Asymptotics to all orders of the Hurwitz zeta function

Abstract

We present several formulae for the large-$t$ asymptotics of the modified Hurwitz zeta function ${\zeta }_1(x,s),x>0,s=\sigma +it,0<\sigma \le 1,t>0,$ 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.

30 pages, 2 figures