A generalized limit theorem for the periodic Hurwitz zeta-function

Abstract

Probabilistic methods are used in the theory of zeta-functions since Bohr and Jessen time (1910-1935). In 1930, they proved the first theorem for the Riemann zeta-function $$\zeta(s)$$, $$s=\sigma+it$$, which is a prototype of modern limit theorems characterizing the behavior of $$\zeta(s)$$ by weakly convergent probability measures. More precisely, they obtained that, for $$\sigma>1$$, there exists the limit $$\lim_{T\to\infty} \frac{1}{T} \mathrm{J} \left\{t\in[0,T]: \log\zeta(\sigma+it)\in R\right\}, $$ where R is a rectangle on the complex plane with edges parallel to the axes, and $$\mathrm{J}A$$ denotes the Jordan measure of a set $$A\subset \mathbb{R}$$. Two years latter, they extended the above result to the half-plane $$\sigma>\frac{1}{2}$$. Ideas of Bohr and Jessen were developed by Wintner, Borchsenius, Jessen, Selberg and other famous mathematicians. Modern versions of the Bohr-Jessen theorems, for a wide class of zeta-functions, were obtained in the works of K. Matsumoto. 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 $$\alpha$$, $$0<\alpha \leqslant 1$$, be a fixed parameter, and $$\mathfrak{a}=\{a_m: m\in \mathbb{N}_0= \mathbb{N}\cup\{0\}\}$$ be a periodic sequence of complex numbers. The periodic Hurwitz zeta-function $$\zeta(s,\alpha; \mathfrak{a})$$ is defined, for $$\sigma>1$$, by the Dirichlet series $$\zeta(s,\alpha; \mathfrak{a})=\sum_{m=0}^\infty \frac{a_m}{(m+\alpha)^s}, $$ and is meromorphically continued to the whole complex plane. Let $$\mathcal{B}(\mathbb{C})$$ denote the Borel $$\sigma$$-field of the set of complex numbers, $$\mathrm{meas}A$$ be the Lebesgue measure of a measurable set $$A\subset \mathbb{R}$$, and let the function $$\varphi(t)$$ for $$t\geqslant T_0$$ have the monotone positive derivative $$\varphi'(t)$$ such that $$(\varphi'(t))^{-1}=o(t)$$ and $$\varphi(2t) \max_{t\leqslant u\leqslant 2t} (\varphi'(u))^{-1}\ll t$$. Then it is obtained in the paper that, for $$\sigma>\frac{1}{2}$$, $$\frac{1}{T} \mathrm{meas}\left\{t\in[0,T]: \zeta(\sigma+i\varphi(t), \alpha; \mathfrak{a})\in A\right\},\quad A\in \mathcal{B}(\mathbb{C}), $$ converges weakly to a certain explicitly given probability measure on $$(\mathbb{C}, \mathcal{B}(\mathbb{C}))$$ as $$T\to\infty$$.