Some integral and asymptotic formulas associated with the Hurwitz Zeta function
The main object of this paper is to present an easily accessible integral representation of the partial sum L"u(x,a)=@?"0"=<"n"=<"x(n+a)^u of the Hurwitz Zeta function @z(-u,a), which entails a number of important implications for L"u(x,a) and @z(-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.