers linearly independent over the field of rational numbers.

1. Introduction

The series of the types

$$\sum _{m=1}^{\infty }\frac{a_m}{m^s}\mathrm{and}\sum _{m=1}^{\infty }{a}_m{\mathrm{e}}^{-{\lambda }_{m}s},s=\sigma +it,$$

where $\left\{{\lambda }_m\right\}$ is a nondecreasing sequence of real numbers and ${lim}_{m\to \infty }{\lambda }_m=+\infty$ are called Dirichlet series. The majority of zeta-functions are meromorphic functions in some half-plane defined by Dirichlet series having a certain arithmetic sense. The most important of zeta-functions is the Riemann zeta-function

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

In [1], Voronin discovered a very interesting and important property of $\zeta \left(s\right)$ to approximate a wide class of analytic functions by shifts $\zeta (s+i\tau )$, $\tau \in \mathbb{R}$, and called it universality. Later, it turned out that some other zeta-functions also are universal in the Voronin sense. This paper is devoted to the universality of zeta-functions of certain cusp forms.

Let

$$SL(2,\mathbb{Z})=\left\{\left(\begin{array}{cc}a& b\\ c& d\end{array}\right):a,b,c,d\in \mathbb{Z},ad-bc=1\right\}$$

be the full modular group. If the function $F\left(z\right)$ is holomorphic in the upper half-plane $\mathrm{Im}z>0$, and for all elements of $SL(2,\mathbb{Z})$ with some $\kappa \in 2\mathbb{N}$ satisfies the functional equation

$$F\left(\frac{az+b}{cz+d}\right) ={(cz+d)}^{\kappa }F\left(z\right),$$

(1)

where $F\left(z\right)$ is called a modular form of weight $\kappa$ for the full modular group. Then, $F\left(z\right)$ has Fourier series expansion

$$F\left(z\right)=\sum _{m=-\infty }^{\infty }c\left(m\right){\mathrm{e}}^{2\pi imz}.$$

If $c\left(m\right)=0$ for all $m⩽0$, then $F\left(z\right)$ is a cusp form of weight $\kappa$. The corresponding zeta-function (or L-function) $\zeta (s,F)$ is defined for $\sigma >\frac{\kappa +1}{2}$ by the Dirichlet series

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

and has the analytic continuation to an entire function. Additionally, we suppose that $F\left(z\right)$ is a simultaneous eigenfunction of all Hecke operators $T_m$

$$T_{m}F\left(z\right)=m^{\kappa -1}\underset{ad=m}{\sum _{a,d>0}}\frac{1}{d^{\kappa }}\sum _{b\left(\mathrm{mod}d\right)}F\left(\frac{az+b}{d}\right),m\in \mathbb{N}.$$

In this case, $c\left(1\right)\ne 0$; therefore, the form $F\left(z\right)$ can be normalized, and thus, we may suppose that $c\left(1\right)=1$.

Now, we suppose that $F\left(z\right)$ is a normalized Hecke-eigen cusp form of weight $\kappa$ for the full modular group. Then, the zeta-function $\zeta (s,F)$ can be written, for $\sigma >\frac{\kappa +1}{2}$, as a product over primes

$$\zeta (s,F)=\prod _p{\left(1-\frac{\alpha \left(p\right)}{p^s}\right)}^{-1}{\left(1-\frac{\beta \left(p\right)}{p^s}\right)}^{-1},$$

where $\alpha \left(p\right)$ and $\beta \left(p\right)$ are conjugate complex numbers satisfying the equality $\alpha \left(p\right)+\beta \left(p\right)=c\left(p\right)$.

In the paper [2], the universality of the function $\zeta (s,F)$ was proved. Let $D_{\kappa }= \left\{s\in \mathbb{C}:\frac{\kappa }{2}<\sigma <\frac{\kappa +1}{2}\right\}$, ${\mathcal{K}}_F$ be the class of compact subsets of the strip $D_{\kappa }$ with connected complements, and $H_{0,F}\left(K\right)$, $K\in {\mathcal{K}}_F$ the class of continuous nonvanishing functions on K that are analytic in the interior of K. Moreover, let $\mathrm{meas}A$ denote the Lebesgue measure of a measurable set $A\subset \mathbb{R}$. Then, in [2], the following theorem was obtained.

Theorem 1.

Suppose that $K\in {\mathcal{K}}_F$ and $f\left(s\right)\in H_{0,F}\left(K\right)$. Then, for every $\epsilon >0$,

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

Theorem 1 shows that there are infinitely many shifts $\zeta (s+i\tau ,F)$ approximating a given function $f\left(s\right)\in H_{0,F}$. In the shifts $\zeta (s+i\tau ,F)$ of Theorem 1, $\tau$ takes arbitrary real values; therefore, the theorem is of continuous type. Further, discrete universality theorems for the function $\zeta (s,F)$ are known. In [3,4], the discrete universality theorems with shifts $\zeta (s+ikh,F)$, $k\in \mathbb{N}$, $h>0$ being a fixed number, were proved. Denote by $H\left(D_{\kappa }\right)$ the space of analytic on $D_{\kappa }$ functions endowed with the topology of uniform convergence on compacta. The paper [5] is devoted to the universality for compositions $\mathsf{\Phi }\left(\zeta \right(s,F\left)\right)$ with certain operators $\mathsf{\Phi }:H\left(D_{\kappa }\right)\to H\left(D_{\kappa }\right)$. The results of the latter paper were applied in [6] for the functional independence of the compositions $\mathsf{\Phi }\left(\zeta \right(s,F\left)\right)$.

Let, for a fixed $l\in \mathbb{N}$,

$${\mathsf{\Gamma }}_0\left(l\right)= \left\{\left(\begin{array}{cc}a& b\\ c& d\end{array}\right) \in SL(2,\mathbb{Z}):c\equiv 0\left(\mathrm{mod}l\right)\right\}$$

denote the Hecke subgroup of the group $SL(2,\mathbb{Z})$. If $F\left(z\right)$ satisfies (1) for all elements of ${\mathsf{\Gamma }}_0\left(l\right)$, then $F\left(z\right)$ is called a cusp form of weight $\kappa$ and level l. The form $F\left(z\right)$ is called a new form if it is not a cusp form of level $l_1\mid l$. In [7], a universality theorem was obtained for zeta-functions of new forms.

The universality theorem of [2] was generalized in [8] for shifts $\zeta (s+i\phi (\tau ),F)$ with differentiable function $\phi \left(\tau \right)$ satisfying the estimates ${\left({\phi }^{\prime }\left(\tau \right)\right)}^{-1}=o\left(\tau \right)$ and $\phi \left(2\tau \right)\underset{\tau ⩽t⩽2\tau }{max}{\left({\phi }^{\prime }\left(t\right)\right)}^{-1}$$\ll \tau$ as $\tau \to \infty$. The discrete version of results of [8] is given in [9]. In [10], the shifts $\zeta (s+i{\gamma }_k,F)$, where $\{{\gamma }_k:k\in \mathbb{N}\}$ is the sequence of nontrivial zeros of $\zeta \left(s\right)$, are used.

The joint universality of zeta- and L-functions is a more complicated problem of analytic number theory. In this case, a collection of analytic functions are simultaneously approximated by a collection of shifts of zeta-functions. The first result in this direction also belongs to Voronin. He considered [11] the functional independence of Dirichlet L-functions $L(s,\chi )$ with pairwise nonequivalent Dirichlet characters $\chi$ and, for this, he obtained their joint universality. The paper [12] is devoted to the joint universality for zeta-functions of new forms twisted by Dirichlet characters, i.e., for the functions

$$\sum _{m=1}^{\infty }\frac{c\left(m\right)\chi \left(m\right)}{m^s},\sigma >\frac{\kappa +1}{2},$$

with pairwise nonequivalent Dirichlet characters ${\chi }_1,\dots ,{\chi }_r$.

Joint universality theorems with generalized shifts $\zeta (s+i{\phi }_j\left(k\right),F)$, $j=1,\dots ,r$, with some differentiable functions ${\phi }_j\left(\tau \right)$ can be found in [13]. Continuous and discrete joint universality theorems for more general zeta-functions are given in [14,15,16].

Our aim is to obtain a joint universality theorem for zeta-functions of normalized Hecke-eigen cusp forms by using different shifts. The first of the denseness results for shifts of a universal function were discussed in [17].

The main result of the paper is the following statement.

Theorem 2.

Suppose that $h_1,\dots ,h_r$ are real algebraic numbers linearly independent over the field of rational numbers $\mathbb{Q}$. For $j=1,\dots ,r$, let $K_j\in {\mathcal{K}}_F$ and $f_j\left(s\right)\in H_{0,F}\left(K_j\right)$. Then, for every $\epsilon >0$,

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

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

For the proof of Theorem 2, we will apply the probabilistic approach based on a limit theorem in the space of analytic functions.

2. Mean Square Estimates

Recall the metric in the space $H\left(D_{\kappa }\right)$. Let $\{K_l:l\in \mathbb{N}\}\subset D_{\kappa }$ be a sequence of compact subsets such that

$$D_{\kappa }=\bigcup _{l=1}^{\infty }{K}_l,$$

$K_l\subset K_{l+1}$ for $l\in \mathbb{N}$, and if $K\subset D_{\kappa }$ is a compact, then $K\subset K_l$ for some l. For example, we can take $K_l$ closed rectangles. Then

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

is a metric in $H\left(D_{\kappa }\right)$ inducing the topology of uniform convergence on compacta.

Let

$$H^r\left(D_{\kappa }\right)= (\underset{r}{\underbrace{H\left(D_{\kappa }\right)\times \cdots \times H\left(D_{\kappa }\right)}}.$$

For ${\underline{g}}_j=(g_{j1},\dots ,g_{jr})\in H^r\left(D_{\kappa }\right)$, $j=1,2$, define

$$\underline{\rho }({\underline{g}}_1,{\underline{g}}_2)=\underset{1⩽j⩽r}{max}\rho (g_{1j},g_{2j}).$$

Then, $\underline{\rho }$ is a metric in $H^r\left(D_{\kappa }\right)$ inducing the product topology.

Let $\theta >\frac{1}{2}$ be a fixed number, and

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

Then, the series

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

in view of the estimate

$$c\left(m\right)\ll m^{\frac{\kappa -1}{2}+\epsilon },$$

is absolutely convergent in every fixed half plane $\sigma >\widehat{\sigma }$. However, for our aim, this convergence is sufficient only for $\sigma >\frac{\kappa }{2}$.

For brevity, let $\underline{h}=(h_1,\dots ,h_r)$,

$$\underline{\zeta }(s+i\underline{h}\tau ,F)= \left(\zeta (s+i{h}_1\tau ,F),\dots ,\zeta (s+i{h}_r\tau ,F)\right)$$

and

$${\underline{\zeta }}_n(s+i\underline{h}\tau ,F)= \left({\zeta }_n(s+i{h}_1\tau ,F),\dots ,{\zeta }_n(s+i{h}_r\tau ,F)\right).$$

Lemma 1.

For all $\underline{h}$,

$$\underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\frac{1}{T}\underset{0}{\overset{T}{\int }}\underline{\rho }\left(\underline{\zeta }(s+i\underline{h}\tau ,F),{\underline{\zeta }}_n(s+i\underline{h}\tau ,F)\right)\mathrm{d}\tau =0.$$

Proof. 

By the definitions of the metrics $\rho$ and $\underline{\rho }$, it suffices to show that, for every $h\in \mathbb{R}$ and compact set $K\subset D_{\kappa }$,

$$\underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\frac{1}{T}\underset{0}{\overset{T}{\int }}\underset{s\in K}{sup}\left|\zeta (s+ih\tau ,F),{\zeta }_n(s+ih\tau ,F)\right)\mathrm{d}\tau =0.$$

It is well known that for fixed $\frac{\kappa }{2}<\sigma <\frac{\kappa +1}{2}$,

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

where ${\ll }_{\sigma }$ means that the implied constant depends on $\sigma$. Therefore,

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

and, for $v\in \mathbb{R}$,

$$\frac{1}{T}\underset{0}{\overset{T}{\int }}{\left|\zeta (\sigma +ih\tau +iv,F)\right|}^2\mathrm{d}v{\ll }_{\sigma ,h}1+\left|v\right|.$$

(2)

Let

$$l_n\left(s\right)=\frac{z}{\theta }\mathsf{\Gamma }\left(\frac{z}{\theta }\right)n^z,$$

where $\mathsf{\Gamma }\left(z\right)$ denotes the Euler gamma-function and $\theta$ is a number from the definition of $v_n\left(m\right)$. Using the Mellin formula

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

we find that

$$exp\left\{-{\left(\frac{m}{n}\right)}^{\theta }\right\} =\frac{1}{2\pi i}\underset{\theta -i\infty }{\overset{\theta +i\infty }{\int }}\frac{1}{\theta }\mathsf{\Gamma }\left(\frac{1}{\theta }\right){\left(\frac{m}{n}\right)}^{-s}\mathrm{d}s.$$

Therefore, in virtue of the definition of the function $v_n\left(m\right)$, we obtain that, for $\sigma >\frac{\kappa }{2}$,

$$\begin{array}{ccc}\hfill {\zeta }_n(s,F)& =& \frac{1}{2\pi i}\sum _{m=1}^{\infty }\frac{c\left(m\right)}{m^s}\underset{\theta -i\infty }{\overset{\theta +i\infty }{\int }}\frac{z}{\theta }\mathsf{\Gamma }\left(\frac{z}{\theta }\right){\left(\frac{m}{n}\right)}^{-z}\frac{\mathrm{d}z}{z}\hfill \end{array}$$

$$\begin{array}{ccc}& =& \frac{1}{2\pi i}\underset{\theta -i\infty }{\overset{\theta +i\infty }{\int }}\left(\frac{l_n\left(z\right)}{z}\sum _{m=1}^{\infty }\frac{c\left(m\right)}{m^{s+z}}\right)\mathrm{d}z\hfill \end{array}$$

$$\begin{array}{ccc}& =& \frac{1}{2\pi i}\underset{\theta -i\infty }{\overset{\theta +i\infty }{\int }}\zeta (s+z,F)l_n\left(z\right)\frac{\mathrm{d}z}{z}.\hfill \end{array}$$

(3)

Let $K\in D_{\kappa }$ be a fixed compact set. Then, there exists $\epsilon >0$ such that, for all $s=\sigma +it\in K$, the inequalities $\frac{\kappa }{2}+2\epsilon <\sigma <\frac{\kappa +1}{2}-\epsilon$ are satisfied. We take, for such $\sigma$,

$${\theta }_1=\frac{\kappa }{2}+\epsilon -\sigma .$$

Then, ${\theta }_1<0$. Therefore, by the residue theorem and (3),

$${\zeta }_n(s,F)-\zeta (s,F)=\frac{1}{2\pi i}\underset{{\theta }_1-i\infty }{\overset{{\theta }_1+i\infty }{\int }}\zeta (s+z,F)l_n\left(z\right)\frac{\mathrm{d}z}{z}.$$

Hence, for all $s\in K$,

$$\begin{array}{ccc}\hfill \zeta (s+ih\tau ,F)-{\zeta }_n(s+ih\tau ,F)& =& \frac{1}{2\pi i}\underset{-\infty }{\overset{\infty }{\int }}\zeta \left(\frac{\kappa }{2}+\epsilon +it+ih\tau +iv,F\right)\frac{l_n\left(\frac{\kappa }{2}+\epsilon -\sigma +iv\right)}{\frac{\kappa }{2}+\epsilon -\sigma +iv}\mathrm{d}v\hfill \\ & =& \frac{1}{2\pi i}\underset{-\infty }{\overset{\infty }{\int }}\zeta \left(\frac{\kappa }{2}+\epsilon +ih\tau +iv,F\right)\frac{l_n\left(\frac{\kappa }{2}+\epsilon -s+iv\right)}{\frac{\kappa }{2}+\epsilon -s+iv}\mathrm{d}v\hfill \\ & \ll & \frac{1}{2\pi i}\underset{-\infty }{\overset{\infty }{\int }}\left|\zeta \left(\frac{\kappa }{2}+\epsilon +ih\tau +iv,F\right)\right|\underset{s\in K}{sup}\left|\frac{l_n\left(\frac{\kappa }{2}+\epsilon -s+iv\right)}{\frac{\kappa }{2}+\epsilon -s+iv}\right|\mathrm{d}v.\hfill \end{array}$$

Thus, in view of (2),

$$\begin{array}{ccc}\hfill & & \frac{1}{T}\underset{0}{\overset{\infty }{\int }}\underset{s\in K}{sup}\left|\zeta (s+ih\tau ,F)-{\zeta }_n(s+ih\tau ,F)\right|\mathrm{d}\tau \hfill \end{array}$$

$$\begin{array}{ccc}& & \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\ll \underset{-\infty }{\overset{\infty }{\int }}\left({\left(\frac{1}{T}\underset{0}{\overset{\infty }{\int }}{\left|\zeta \left(\frac{\kappa }{2}+\epsilon +ih\tau +iv\right)\right|}^2\mathrm{d}\tau \right)}^{1/2}\underset{s\in K}{sup}\left|\frac{l_n\left(\frac{\kappa }{2}+\epsilon -s+iv\right)}{\frac{\kappa }{2}+\epsilon -s+iv}\right|\right)\mathrm{d}v\hfill \end{array}$$

$$\begin{array}{ccc}& & \hspace{1em}\hspace{1em}{\ll }_{\epsilon ,h,K}{n}^{-\epsilon }\underset{-\infty }{\overset{\infty }{\int }}(1+|v\left|\right)exp\{-c_1\left|v\right|\}\mathrm{d}v{\ll }_{\epsilon ,h,K}{n}^{-\epsilon }\hfill \end{array}$$

(4)

Here, we used the estimate

$$\mathsf{\Gamma }\left(\frac{1}{\theta }\left(\frac{\kappa }{2}+\epsilon -s+iv\right)\right) \ll exp\left\{-\frac{c}{\theta }|v-t|\right\} {\ll }_{\kappa } exp\{-c_1\left|v\right|\},c_1>0.$$

Estimate (4) proves the lemma. □

Let $\mathbb{P}$ be the set of all prime numbers, and ${\gamma }_p=\{s\in \mathbb{C}:|s|=1\}$ for all $p\in \mathbb{P}$. Define the set

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

Then, the torus $\mathsf{\Omega }$ with product topology and pointwise multiplication is a compact topological Abelian group. Therefore, on $(\mathsf{\Omega },\mathcal{B}(\mathsf{\Omega }\left)\right)$ ($\mathcal{B}\left(\mathbb{X}\right)$ is the Borel $\sigma$-field of the space $\mathbb{X}$), the probability Haar measure $m_H$ can be defined. Moreover, let

$$\underline{\mathsf{\Omega }}={\mathsf{\Omega }}_1\times \cdots \times {\mathsf{\Omega }}_r,$$

where ${\mathsf{\Omega }}_j=\mathsf{\Omega }$ for all $j=1,\dots ,r$. Once again, $\underline{\mathsf{\Omega }}$ is a compact topological Abelian group. Therefore, on $(\underline{\mathsf{\Omega }},\mathcal{B}\left(\underline{\mathsf{\Omega }}\right))$ the probability Haar measure ${\underline{m}}_H$ exists. This gives the probability space $(\underline{\mathsf{\Omega }},\mathcal{B}\left(\underline{\mathsf{\Omega }}\right),{\underline{m}}_H)$. Denote by $m_{jH}$ the Haar measure on $({\mathsf{\Omega }}_j,\mathcal{B}\left({\mathsf{\Omega }}_j\right))$, $j=1,\dots ,r$. Then, ${\underline{m}}_H$ is the product of the measures $m_{1H},\dots ,m_{rH}$. Now, denote by $\underline{\omega }=({\omega }_1,\dots ,{\omega }_r)$ the elements of $\underline{\mathsf{\Omega }}$, where ${\omega }_j\in {\mathsf{\Omega }}_j$, $j=1,\dots ,r$. Let ${\omega }_j\left(p\right)$ be the pth component of an element ${\omega }_j\in {\mathsf{\Omega }}_j$, $j=1,\dots ,r$, $p\in \mathbb{P}$. Extend elements ${\omega }_j\left(p\right)$ to the set $\mathbb{N}$ by the formula

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

and define $H\left(D_{\kappa }\right)$-valued random element

$$\zeta (s,{\omega }_j,F)=\sum _{m=1}^{\infty }\frac{c\left(m\right){\omega }_j\left(m\right)}{m^s},j=1,\dots ,r.$$

The later series is uniformly convergent on compact subsets of $D_{\kappa }$ for almost all ${\omega }_j$. Moreover, for fixed $\sigma \in \left(\frac{\kappa }{2},\frac{\kappa +1}{2}\right)$

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

(5)

for almost all ${\omega }_j$, $j=1,\dots ,r$ [18]. Define one more series

$${\zeta }_n(s,{\omega }_j,F)=\sum _{m=1}^{\infty }\frac{c\left(m\right){\omega }_j\left(m\right)v_n\left(m\right)}{m^s},j=1,\dots ,r,$$

which also, as ${\zeta }_n(s,F)$, are absolutely convergent for $\sigma >\frac{\kappa }{2}$. Let

$$\underline{\zeta }(s+i\underline{h}\tau ,\underline{\omega },F)= \left(\zeta (s+i{h}_1\tau ,{\omega }_1,F),\dots ,\zeta (s+i{h}_r\tau ,{\omega }_1,F)\right)$$

and

$${\underline{\zeta }}_n(s+i\underline{h}\tau ,\underline{\omega },F)= \left({\zeta }_n(s+i{h}_1\tau ,{\omega }_1,F),\dots ,{\zeta }_n(s+i{h}_r\tau ,{\omega }_r,F)\right).$$

Then, repeating the proof of Lemma 1 and using estimate (5), we arrive to the following statement.

Lemma 2.

For all $\underline{h}$ and almost all $\underline{\omega }$,

$$\underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\frac{1}{T}\underset{0}{\overset{T}{\int }}\underline{\rho }\left(\underline{\zeta }(s+i\underline{h}\tau ,\underline{\omega },F),{\underline{\zeta }}_n(s+i\underline{h}\tau ,\underline{\omega },F)\right)\mathrm{d}\tau =0.$$

3. Limit Theorems

On the probability space $(\underline{\mathsf{\Omega }},\mathcal{B}\left(\underline{\mathsf{\Omega }}\right),{\underline{m}}_H)$, define $H\left(D_{\kappa }\right)$-valued random element

$$\underline{\zeta }(s,\underline{\omega },F)= \left(\zeta (s,{\omega }_1,F),\dots ,\zeta (s,{\omega }_1,F)\right)$$

and denote by $P_{\underline{\zeta },F}$ its distribution, i.e.,

$$P_{\underline{\zeta },F}\left(A\right)={\underline{m}}_H\left\{\underline{\omega }\in \underline{\mathsf{\Omega }}:\underline{\zeta }(s,\underline{\omega },F)\in A\right\},A\in \mathcal{B}\left(H^r\left(D_{\kappa }\right)\right).$$

Theorem 3.

Suppose that $h_1,\dots ,h_r$ are real algebraic numbers linearly independent over $\mathbb{Q}$, and

$$P_{T,F}\left(A\right)\stackrel{def}{=}\frac{1}{T}\mathrm{meas}\left\{\tau \in [0,T]:\underline{\zeta }(s+i\underline{h}\tau ,F)\in A\right\},A\in \mathcal{B}\left(H^r\left(D_{\kappa }\right)\right).$$

Then, $P_{T,F}$ converges weakly to $P_{\underline{\zeta },F}$ as $T\to \infty$.

We divide the proof of Theorem 3 into several lemmas.

Lemma 3.

Suppose that ${\lambda }_1,\dots ,{\lambda }_r$ are algebraic numbers such that the system $log{\lambda }_1,\dots ,log{\lambda }_r$ is linearly independent over $\mathbb{Q}$. Then, for arbitrary algebraic numbers ${\beta }_0,{\beta }_1,\dots ,{\beta }_r$ that are not all zeros, the inequality

$$|{\beta }_0+{\beta }_{1}log{\lambda }_1+\cdots +{\beta }_{r}log{\lambda }_r| >h^{-c}$$

holds. Here, h denotes the height of the numbers ${\beta }_0,{\beta }_1,\dots ,{\beta }_r$, and c is an effective constant depending on r, ${\lambda }_1,\dots ,{\lambda }_r$ and maximum of degrees of the numbers ${\beta }_0,{\beta }_1,\dots ,{\beta }_r$.

The lemma is a Baker result on linear forms of logarithm; see, for example, ref. [19]. For $A\in \mathcal{B}\left(\underline{\mathsf{\Omega }}\right)$, define

$$Q_T\left(A\right)=\frac{1}{T}\mathrm{meas}\left\{\tau \in [0,T]:\left(\left(p^{-i{h}_1\tau }:p\in \mathbb{P}\right),\dots ,\left(p^{-i{h}_r\tau }:p\in \mathbb{P}\right)\right) \in A\right\}.$$

Lemma 4.

Let ${\lambda }_1,\dots ,{\lambda }_r$ be the same as in Theorem 3. Then, $Q_T$ converges weakly to the Haar measure ${\underline{m}}_H$ as $T\to \infty$.

Proof. 

We apply the Fourier transform method. Denote by $g_T({\underline{k}}_1,\dots ,{\underline{k}}_r)$, ${\underline{k}}_j=\{k_{pj}:k_{pj}\in \mathbb{Z},p\in \mathbb{P}\}$, $j=1,\dots ,r$ the Fourier transform of $Q_T$. By the definition of $Q_T$, we have

$$\begin{array}{ccc}{g}_T({\underline{k}}_1,\dots ,{\underline{k}}_r)\hfill & =\hfill & \underset{\underline{\mathsf{\Omega }}}{\int }{\prod }_{j=1}^r{\displaystyle \underset{p\in \mathbb{P}}{{\prod }^{*}}}{\omega }_j^{k_{pj}}\left(p\right)\mathrm{d}{Q}_T\hfill \\ \\ \hfill & \hfill & \frac{1}{T}\underset{0}{\overset{T}{\int }}exp\left\{-i\tau {\sum }_{j=1}^k{\displaystyle \underset{p\in \mathbb{P}}{{\sum }^{*}}}{h}_j{k}_{pj}logp\right\}\mathrm{d}\tau ,\hfill \end{array}$$

(6)

where the star shows that only a finite number of integers $k_{pj}$ are not zero. Obviously,

$$g_T(\underline{0},\dots ,\underline{0})=1.$$

(7)

Now, suppose that $({\underline{k}}_1,\dots ,{\underline{k}}_r)\ne (\underline{0},\dots ,\underline{0})$. Then, there exists a prime number p such that $k_{pj}\ne 0$ for some j. Therefore,

$${\beta }_p\stackrel{def}{=}\sum _{j=1}^r{h}_j{k}_{pj}\ne 0$$

because the numbers $h_1,\dots ,h_r$ are linearly independent over Q. Thus, in view of Lemma 3,

$$B_{{\underline{k}}_1,\dots ,{\underline{k}}_r}\stackrel{def}{=}\sum _{j=1}^k\underset{p\in \mathbb{P}}{{\sum }^{*}}{h}_j{k}_{pj}logp=\underset{p\in \mathbb{P}}{{\sum }^{*}}{\beta }_{p}logp\ne 0.$$

This and (6) imply

$$g_T({\underline{k}}_1,\dots ,{\underline{k}}_r)=\frac{1-exp\left\{-iT{B}_{{\underline{k}}_1,\dots ,{\underline{k}}_r}\right\}}{iT{B}_{{\underline{k}}_1,\dots ,{\underline{k}}_r}}.$$

Therefore, by (7),

$$\underset{T\to \infty }{lim}{g}_T({\underline{k}}_1,\dots ,{\underline{k}}_r)\stackrel{def}{=}\left\{\begin{array}{cc}1\hfill & \mathrm{if}({\underline{k}}_1,\dots ,{\underline{k}}_r)=(\underline{0},\dots ,\underline{0}),\hfill \\ 0\hfill & \mathrm{if}({\underline{k}}_1,\dots ,{\underline{k}}_r)\ne (\underline{0},\dots ,\underline{0}),\hfill \end{array}\right.$$

and this proves the lemma. □

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

$$P_{T,n,F}\left(A\right)=\frac{1}{T}\mathrm{meas}\left\{\tau \in [0,T]:{\underline{\zeta }}_n(s+i\underline{h}\tau ,F)\in A\right\}$$

and

$$P_{T,n,\underline{\mathsf{\Omega }},F}\left(A\right)=\frac{1}{T}\mathrm{meas}\left\{\tau \in [0,T]:{\underline{\zeta }}_n(s+i\underline{h}\tau ,\underline{\omega },F)\in A\right\}.$$

Moreover, let the mapping $u_n:\underline{\mathsf{\Omega }}\to H^r\left(D_{\kappa }\right)$ be given by

$$u_{n,F}\left(\underline{\omega }\right)={\underline{\zeta }}_n(s,\underline{\omega },F),$$

and $V_{n,F}={\underline{m}}_H{u}_{n,F}^{-1}$, where

$$V_{n,F}\left(A\right)={\underline{m}}_H\left(u_{n,F}^{-1}A\right),A\in \mathcal{B}\left(H^r\left(D_{\kappa }\right)\right).$$

Since the series for ${\zeta }_n(s,{\omega }_j,F)$ are absolutely convergent for $\sigma >\frac{\kappa }{2}$, the mapping $u_{n,F}$ is continuous. Moreover, by the definitions of $Q_T$ and $P_{T,n,F}$, we have $P_{T,n,F}=Q_T{u}_{n,F}^{-1}$. This equality, continuity of $u_{n,F}$, Lemma 4, the well-known properties of weak convergence, and the invariance of the Haar measure ${\underline{m}}_H$ lead to the following lemma.

Lemma 5.

Let $h_1,\dots ,h_r$ be the same as Theorem 3. Then, $P_{T,n,F}$ and $P_{T,n,\underline{\mathsf{\Omega }},F}$ both converge weakly to the measure $V_{n,F}$ as $T\to \infty$.

Additionally to $P_{T,F}$, define

$$P_{T,\underline{\mathsf{\Omega }},F}\left(A\right)=\frac{1}{T}\mathrm{meas}\left\{\tau \in [0,T]:\underline{\zeta }(s+i\tau ,\underline{\omega },F)\in A\right\},A\in \mathcal{B}\left(H^r\left(D_{\kappa }\right)\right).$$

Lemma 6.

Let $h_1,\dots ,h_r$ be the same as Theorem 3. Then, on $(H^r\left(D_{\kappa }\right),\mathcal{B}\left(H^r\left(D_{\kappa }\right)\right))$, there exists a probability measure $P_F$ such that $P_{T,F}$ and $P_{T,\underline{\mathsf{\Omega }},F}$ both converge weakly to $P_F$ as $T\to \infty$.

Proof. 

Since the series for ${\zeta }_n(s,F)$ is absolutely convergent, by a standard way it follows—see, for example [14,18]—that the sequence $\{V_{n,F}:m\in \mathbb{N}\}$ is tight, i.e., for every $\epsilon >0$, there exists a compact set $K\subset H^r\left(D_{\kappa }\right))$ such that

$$V_{n,F}\left(K\right)>1-\epsilon$$

for all $n\in \mathbb{N}$. Hence, by the Prokhorov theorem, see [20], the sequence $\left\{V_{n,F}\right\}$ is relatively compact, i.e., each of its subsequences contains a subsequence $\left\{V_{n_k,F}\right\}$ such that $V_{n_k,F}$ converges weakly to a certain probability measure $P_F$ on $(H^r\left(D_{\kappa }\right),\mathcal{B}\left(H^r\left(D_{\kappa }\right)\right))$ as $k\to \infty$.

Let ${\xi }_T$ be a random variable defined on a certain probability space with measure $\nu$ and uniformly distributed on $[0,T]$. Define the $H^r\left(D_{\kappa }\right)$-valued random element

$${\underline{X}}_{T,n,F}={\underline{X}}_{T,n,F}\left(s\right)={\underline{\zeta }}_n(s+i\underline{h}{\xi }_T,F)$$

and denote by ${\underline{X}}_{n,F}={\underline{X}}_{n,F}\left(s\right)$ the $H^r\left(D_{\kappa }\right)$-valued random element having the distribution $V_{n,F}$. Then, by Lemma 5, we have

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

(8)

where $\underset{T\to \infty }{\overset{\mathcal{D}}{\to }}$ means the convergence in distribution. Moreover, since $V_{n_k,F}$ converges weakly to $P_F$, the relation

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

(9)

is true. Let

$${\underline{X}}_{T,F}={\underline{X}}_{T,F}\left(s\right)=\underline{\zeta }(s+i\underline{h}{\xi }_T,F).$$

Then, using Lemma 1, we find that for every $\epsilon >0$,

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}& \underset{T\to \infty }{lim\; sup}\nu \left\{\underline{\rho }\left({\underline{X}}_{T,F},{\underline{X}}_{T,n,F}\right) ⩾\epsilon \right\}\hfill \\ & ⩽\underset{n\to \infty }{lim}\underset{T\to \infty }{lim\; sup}\frac{1}{\epsilon T}\underset{0}{\overset{T}{\int }}\underline{\rho }\left(\underline{\zeta }(s+i\underline{h}\tau ,F),{\underline{\zeta }}_n(s+i\underline{h}\tau ,F)\right)\mathrm{d}\tau =0.\hfill \end{array}$$

The later equality together with (8) and (9), and Theorem 4.2 of [20] lead to the relation

$$X_{T,F}\underset{T\to \infty }{\overset{\mathcal{D}}{\to }}{P}_F.$$

(10)

This proves that $P_{T,F}$ converges weakly to $P_F$ as $T\to \infty$.

The relation (10) shows that the limit measure $P_F$ is independent of the subsequence $\left\{n_k\right\}$. Therefore, we have

$${\underline{X}}_{n,F}\underset{n\to \infty }{\overset{\mathcal{D}}{\to }}{P}_F.$$

(11)

Define the $H^r\left(D_{\kappa }\right)$-valued random elements

$${\underline{X}}_{T,n,\underline{\mathsf{\Omega }},F}={\underline{X}}_{T,n,\underline{\mathsf{\Omega }},F}\left(s\right)={\underline{\zeta }}_n(s+i\underline{h}{\xi }_T,\underline{\omega },F)$$

an

$${\underline{X}}_{T,\underline{\mathsf{\Omega }},F}={\underline{X}}_{T,\underline{\mathsf{\Omega }},F}\left(s\right)=\underline{\zeta }(s+i\underline{h}{\xi }_T,\underline{\omega },F).$$

Then, repeating the above arguments using Lemmas 2 and 5, and relation (11), we obtain that

$$X_{T,n,F}\underset{T\to \infty }{\overset{\mathcal{D}}{\to }}{P}_F,$$

and this is equivalent to weak convergence of $P_{T,\underline{\mathsf{\Omega }},F}$ to $P_F$ as $T\to \infty$. The lemma is proved. □

To prove Theorem 3, it remains to show that $P_F=P_{\underline{\zeta },F}$. For this, we will apply some elements of the ergodic theory. For brevity, let

$${\underline{h}}_{\tau }= \left(\left(p^{-i{h}_1\tau }:p\in \mathbb{P}\right),\dots ,\left(p^{-i{h}_r\tau }:p\in \mathbb{P}\right)\right),\tau \in \mathbb{R}.$$

Define the transformation of $\underline{\mathsf{\Omega }}$

$${\phi }_{\tau }\left(\underline{\omega }\right)={\underline{h}}_{\tau }\underline{\omega },\underline{\omega }\in \underline{\mathsf{\Omega }}.$$

Since the Haar measure ${\underline{m}}_H$ is invariant, the transformation ${\phi }_{\tau }$ is measure-preserving and $\{{\phi }_{\tau }:\tau \in \mathbb{R}\}$ is a one-parameter group. A set $A\in \mathcal{B}\left(\underline{\mathsf{\Omega }}\right)$ is called invariant with respect to the group $\left\{{\phi }_{\tau }\right\}$ if the sets A and ${\phi }_{\tau }\left(A\right)$, $\tau \in \mathbb{R}$, differ one from another at most by a set of ${\underline{m}}_H$-measure zero.

Lemma 7.

Let $h_1,\dots ,h_r$ be the same as Theorem 3. Then, the group $\left\{{\phi }_{\tau }\right\}$ is ergodic, i.e., the σ-field of invariant sets consists of sets having ${\underline{m}}_H$-measure 1 or 0.

Proof. 

The characters $\chi$ of the group $\mathsf{\Omega }$ are of the form

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

This fact already was used in the proof of Lemma 4. Let A be an arbitrary invariant set, $I_A$ its indicator function, and $\chi$ be a nontrivial character. Preserving the notation of the proof of Lemma 4, we have $({\underline{k}}_1,\dots ,{\underline{k}}_r)\ne (\underline{0},\dots ,\underline{0})$ and $B_{{\underline{k}}_1,\dots ,{\underline{k}}_r}\ne 0$. Therefore, there exists ${\tau }_0\in \mathbb{R}$ such that

$$\chi \left({\underline{h}}_{\tau }\right)=exp\left\{-i{\tau }_0{B}_{{\underline{k}}_1,\dots ,{\underline{k}}_r}\right\} \ne 1.$$

(12)

Moreover, in view of the invariance of A, we have

$$I_A\left({\underline{h}}_{{\tau }_0}\underline{\omega }\right)=I_A\left(\underline{\omega }\right)$$

(13)

for almost all $\underline{\omega }\in \underline{\mathsf{\Omega }}$. Denote by ${\widehat{I}}_A$ the Fourier transform of $I_A$. Then, by (13),

$${\widehat{I}}_A\left(\chi \right)=\chi \left({\underline{h}}_{{\tau }_0}\right)\underset{\underline{\mathsf{\Omega }}}{\int }{I}_A\left({\underline{h}}_{{\tau }_0}\underline{\omega }\right)\chi \left(\underline{\omega }\right)\mathrm{d}{\underline{m}}_H=\chi \left({\underline{h}}_{{\tau }_0}\right){\widehat{I}}_A\left(\chi \right).$$

This and (12) show that

$${\widehat{I}}_A\left(\chi \right)=0.$$

(14)

Now, let ${\chi }_0$ denote the trivial character of $\underline{\mathsf{\Omega }}$, and suppose that ${\widehat{I}}_A\left({\chi }_0\right)=\alpha$. Then, in view of (14), we find that

$${\widehat{I}}_A\left(\chi \right)=\alpha \underset{\underline{\mathsf{\Omega }}}{\int }\chi \left(\underline{\omega }\right)\mathrm{d}{\underline{m}}_H=\widehat{\alpha }\left(\chi \right).$$

Hence, $I_A\left(\underline{\omega }\right)=\alpha$ for almost all $\underline{\omega }\in \underline{\mathsf{\Omega }}$. Since $I_A$ is the indicator function, $I_A\left(\underline{\omega }\right)=1$ or $I_A\left(\underline{\omega }\right)=0$ for almost all $\underline{\omega }$. Thus, ${\underline{m}}_H\left(A\right)=1$ or ${\underline{m}}_H\left(A\right)=0$, and the lemma is proved. □

Proof of Theorem 3.

We have mentioned that it suffices to show that $P_F=P_{\underline{\zeta },F}$. By Lemma 6 and the equivalent of weak convergence in terms of continuity sets, we have

$$\underset{T\to \infty }{lim}{P}_{T,\underline{\mathsf{\Omega }},F}\left(A\right)=P_F\left(A\right)$$

(15)

for a continuity set A of the measure $P_F$, i.e., $P_F(\partial A)=0$, where $\partial A$ is the boundary of A. On the probability space $(\underline{\mathsf{\Omega }},\mathcal{B}\left(\underline{\mathsf{\Omega }}\right),{\underline{m}}_H)$, define the random variable

$$\xi \left(\underline{\omega }\right)= \left\{\begin{array}{c}1\mathrm{if}\underline{\zeta }(s,\underline{\omega },F)\in A,\hfill \\ 0\mathrm{otherwise}.\hfill \end{array}\right.$$

Lemma 7 implies the ergodicity of the random process $\xi \left({\phi }_{\tau }\left(\underline{\omega }\right)\right)$. Therefore, by the classical Birkhoff–Khintchine ergodic theorem, see, for example [21],

$$\begin{array}{c}\hfill \underset{T\to \infty }{lim}\frac{1}{T}\underset{0}{\overset{T}{\int }}\xi \left({\phi }_{\tau }\left(\underline{\omega }\right)\right)\mathrm{d}\tau =\mathbb{E}\xi =P_{\underline{\zeta },F}\left(A\right),\end{array}$$

(16)

where $\mathbb{E}\xi$ is the expectation of $\xi$.

However, by the definitions of ${\phi }_{\tau }$ and $\xi$,

$$\begin{array}{c}\hfill \frac{1}{T}\underset{0}{\overset{T}{\int }}\xi \left({\phi }_{\tau }\left(\underline{\omega }\right)\right)\mathrm{d}\tau =\frac{1}{T}\mathrm{meas}\left\{\tau \in [0,T]:\underline{\zeta }(s+i\underline{h}\tau ,\underline{\omega },F)\in A\right\} =P_{T,\underline{\mathsf{\Omega }},F}\left(A\right).\end{array}$$

This and (16) show that

$$\underset{T\to \infty }{lim}{P}_{T,\underline{\mathsf{\Omega }},F}\left(A\right)=P_{\underline{\zeta },F}\left(A\right).$$

Therefore, by (15), we obtain that $P_F\left(A\right)=P_{\underline{\zeta },F}\left(A\right)$ for all continuity sets A of $P_F\left(A\right)$. Hence, $P_F=P_{\underline{\zeta },F}$, and the theorem is proved. □

4. Proof of Theorem 2

Recall that the support of the measure $P_{\underline{\zeta },F}$ is a minimal closed set $S_F\subset H^r\left(D_{\kappa }\right)$ such that $P_{\underline{\zeta },F}\left(S_F\right)=1$.

Lemma 8.

The support of the measure $P_{\underline{\zeta },F}$ is the set ${\left(\left\{g\in H\left(D_{\kappa }\right):g\left(s\right)\ne 0org\left(s\right)\equiv 0\right\}\right)}^r$.

Proof. 

Since the space $H^r\left(D_{\kappa }\right)$ is separable, we have [20],

$$\mathcal{B}\left(H^r\left(D_{\kappa }\right)\right) = (\underset{r}{\underbrace{\mathcal{B}\left(H\left(D_{\kappa }\right)\right)\times \cdots \times \mathcal{B}\left(H\left(D_{\kappa }\right)\right)}}.$$

Therefore, it suffices to consider the measure $P_{\underline{\zeta },F}$ on the rectangular sets

$$A=A_1\times \cdots \times A_r,A_1,\dots ,A_r\in H\left(D_{\kappa }\right).$$

Let $\zeta (s,\omega ,F)$ be the $H\left(D_{\kappa }\right)$-valued random element defined on the probability space $(\mathsf{\Omega },\mathcal{B}\left(\mathsf{\Omega }\right),m_H)$, where $m_H$ is the Haar measure. Then, it is known [10] that the support of the distribution of $\zeta (s,\omega ,F)$ is the set $\{g\in H\left(D_{\kappa }\right):g\left(s\right)\ne 0\mathrm{or}g\left(s\right)\equiv 0\}$. Thus, the same set is the support of the distributions of $\zeta (s,{\omega }_j,F)$, $j=1,\dots ,r$. Since the measure ${\underline{m}}_H$ is the product of the measures $m_{jH}$, $j=1,\dots ,r$, we have

$${\underline{m}}_H\left\{\underline{\omega }\in \underline{\mathsf{\Omega }}:\underline{\zeta }(s,\underline{\omega },F)\in A\right\} =\prod _{j=1}^r{m}_{jH}\left\{{\omega }_j\in {\mathsf{\Omega }}_j:\zeta (s,{\omega }_j,F)\in A_j\right\}.$$

This equality, the minimality of the support, and the support of the distributions of $\zeta (s,{\omega }_j,F)$ prove the lemma. □

Proof of Theorem 2.

By the Mergelyan theorem on the approximation of analytic functions by polynomials [22], there exist polynomials $p_1\left(s\right),\dots ,p_r\left(s\right)$ such that

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

(17)

Define the set

$$G_{\epsilon }= \left\{(g_1,\dots ,g_r)\in H^r\left(D_{\kappa }\right):\underset{1\le j\le r}{sup}\underset{s\in K_j}{sup}\left|g_j\left(s\right)-e^{p_j\left(s\right)}\right| <\frac{\epsilon }{2}\right\}.$$

In view of Lemma 8, the set $G_{\epsilon }$ is an open neighborhood of an element $(e^{p_1\left(s\right)},\dots ,e^{p_r\left(s\right)})$ in support of the measure $P_{\underline{\zeta },F}$. Hence,

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

(18)

This, Theorem 3 and the equivalent of weak convergence in terms of open sets, and the definitions of $P_{T,F}$ and $G_{\epsilon }$ prove the theorem with “lim inf”. Define one more set

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

There $\partial {\widehat{G}}_{{\epsilon }_1}\bigcap \partial {\widehat{G}}_{{\epsilon }_2}=\varnothing$ for ${\epsilon }_1\ne {\epsilon }_2$. This shows that $P_{\underline{\zeta },F}\left(\partial {\widehat{G}}_{\epsilon }\right)=0$ for all but, for those countable, many $\epsilon >0$. Moreover, (17) and (18) imply that $P_{\underline{\zeta },F}\left({\widehat{G}}_{\epsilon }\right)>0$. This, Theorem 3 and the equivalent of weak convergence of probability measures in terms of continuity sets, and the definitions of $P_{T,F}$ and ${\widehat{G}}_{\epsilon }$ prove the theorem with “lim”. □