The Ramanujan Journal

, Volume 36, Issue 1–2, pp 43–68 | Cite as

On Eulerian log-gamma integrals and Tornheim–Witten zeta functions

  • David H. BaileyEmail author
  • David Borwein
  • Jonathan M. Borwein
Research Paper


Stimulated by earlier work by Moll and his coworkers (Amdeberhan et al., Proc. Am. Math. Soc., 139(2):535–545, 2010), we evaluate various basic log Gamma integrals in terms of partial derivatives of Tornheim–Witten zeta functions and their extensions arising from evaluations of Fourier series. In particular, we fully evaluate
$$\mathcal{LG}_n=\int_0^1 \log^n\varGamma(x) \,\mathrm{d}x $$
for 1≤n≤4 and make some comments regarding the general case. The subsidiary computational challenges are substantial, interesting and significant in their own right.


Gamma function Log gamma function Riemann zeta function Tornheim–Weitten zeta function Integration 

Mathematics Subject Classification

11M35 11M32 33B15 



Thanks are due to Victor Moll for suggesting we revisit this topic and to Richard Crandall for his significant help with computation of ω’s partial derivatives and for his insightful solutions to various other of our computational problems.


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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • David H. Bailey
    • 1
    Email author
  • David Borwein
    • 2
  • Jonathan M. Borwein
    • 3
    • 4
  1. 1.Lawrence Berkeley National LaboratoryBerkeleyUSA
  2. 2.Department of MathematicsUniversity of Western OntarioLondonCanada
  3. 3.Centre for Computer-assisted Research Mathematics and its Applications (CARMA), School of Mathematical and Physical SciencesUniversity of NewcastleCallaghanAustralia
  4. 4.King Abdul-Aziz UniversityJeddahSaudia Arabia

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