Abelian and Tauberian results on Dirichlet series
type: Item
instance of: scholarly article
title: Abelian and Tauberian results on Dirichlet series
zbMATH Open document ID: 1087.11056
DOI: 10.1007/BF03322893
author: N. E. Zubov
published in: Results in Mathematics
publication date: 2006-03-07T00:00:00Z
review text: Consider the sequence \(\{a_n\}\) of complex numbers and an asymptotic formula \[ \sum_{n\leq x} a_n= xP(\log x)+ O(x^\beta L(x)),\quad P(y)= \sum^k_{m=0} p_m y^m, \] where \(L\) is a slowly oscillating function. Conditions are supplied under which the coefficients, \(C_r\), of the Laurent expansion \[ \sum^\infty_{n=1} {a_n\over n^s}= \sum^k_{m=0} {p_m m!s\over (s-1)^{m+1}}+ \sum^\infty_{r=0} {(-1)^r C_r\over r!} (s- 1)^r \] can be obtained. Consequently, an asymptotic formula is obtained for \[ \sum_{n\leq x} {a_n\over n}\log^r n. \] This provides a generalization of the results of \textit{W. E. Briggs} [Mathematika 9, 49--53 (1962; Zbl 0166.05401)]. Several examples for specific arithmetic functions are included.
Mathematics Subject Classification ID: 11M36
Mathematics Subject Classification ID: 11M45
Mathematics Subject Classification ID: 11A25
zbMATH DE Number: 5009751
zbMATH Keywords: Tauberian theorems
zbMATH Keywords: Dirichlet series
zbMATH Keywords: arithmetic functions
reviewed by: Robert G. Buschman
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5009751