The evaluation of Tornheim double sums. Part2

Details The evaluation of Tornheim double sums. Part2 The evaluation of Tornheim double sums. Part2

2008-11-04

arxiv.org/abs/0811.0557v1

Mathematics

Number Theory

38 pages, AMSLaTeX; submitted to Journal of Number Theory. "Tornheim.m", a Mathematica 6.0 package to work with Tornheim sums,

Scientific paper

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 % \int_0^1 loggamma(q) B_{k}(q) Cl_{j+1} (2 \pi q) 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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