Some integral and asymptotic formulas associated with the Hurwitz Zeta function

摘要

The main object of this paper is to present an easily accessible integral representation of the partial sum Lu(x,a)=∑0≦n≦x(n+a)u of the Hurwitz Zeta function ζ(−u,a), which entails a number of important implications for Lu(x,a) and ζ(−u,a) and for their derivatives. Furthermore, a new proof is given for a fundamental summation formula (contained in Theorem 3 below), which is shown to possess numerous further consequences and applications. Relevant connections of the results obtained here with those considered in many other earlier and recent works are also discussed.