A generalized limit theorem for the periodic Hurwitz zeta-function

A. Rimkevičienė

Čebyševskij sbornik, Tome 20 (2019) no. 1, pp. 261-271.

Résumé

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 $T\to\infty$.

Détail

Keywords: Haar measure, Hurwitz zeta-function, limit theorem, periodic Hurwitz zeta-function, weak convergence.

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