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Kyushu Journal of Mathematics
Online ISSN : 1883-2032
Print ISSN : 1340-6116
ISSN-L : 1340-6116
67-1
THE BARNES G-FUNCTION AND THE CATALAN CONSTANT
Xiaohan WANG
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2013 Volume 67 Issue 1 Pages 105-116

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Abstract

The Catalan constant, Gn=1[(−1)k/(2n−1)2], has attracted much attention and has appeared in various contexts. It is the special value at s=2 of the Dirichlet L-function associated with the Gaussian field with its character odd. In this paper, we appeal to the Barnes G-function and a form of the functional equation for the Riemann zeta-function,to deduce an expression for the integral in t/sin t and one for G. These give rise to a corollary which reveals the intrinsicity of the Catalan constant G to the Barnes G-function. We also use one of Ramanujan's formulas to derive a theorem.

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© 2013 Faculty of Mathematics, Kyushu University
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