On some integrals involving the Hurwitz zeta function. I
A series of integral formulae involving the Hurwitz zeta function are established. After reviewing some useful integral representations of the Hurwitz zeta function, as Hermite's and Yu's and Williams' ones, the authors do the same with a number of known definite integrals involving the Hurwitz zeta function, as Mikolas' identity for the integral of the product of two of them. They also mention the numerous physical applications as summarized in a book by the reviewer [\textit{E. Elizalde}, Ten physical applications of spectral zeta functions (1995; Zbl 0855.00002)]. After this introduction, the authors deal with the Fourier expansion of \(\zeta(z,q)\) and employ it to evaluate definite integrals of the form \(\int^1_0f(q) \zeta(z,q)dq\). A very useful theorem for the integral of the product of two Hurwitz zeta functions is proven in the next section, and classical relations for the Bernoulli polynomials are obtained as corollaries. Several explicit applications are then given, in particular to integrals of Bernoulli polynomials, \(\ln\Gamma (q)\) and \(\ln\sin q\), to an expression for Catalan's constant, to Clausen and related functions, and to Eisenstein series. Finally, to be remarked is the fact that most of the formulae involving elementary functions and \(\ln\Gamma(q)\) can be considered, with the help of equalities of the form \(\ln\Gamma (q)+\ln \Gamma (1-q)= \ln\pi-\ln \sin(\pi q)\), to be specail cases of the already mentioned theorem, which relates, in a linear way, all the above mentioned examples.
scientific article; zbMATH DE number 1785613