An integral involving the generalized zeta function

Abstract

A general value for ∫abdtlogΓ(t), for a, b positive reals, is derived in terms of the Hurwitz ζ function. That expression is checked for a previously known special integral, and the case where a is a positive integer and b is half an odd integer is considered. The result finds application in calculating the numerical value of the derivative of the Riemann zeta function at the point −1, a quantity that arises in the evaluation of determinants of Laplacians on compact Riemann surfaces.

Keywords

• zeta function
• determinants
• Riemann surfaces
• string theories.