The analytic continuation and the order estimate of multiple Dirichlet series.
type: Item
instance of: scholarly article
title: The analytic continuation and the order estimate of multiple Dirichlet series.
zbMATH Open document ID: 1050.11082
author: Kohji Matsumoto
author: Yoshio Tanigawa
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2004-03-16T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/224084
review text: The present paper deals with multiple Dirichlet series. For \(1\leq k\leq r\), let \[ \varphi_k(s)=\sum_{n=1}^\infty{a_k(n)\over n^s} \] be a Dirichlet series with complex coefficients, being absolutely convergent for \(\sigma:=\text{ Re}\,s>\alpha_k>0\), having a meromorphic continuation to the whole complex plane except for at most a simple pole at \(s=\alpha_k\), and satisfying the order estimate \(\varphi_k(\sigma+it)=O(| t|^A)\) in any fixed strip \(\sigma_1\leq \sigma\leq \sigma_2\), where \(A\) is a non-negative constant, depending only on \(\sigma_1,\sigma_2\). The authors introduce the multiple Dirichlet series \[ \Phi_r((s_1,\ldots,s_r);(\varphi_1,\ldots,\varphi_r)) \] \[ =\sum_{m_1=1}^\infty\sum_{m_2=1}^\infty\ldots\sum_{m_r=1}^\infty{a_1(m_1)\over m_1^s}{a_2(m_2)\over (m_1+m_2)^{s_2}}\cdot\ldots\cdot{a_r(m_r)\over (m_1+\ldots+m_r)^{s_r}} \] associated with \(\varphi_1,\ldots,\varphi_r\), where \(s_k=\sigma_k+it_k\). It is proved that \(\Phi_r((s_1,\ldots,s_r);(\varphi_1,\ldots,\varphi_r))\) can be continued meromorphically and its possible singularities are located on subsets defined by one of the equations \[ s_j+\ldots +s_r=\alpha_j+\delta_{j+1}\alpha_{j+1}+\ldots +\delta_r\alpha_r-n, \] where \(1\leq j\leq r\), \(n\) is a non-negative integer, and \(\delta_k\) is equal to zero or one; if \(j=r\), then the equation above is to be read as \(s_r=\alpha_r-n\). The authors give some restrictions on the possible values for \(n\), depending on \(j\) and \(r\). It is also proved that \(\Phi_r((s_1,\ldots,s_r);(\varphi_1,\ldots,\varphi_r))\) is an entire function if all \(\varphi_k(s)\) are entire. Finally, the authors prove an order estimate for \(\Phi_2((s_1,s_2);(\varphi_1,\varphi_2))\) in the region \(\sigma_1+\sigma_2>\alpha_1+\eta,\alpha_2+\eta\) for any positive \(\eta\) (except for points near the set of singularities). The method of proof relies on the classical Mellin-Barnes formula and the techniques in in [Analytic and probabilistic methods in numb…
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 33B15
zbMATH DE Number: 2058869
zbMATH Keywords: multiple Dirichlet series
reviewed by: Jörn Steuding
MaRDI profile type: Publication
OpenAlex ID: W2321457838
cites work: Analytic continuation of multiple zeta-functions and their values at non-positive integers
cites work: Q3152843
cites work: Multiple zeta values, poly-Bernoulli numbers, and related zeta functions
cites work: On multiple \(L\)-values
cites work: Multiple polylogarithms, cyclotomy and modular complexes
cites work: Volumes of hyperbolic manifolds and mixed Tate motives
cites work: The square mean of Dirichlet series associated with cusp forms
cites work: On analytic properties of a multiple \(L\)-function
cites work: A multiple character sum and a multiple \(L\)-function
cites work: Asymptotic expansions of double zeta-functions of Barnes, of Shintani, and Eisenstein series
cites work: The analytic continuation and the asymptotic behaviour of certain multiple zeta-functions. I.
cites work: Q4409710
cites work: Analytic continuation of multiple zeta functions
DOI: 10.5802/JTNB.402
description: scientific article; zbMATH DE number 2058869