The Ramanujan Journal

, Volume 22, Issue 1, pp 55–99

The evaluation of Tornheim double sums. Part 2


DOI: 10.1007/s11139-009-9181-1

Cite this article as:
Espinosa, O. & Moll, V.H. Ramanujan J (2010) 22: 55. doi:10.1007/s11139-009-9181-1


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 Bk(q) is a Bernoulli polynomial and Cl l+1(x) is a Clausen function.


Hurwitz zeta function Tornheim sum Witten zeta function 

Mathematics Subject Classification (2000)

33E20 11M06 11M35 

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Departamento de FísicaUniversidad Téc. Federico Santa MaríaValparaísoChile
  2. 2.Department of MathematicsTulane UniversityNew OrleansUSA

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