J. Korean Math. Soc. 2020; 57(6): 1535-1549
Online first article June 1, 2020 Printed November 1, 2020
https://doi.org/10.4134/JKMS.j190789
Copyright © The Korean Mathematical Society.
Ho-Hyeong Lee, Jong-Do Park
Kyung Hee University; Kyung Hee University
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
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