Series involving the Zeta function and multiple Gamma functions

摘要

The theory of multiple Gamma functions, which was recently revived in the study of the determinants of the Laplacians, was applied in several earlier works in order to evaluate some families of series involving the Riemann Zeta function as well as to compute the determinants of the Laplacians. Here, in the present paper, the authors address the converse problem and apply various (known or new) formulas for series associated with the Zeta and related functions with a view to developing the corresponding theory of multiple Gamma functions and then using these series to compute the determinants of the Laplacians on the n-dimensional unit sphere Sn (n=5,6,7) explicitly.