Asymptotics to all orders of the Hurwitz zeta function

TL;DR

This paper derives comprehensive all-order asymptotic formulas for the modified Hurwitz zeta function, extending classical results and including special cases like the Riemann zeta function.

Contribution

It provides new all-order asymptotic expansions for the Hurwitz zeta function valid for large t, generalizing previous results and connecting to classical cases.

Findings

Derived formulas valid to all orders for large t

Unified asymptotic expressions for Hurwitz and Riemann zeta functions

Extended classical asymptotics to broader parameter ranges

Abstract

We present several formulae for the large-$t$ asymptotics of the modified Hurwitz zeta function $\zeta_1(x,s),x>0,s=\sigma+it,0<\sigma\leq1,t>0,$ which are valid to all orders. In the case of $x=0$, these formulae reduce to the asymptotic expressions recently obtained for the Riemann zeta function, which include the classical results of Siegel as a particular case.