A multiple character sum and a multiple \(L\)-function
type: Item
instance of: scholarly article
title: A multiple character sum and a multiple \(L\)-function
zbMATH Open document ID: 1034.11053
DOI: 10.1007/BF02638381
author: Yanyan Li
published in: Archiv der Mathematik
publication date: 2003-03-27T00:00:00Z
review text: The main result of the paper is an asymptotic formula for the Riesz mean \[ {1\over k!}\sum_{n\leq x}(x-n)^kh_j(n),\leqno(1) \] where \[ h_j(n) = \sum_{m_1m_2\cdots m_{j-1}| n, m_1<\cdots <m_{j-1}<n/(m_1\cdots m_{j-1})} \chi_1(m_1)\cdots\chi_{j-1}(m_{j-1}) \chi_j\bigl({n\over m_1\cdots m_{j-1}}\bigr), \] with \(\chi_i\;(i = 1,\cdots, j)\) Dirichlet characters to the same modulus \(q\). The sum is connected to the multiple \(L\)-function \[ L_j(s) = \sum_{0<n_1<\cdots<n_j} {\chi_1(n_1)\cdots \chi_j(n_j)\over (n_1\cdots n_j)^s} = \sum_{n=1}^\infty {h_j(n)\over n^s}\quad(\Re s > 1). \] The asymptotics for (1) (the reader is referred to the paper for details) are derived by the Perron inversion formula in the form suitable for the evaluation of Riesz means. The main term comes from the poles of the analytic continuation of \(L_j(s)\). However, the lack of the functional equation for \(L_j(s)\) makes its estimation difficult. The paper ends with some explicit examples of the asymptotic formulas for (1) when \(j = 3\) and \(j = 4\).
reviewed by: Aleksandar Ivić
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11M41
zbMATH DE Number: 1889274
zbMATH Keywords: mean values
zbMATH Keywords: Dirichlet characters
zbMATH Keywords: Riesz mean
zbMATH Keywords: multiple \(L\)-function
zbMATH Keywords: Bernoulli polynomials
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1889274