Vol. 168, No. 2, 1995

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Values of the Riemann zeta function and integrals involving log(2sinh𝜃 2) and log(2sin𝜃 2)

Zhang Nan-Yue and Kenneth S. Williams

Vol. 168 (1995), No. 2, 271–289
Abstract

Integrals involving the functions log(2sinh(𝜃∕2)) and log(2sin(𝜃∕2)) are studied, particularly their relationship to the values of the Riemann zeta function at integral arguments. For example general formulae are proved which contain the known results

                    ∫ π
3 log2(2sin(𝜃∕2))d𝜃 = 7π3∕108,
0
∫ π3
𝜃 log2(2sin(𝜃∕2))d𝜃 = 17π4∕6480,
∫  π                0
3(log4(2 sin(𝜃∕2)) − 3𝜃2log2(2sin (𝜃∕2)))d𝜃 = 253π5∕3240,
0                  2
∫ π3     4            𝜃3                      6
(𝜃log (2 sin(𝜃∕2)) − 2-log(2sin (𝜃∕2)))d𝜃 = 313π ∕408240,
0

as special cases.

Mathematical Subject Classification 2000
Primary: 11Y60
Secondary: 11M06
Milestones
Received: 5 November 1992
Revised: 16 November 1993
Published: 1 April 1995
Authors
Zhang Nan-Yue
Kenneth S. Williams