Asymptotic behavior of the inverse of tails of Hurwitz zeta function

J. Korean Math. Soc. Published online June 1, 2020

Ho-Hyeong Lee and Jong-Do Park
Kyung Hee University

Abstract : This paper deals with the inverse of tails of Hurwitz zeta function.
More precisely, for any positive integer $s\geq2$, we give an algorithm for finding a simple form of $f_s(n)$ such that
$$
\lim_{n\rightarrow\infty}\left\{\left(\sum^\infty_{k=n}\frac{1}{(k+a)^s}\right)^{-1}-f_s(n)\right\}=0.
$$
We show that $f_s(n)$ is a polynomial in $n-a$ of order $s-1$.
All coefficients of $f_s(n)$ are represented in terms of Bernoulli numbers.