Integrals involving a function associated with the Euler-Maclaurin summation formula

Abstract

The authors first evaluate sums of certain series involving the Riemann Zeta function by using the theory of the double Gamma function. With or without the use of these results, the authors then establish seberal integrals involving a certain periodic function which is associated, for example, with the Euler-Maclaurin summation formula. Finally, by introducing some new mathematical constants, several families of definite integrals involving logarithms of Gamma functions are considered. Relevant connections of the results presented here with those given in earlier works are also pointed out.