, Volume 22, Issue 1, pp 55-99
Date: 19 Mar 2010

The evaluation of Tornheim double sums. Part 2

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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 $$\int_{0}^{1}\log\Gamma(q)B_{k}(q)\operatorname{Cl}_{l+1}(2\pi q)\,dq,\vspace*{-3pt}$$ with k+l=N, where B k (q) is a Bernoulli polynomial and Cl  l+1(x) is a Clausen function.

The second author acknowledges the partial support of NSF-DMS 0713836.