On the Lerch zeta-function
Let \(s= \sigma+it\) be a complex variable, and let \(\mathbb{R}\) and \(\mathbb{Z}\) denote the sets of all real numbers and all integer numbers, respectively. Then the Lerch zeta-function is defined by \[ L(\lambda, \alpha,s) =\sum^\infty_{m=0} {e^{2 \pi i\lambda m} \over (m+ \alpha)^s} \quad \text{for} \quad \sigma>1, \] where \(\lambda \in\mathbb{R}\) and \(\alpha>0\). It is clear that when \(\lambda \notin \mathbb{N}\) the function \(L(\lambda, \alpha,s)\) can be continued analytically to the \(s\)-plane, where it represents an entire function. The reviewer [\textit{W. Zhang}, On the mean square value formula of Lerch zeta-function, Adv. Math., Beijing 22, 367--369 (1993; Zbl 0789.11050)], proved an asymptotic formula for the mean square \[ I(s; \lambda) =\int^1_0 \bigl| L (\lambda, \alpha,s) -\alpha^{-s} |^2 d\alpha. \] In this paper, the author studies the statistical properties of the Lerch function, and proves a limit theorem in the sense of the weak convergence of probability measures in the complex plane \(\mathbb{C}\).
weak convergence of probability measures
scientific article; zbMATH DE number 1135555