International Journal of Mathematics and Mathematical Sciences
Volume 13 (1990), Issue 3, Pages 453-460
doi:10.1155/S0161171290000679
An integral involving the generalized zeta function
Dept. of Structure and Constituents of Matter, Faculty of Physics, University of Barcelona, Diagonal 647, Barcelona 08028, Spain
Received 21 February 1989; Revised 5 May 1989
Copyright © 1990 E. Elizalde and A. Romeo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.