# Error bounds for the asymptotic expansion of the Hurwitz zeta function

## TL;DR

This paper derives new, sharp error bounds for the asymptotic expansion of the Hurwitz zeta function at large arguments, with applications to related special functions.

## Contribution

It introduces novel representations for the remainder term, enabling precise error bounds for the Hurwitz zeta function and related functions.

## Key findings

- New representations for the remainder term of the asymptotic expansion
- Sharp and realistic error bounds established
- Applications to polygamma, gamma, Barnes G-functions, and derivatives

## Abstract

In this paper, we reconsider the large-$a$ asymptotic expansion of the Hurwitz zeta function $\zeta(s,a)$. New representations for the remainder term of the asymptotic expansion are found and used to obtain sharp and realistic error bounds. Applications to the asymptotic expansions of the polygamma functions, the gamma function, the Barnes $G$-function and the $s$-derivative of the Hurwitz zeta function $\zeta(s,a)$ are provided. A detailed discussion on the sharpness of our error bounds is also given.