Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2020; 57(6): 1535-1549

Published online November 1, 2020 https://doi.org/10.4134/JKMS.j190789

Copyright © The Korean Mathematical Society.

Asymptotic behavior of the inverse of tails of Hurwitz zeta function

Ho-Hyeong Lee, Jong-Do Park

Kyung Hee University; Kyung Hee University

Abstract

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

Keywords: Hurwitz zeta function, Riemann zeta function, gamma function, Bernoulli number, convergent series

MSC numbers: Primary 33B15, 11M35, 11B68

Supported by: This work was supported by NRF-2018R1D1A1B07050044 from National Research Foundation of Korea