A two variable Riemann zeta-function
type: Item
instance of: scholarly article
title: A two variable Riemann zeta-function
zbMATH Open document ID: 0623.10028
DOI: 10.1016/0022-314X(87)90063-1
author: B. George
published in: Journal of Number Theory
publication date: 1987-01-01T00:00:00Z
review text: This paper is essentially a pendant to two earlier papers of the author [J. Lond. Math. Soc., II. Ser. 20, 101--108 (1979; Zbl 0401.12017); ibid. 11, 256 (1975; Zbl 0335.10017)]. The numbers \(A^*_{m,n}\) may be defined by \[ A^*_{m,n}=\frac{1}{n+1}\sum^{n+1}_{j=1}(-1)^ j \left( \begin{matrix} n+1\\ j\end{matrix} \right) B_{m+j} B_{n+1-j}\quad (m,n\geq 0) \] where \(B_ n\) is the \(n\)th Bernoulli number (in the notation with \(B_{2m+1}=0\) for \(m\geq 1)\). The author suggests that the \(A^*_{m,n}\) are a genuinely two variable analogue of the Bernoulli numbers. In this direction he proves the following results. Define the ''poly-Dirichlet series'' \(Y(w,z)\), for \(\text{Re}\,w>2\), \(\text{Re}\,z>2\), by \[ Y(w,z) = (z-1)\sum^{\infty}_{m\leq n, m=1}\sum^{\infty}_{n=1}m^{-w+1} n^{- z}-(w-1)\sum^{\infty}_{m>n, m=1}\sum^{\infty}_{n=1}m^{-w} n^{-z+1}. \] Then \(Y(w,z)\) is analytically continuable to an entire function of \((w,z)\in\mathbb C\times\mathbb C\) and satisfies the equation \(Y(w,z)+Y(z,w)=Z(z+w-1)\) where \(Z(s)=(s-1)\zeta (s)\), \(\zeta\) the Riemann zeta-function. Furthermore \[ Y(1-m,1-n) = (-1)^{m+1} A^*_{m,n}+(-1)^ m B_ m \delta_{n0} \] where \(\delta_{nm}\) is the Kronecker delta. Various relationships and identities for the \(A^*_{m,n}\) are also exhibited. At the paper's conclusion, the author raises the question whether there exists a ``poly-Dirichlet series'' entire function \(X\) of two complex variables satisfying \(X(z,w)=X(w,z)\) and \(X(1-m,1-n)=A^*_{m,n}\).
Mathematics Subject Classification ID: 11M35
Mathematics Subject Classification ID: 11B68
Mathematics Subject Classification ID: 11M99
Mathematics Subject Classification ID: 32A15
Mathematics Subject Classification ID: 11S80
Mathematics Subject Classification ID: 32D99
zbMATH DE Number: 4010612
zbMATH Keywords: zeta-functions
zbMATH Keywords: functional equation
zbMATH Keywords: analytic continuation
zbMATH Keywords: two-variable Riemann zeta-function
zbMATH Keywords: two variable analogue of the Bernoulli numbers
zbMATH Keywords: poly-Dirichlet series
zbMATH Keywords: entire function
reviewed by: Sanford L. Segal
MaRDI profile type: Publication
cites work: Q4101884
cites work: Dirichlet series related to the Riemann zeta function
cites work: Q4161937
cites work: AN APPLICATION OF p-ADIC CONVOLUTIONS
cites work: Q3901719
cites work: Convolutions on the Ring of p -Adic Integers
cites work: A Spectral Valuation on the Ring of p -Adic Integers
cites work: Spectral Properties of Tame Functions on the Ring of p -Adic Integers
full work available at URL: 0022-314x(87)90063-1
OpenAlex ID: W1985992634
description: scientific article; zbMATH DE number 4010612