The evaluation of Tornheim double sums. Part2

2008/11/04 by Olivier Espinosa, Espinosa, Olivier, Victor H. Moll +1
Mathematics · #11M35 #33E20 #40C10 #Advanced Mathematical Identities #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT) #math.CA #math.NT #msc:11M35 #msc:33E20 #msc:40C10

paper · pdf · doi:10.48550/arxiv.0811.0557

38 pages, AMSLaTeX; submitted to Journal of Number Theory. "Tornheim.m", a Mathematica 6.0 package to work with Tornheim sums, is available for download from http://www.math.tulane.edu/~vhm/packages.html

arxiv created 2008/11/04 · openalex publication_date 2008/11/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We provide an explicit formula for the Tornheim double series T(a,0,c) in terms of an integral involving the Hurwitz zeta function. For integer values of the parameters, a=m, c=n, we show that in the most interesting case of even weight N:=m+n the Tornheim sum T(m,0,n) can be expressed in terms of zeta values and the family of integrals % ∫₀¹ loggamma(q) B_k(q) Cl_j+1 (2 πq) dq, % with k+j = N, where B_k(q) is a Bernoulli polynomial and \Cl_j+1(x) is a Clausen function.

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