The translation \(\frac 12\) in the theory of Dirichlet series
type: Item
instance of: scholarly article
title: The translation \(\frac 12\) in the theory of Dirichlet series
zbMATH Open document ID: 1026.42011
published in: Real Analysis Exchange
publication date: 2003-01-05T00:00:00Z
review text: Many results in the theory of Dirichlet series lead to a conclusion on the behavior of this or related series at a point shifted by \(1/2.\) It is worth giving a couple of (known) examples from the paper. First, if \(\sum|a_n|^2<\infty,\) then \(\sum a_n n^{-1/2+it}\) converges for almost all \(t.\) Next, if \(\sum a_n n^{-s}\) and \(\sum b_n n^{-s}\) converge at \(s=0,\) then the product of these series converges at \(s=1/2.\) Both results are sharp. In the paper this phenomenon is studied in detail. Besides considering results related to above examples, the authors show the non-existence of Rudin-Shapiro type Dirichlet polynomials.
reviewed by: Elijah Liflyand
Mathematics Subject Classification ID: 42A32
Mathematics Subject Classification ID: 40F05
Mathematics Subject Classification ID: 42A05
Mathematics Subject Classification ID: 42A20
Mathematics Subject Classification ID: 42A61
Mathematics Subject Classification ID: 60D05
zbMATH DE Number: 1848909
zbMATH Keywords: Dirichlet series
zbMATH Keywords: almost everywhere convergence
zbMATH Keywords: abscissa of convergence
zbMATH Keywords: Rudin-Shapiro like polynomials
zbMATH Keywords: \(p\)-Sidon set
author: Sergei Konyagin
author: Hervé Queffélec
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1848909