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An application of the theory of the double gamma function. (English) Zbl 1013.11053

Authors’ summary: The main purpose of this paper is first to show how one can apply the theory of the double Gamma function, which has recently been revived in the study of the determinants of Laplacians, to evaluate some classes of series involving the zeta-function. The determinants of Laplacians on the \(n\)-sphere \(S^n\) (\(n=1,2,3)\) are computed by using our evaluations of series involving the zeta-function. Relevant connections of the results presented here with those given in earlier works are also pointed out.

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
33B15 Gamma, beta and polygamma functions
11M35 Hurwitz and Lerch zeta functions
33C55 Spherical harmonics
58J52 Determinants and determinant bundles, analytic torsion
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