H. Bohr's vision of ordinary Dirichlet series; old and new results
type: Item
instance of: scholarly article
title: H. Bohr's vision of ordinary Dirichlet series; old and new results
zbMATH Open document ID: 0881.11068
published in: The Journal of Analysis
publication date: 1998-02-16T00:00:00Z
review text: The author discusses briefly a theorem of Harald Bohr from 1913: \(\sum_{n=1}^\infty a_nn^{-s}\) has a finite abscissa of uniform convergence \(\sigma_u\) implies \(\sum_{p\text{ prime}}|a_p|p^{-s}\) converges for \(\sigma>\sigma_u\), and a theorem of Hewitt and Williamson about ordinary Dirichlet series representations of the reciprocals of absolutely convergent Dirichlet series. He proves a substantial generalization of Bohr's result which leads to a probabilistic estimate connected with the determination of the abscissa of uniform convergence. The author also obtains a new proof of the result of \textit{E. Hille} and \textit{H. F. Bohnenblust} [Ann. Math., II. Ser. 32, 600-622 (1931; Zbl 0001.12901)] that Bohr's estimate \(\sigma_a- \sigma_u\leq 1/2\) (where \(\sigma_a\) is the abscissa of absolute convergence and \(\sigma_u\) as above) is best possible. This is based on a new result improving an earlier one of his. He also gives a novel proof of the result of Hewitt and Williamson using Bohr's ``vision''. The paper closes with some open questions.
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 40A05
Mathematics Subject Classification ID: 11K99
Mathematics Subject Classification ID: 43A46
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 30B50
zbMATH DE Number: 1006728
zbMATH Keywords: Dirichlet series
zbMATH Keywords: probabilistic estimate
zbMATH Keywords: abscissa of uniform convergence
zbMATH Keywords: open questions
author: Hervé Queffélec
reviewed by: Sanford L. Segal
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1006728