Error bounds for the asymptotic expansion of the Hurwitz zeta function

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.Comment: 16 pages. arXiv admin note: text overlap with arXiv:1606.07961, accepted for publication in Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science