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Certain classes of series associated with the zeta function and multiple gamma functions. (English) Zbl 0969.11030

The authors apply the theory of multiple Gamma functions, which was recently reviewed in the study of the determinants of Laplacians, in order to evaluate some families of series involving the Riemann zeta-function. By introducing a (presumably new) mathematical constant, they also systematically evaluate this constant and some definite integrals of the triple Gamma function. Various classes of series associated with the zeta function, some of which can also be used to compute the determinant of the Laplacian on the four-dimensional unit sphere \(S^4\) explicitly, are expressed in closed forms.

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
33B15 Gamma, beta and polygamma functions
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