Sums involving the Hurwitz zeta function

We shall prove a general closed formula for integrals considered by Ramanuj an, from which we derive our former results on sums involving Hurwitz zeta- function in terms not only of the derivatives of the Hurwitz zeta-function, but also of the multiple gamma function, thus covering all possible formul as in this direction. The transition from the derivatives of the Hurwitz ze ta-function to the multiple gamma function and vice versa is proved to be e ffected essentially by the orthogonality relation of Stirling numbers.