Joint value-distribution theorems for the Lerch zeta-functions
type: Item
instance of: scholarly article
title: Joint value-distribution theorems for the Lerch zeta-functions
zbMATH Open document ID: 0938.11046
DOI: 10.1007/BF02465899
author: Kohji Matsumoto
author: Antanas Laurinčikas
published in: Lithuanian Mathematical Journal
publication date: 1999-06-10T00:00:00Z
review text: Für reelle \(\lambda,\alpha\) mit \(0<\lambda< 1\), \(0< \alpha\leq 1\) sei \(L(\lambda, \alpha,s)\) \((s\in \mathbb{C})\) die Lerchsche Zetafunktion. \(B(S)\) sei die Klasse der Borelschen Mengen von \(S\). \(H\) sei der Raum der in \(\operatorname {Re}(s)> \frac 12\) analytischen Funktionen, der mit der Topologie der gleichmäßigen Konvergenz auf kompakten Mengen versehen ist. Für eine Lebesgue-meßbare Menge \(M\subset \mathbb{R}\) sei \(\operatorname {mes}M\) das Lebesguemaß von \(M\). Die Autoren beweisen als Verallgemeinerung einiger Resultate von \textit{R. Garunkštis} und \textit{A. Laurinčikas} [Lith. Math. J. 36, 337-346 (1996; Zbl 0902.11035)] die folgenden Sätze. Satz 1. Es sei \(0< \lambda_j< 1\), \(0< \alpha_j\leq 1\) und \(\sigma_j> \frac 12\) für \(j=1,\dots, n\). Dann existiert ein Wahrscheinlichkeitsmaß \(P\) auf \((\mathbb{C}^n, B(\mathbb{C}^n))\), so daß \[ T^{-1} \operatorname {mes} \{\tau\in [0,T]: (L(\lambda_j, \alpha_j, \sigma_j+ i\tau))_{t\leq j\leq n}\in A\} \qquad (A\in b(\mathbb{C}^n)) \] schwach gegen \(P\) für \(T\to\infty\) konvergiert. Satz 2. Es sei \(0< \alpha_j< 1\) und \(\alpha_j\) transzendent für \(j=1,\dots, n\) oder \(0< \alpha_j\leq 1\) und \(\alpha_j\) rational für \(j=1,\dots, n\). Dann läßt sich ein Wahrscheinlichkeitsmaß \(Q\) konstruieren, so daß \[ T^{-1} \operatorname {mes} \{\tau\in [0,t]: (L(\lambda_j, \alpha_j, s+i\tau))_{1\leq j\leq n}\in A\} \qquad (A\in B(H^n(D))) \] schwach gegen \(Q\) für \(T\to \infty\) konvergiert.
Mathematics Subject Classification ID: 11M35
zbMATH DE Number: 1274174
zbMATH Keywords: joint value-distribution
zbMATH Keywords: Lerch zeta-functions
zbMATH Keywords: weak convergence
zbMATH Keywords: probability measure
reviewed by: Wolfgang Haneke
MaRDI profile type: Publication
cites work: Two New Proofs of Lerch's Functional Equation
cites work: Q5560061
cites work: An explicit form of limit distribution with weight for the Lerch zeta-function in the space of analytic functions
cites work: Q4234451
cites work: On the Lerch zeta-function
cites work: An application of Mellin-Barnes' type integrals to the mean square of Lerch zeta-functions
cites work: Asymptotic equalities for the Lipschitz-Lerch zeta-function
cites work: Q3035386
cites work: A limit theorem for the Lerch zeta-function in the space of analytic functions
cites work: Q4234456
cites work: The universality of the Lerch zeta-function
cites work: On the Lerch zeta-function with rational parameters
cites work: Hilbert's Inequality
cites work: Q4177788
cites work: Q4272283
full work available at URL: https://doi.org/10.1007/bf02465899
OpenAlex ID: W2086370562
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description: scientific article; zbMATH DE number 1274174