Analytic continuation of modified L-function
type: Item
instance of: scholarly article
title: Analytic continuation of modified L-function
zbMATH Open document ID: 0566.10035
DOI: 10.1007/BF00970404
author: Eugenijus Stankus
published in: Lithuanian Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: By a generalized system of prime numbers (g-primes) a sequence \(q_ n\) is meant which satisfies the conditions \(1<q_ 1\leq q_ 2\leq...\), \(q_ n\to \infty\). The multiplicative semigroup generated by the sequence \(\{q_ n\}\) is the set \(1=n_ 1<n_ 2\leq n_ 3\leq..\). which is called a sequence of g-integers. \textit{C. Ryavec} [Ill. J. Math. 17, 608-618 (1973; Zbl 0266.10042)] studied the class of g-primes of the form \(\{\) vp\(\}\), where p are rational primes and \(v>1/2\). There it was proved that the modified zeta-function \[ \zeta (s,v):=\prod_{p}(1-(vp)^{- s})^{-1}=\sum^{\infty}_{j=1}n_ j^{-s},\quad Re(s)>1 \] can be continued analytically to the half-plane \(Re(s)>0\) outside of a ''comb- like'' set which depends on the zeros of the Riemann zeta-function. In this note a more general situation is considered: Let \(\chi\) be a Dirichlet character mod k, v a natural number. Then the author studies the function \[ L(s,v,\chi):=\prod_{p}(1-\chi(vp)^{-s})^{-1} = \sum^{\infty}_{j=1}\chi(n_ j) n_ j^{-s}. \] Here the product over all rational primes converges for \(Re(s)>1\). It is proved by the method of Ryavec that the series \(L(s,v,\chi)\) can also be continued analytically to the half-plane with ''slits'' with respect to the zeros of the Dirichlet L-functions. Then an asymptotic formula is obtained for the number of g-integers \(n_ j\), \(n_ j\equiv \ell (mod k)\) with \((k,v)=(1,k)=1\), not exceeding x. Because the author gets the sharp remainder term \(O(x \exp(-c\sqrt{\log x})\), \(c>0\) he has studied carefully the dependence of the remainder term from the involved moduli, including Siegel zeros.
Mathematics Subject Classification ID: 11M35
Mathematics Subject Classification ID: 11N80
zbMATH DE Number: 3902763
zbMATH Keywords: analytic continuation
zbMATH Keywords: modified L-function
zbMATH Keywords: generalized system of prime numbers
zbMATH Keywords: Dirichlet character
zbMATH Keywords: asymptotic formula
zbMATH Keywords: number of g-integers
zbMATH Keywords: sharp remainder term
zbMATH Keywords: Siegel zeros
MaRDI profile type: Publication
cites work: Q5335427
cites work: Q5619868
cites work: Analyse de la loi asymptotique de la distribution des nombres premiers généralises
cites work: A class of modified \(\zeta\) and \(L\)-functions
cites work: Über eine Klasse modifizierter \(\zeta\)- und \(L\)-Funktionen
cites work: The analytic continuation of Euler products with applications to asymptotic formulae
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description: scientific article; zbMATH DE number 3902763