Обобщенная предельная теорема для периодической дзета-функции Гурвица

С времен Бора и Йессена (1910-1935) в теории дзета-функций прмменяются вероятностные методы. В 1930 г. они доказали первую теорему для дзета-функции Римана $$\zeta(s), $$ $$s=\sigma+it,$$ которая является прототипом современных предельных теорем, характеризующих поведение дзета-функции при помощи слабой сходимости вероятностных мер. Более точно, они получили, что при $$\sigma>1$$ существует предел $$lim_{T\to\infty} \frac{1}{T} \mathrm{J} \left\{t\in[0,T]: \log\zeta(\sigma+it)\in R\right\},$$ где R - прямоугольник на комплексной плоскости со сторонами, паралельными осям, а $$\mathrm{J}A$$ обозначает меру Жордана множества $$A\subset \mathbb{R}.$$ Два года спустя они распространили приведенный результат на полуплоскость $$\sigma>\frac{1}{2}.$$ Идеи Бора и Йессена были развиты в работах Винтнера, Борщсениуса, Йессена, Сельберга и других известных математиков. Современные версии теорем Бора-Йессена для широкого класса дзета-функций были получены в работах К. Матсумото. В основном теория Бора-Йессена применялась для дзета-функций, имеющих эйлерово произведение по простым числам. В настоящей статье доказывается предельная теорема для дзета-функций, не имеющих эйлерова произведения и являющихся обобщением классичесской дзета-функции Гурвица. Пусть $$\alpha, 0<\alpha \leqslant 1, $$ фиксированный параметр, а $$\mathfrak{a}=\{a_m: m\in \mathbb{N}_0= \mathbb{N}\cup\{0\}\}$$ - периодическая последовательность комплексных чисел. Тогда периодическая дзета-функция Гурвица $$\zeta(s,\alpha; \mathfrak{a})$$ в полуплоскости $$\sigma>1$$ определяется рядом Дирихле $$\zeta(s,\alpha; \mathfrak{a})=\sum_{m=0}^\infty frac{a_m}{(m+\alpha)^s}$$ и мероморфно продолжается на всю комплексную плоскость. Пусть $$\mathcal{B}(\mathbb{C})$$ - борелевское $$\sigma$$-поле комплексной плоскости, $$\mathrm{meas}A$$ - мера Лебега измеримого множества $$A\subset \mathbb{R},$$ а функция $$\varphi(t)$$ при $$ t\geqslant T_0$$ имеет монотонную положительную производную $$\varphi'(t), $$ при $$t\to\infty$$ удовлетворяющую оценкам $$(\varphi'(t))^{-1}=o(t)$$ и $$\varphi(2t) \max_{t\leqslant u\leqslant 2t} (\varphi'(u))^{-1}\ll t. $$ Тогда в статье получено, что при $$\sigma>\frac{1}{2}$$ $$ \frac{1}{T} \mathrm{meas}\left\{t\in[0,T]: \zeta(\sigma+i\varphi(t), \alpha; \mathfrak{a})\in A\right\},\quad A\in \mathcal{B}(\mathbb{C}), $$ при $$T\to\infty$$ слабо сходится к некоторой в явном виде заданной вероятностной мере на $$(\mathbb{C}, \mathcal{B}(\mathbb{C})).$$

The theory of Bohr and Jessen is applicable, in general, for zeta-functions having Euler's product over primes. In the present paper, a limit theorem for a zeta-function without Euler's product is proved. This zeta-function is a generalization of the classical Hurwitz zeta-function. Let , 0 < 1, be a fixed parameter, and a = { : ∈ N 0 = N∪{0}} be a periodic sequence of complex numbers. The periodic Hurwitz zeta-function ( , ; a) is defined, for > 1, by the Dirichlet series , and is meromorphically continued to the whole complex plane. Let ℬ(C) denote the Borel -field of the set of complex numbers, meas be the Lebesgue measure of a measurable set ⊂ R, and let the function ( ) for 0 have the monotone positive derivative ′ ( ) such that ( ′ ( )) −1 = ( ) and (2 ) max 2 ( ′ ( )) −1 ≪ . Then it is obtained in the paper that, for converges weakly to a certain explicitly given probability measure on (C, ℬ(C)) as → ∞.
In honor of Professor Antanas Laurinčikas on the occasion of his 70th birthday

Introduction
The idea of application of probabilistic methods in the theory of zeta-functions is due to Bohr and Jessen. In [2], they proved a theorem for the Riemann zeta-function which is a prototype of a modern limit theorems on weakly convergent probability measures. Denote by J the Jordan measure of a measurable set ⊂ R, and let be a rectangle on the complex plane with edges parallel to the axis. Then they proved that, for > 1, there exists the limit Two years later, Bohr and Jessen extended [3] the above result to the half-plane > 1 2 . In this case, a problem arises because of possible zeros of ( ). Therefore, they defined the set where runs over all zeros of ( ) in the region { ∈ C : 1 2 < < 1}, and proved that there exists the limit lim In the sixth decade of the last century, the theory of weak convergence of probability measures was created. Therefore, it became possible to state Bohr-Jessen type theorems in the sense of weakly convergent probability measures, for results, see [6] and [8].
The present note is devoted to limit theorems for the periodic Hurwitz zeta-function. Let , 0 < 1 be a fixed parameter, and let a = { : be a periodic sequence of complex numbers with minimal period ∈ N. The periodic Hurwitz zeta-function ( , ; a) was introduced in [7], and is defined, for > 1, by the Dirichlet series .
If ≡ 1, then ( , ; a) becomes the classical Hurwitz zeta-function which has a meromorphic continuation to the whole complex plane with the unique simple pole at the point = 1 with residue 1. The periodicity of the sequence a implies, for > 1, the equality

А. Римкявичене
Therefore, the function ( , ; a) also can be continued meromorphically to the whole complex plane with the unique simple pole at the point = 1 with residue If = 0, then the periodic Hurwitz zeta-function is entire.
In [4], [9] and [11], limit theorems on weakly convergent probability measures on the complex plane for the function ( , ; a) were proved. Denote by ℬ( ) the Borel -field of the space . Then, for example, it was obtained in [10] that if the parameter is transcendental and > 1 2 is fixed, then, on (C, ℬ(C)), there exists a probability measure such that converges weakly to as → ∞. Moreover, the measure is given explicitly. The aim of this note is a generalization of the above theorem for The main result of this note is the following theorem. Theorem 1. Suppose that the parameter is transcendental, > 1 2 is fixed and ∈ ( 0 ). Then , , ;a converges weakly to the measure , , ;a as → ∞.
Proof. We apply the Fourier transform method. Let the sign " ′ " mean that only a finite number of integers are distinct from zero. Denote by ( ), = ( : ∈ Z, ∈ N 0 ) the Fourier transform of , . Then the definition of , implies that Clearly, Since is transcendental, the set {log( + ) : ∈ N 0 } is linearly independent over the field of rational numbers, thus the finite sum If the function ′ ( ) is decreasing, then ( ′ ( )) −1 is increasing. Thus, by the mean value theorem for integrals, as → ∞, where 0 . Similarly, we find that If the function ′ ( ) is increasing, then ( ′ ( )) −1 is decreasing, and we obtain by similar arguments that ∫︁ Now, the estimates (4)- (6), and equalities (3) and (1) show that The right-hand side of the latter equality is the Fourier transform of the Haar measure . This and a continuity theorem for probability measures on compact groups prove the lemma. P Now, we will deal with absolutely convergent Dirichlet series. Let .