The inverse of tails of Riemann zeta function, Hurwitz zeta function and Dirichlet L-function

Abstract

In this paper, we derive the asymptotic formulas $ B^*_{r, s, t}(n) $ such that $ \mathop{\lim} \limits_{n \rightarrow \infty} \left\{ \left( \sum\limits^{\infty}_{k = n} \frac{1}{k^r(k+t)^s} \right)^{-1} - B^*_{r,s,t}(n) \right\} = 0, $ where $ Re(r+s) > 1 $ and $ t \in \mathbb{C} $. It is evident that the asymptotic formulas for the inverses of the tails of both the Riemann zeta function and the Hurwitz zeta function on the half-plane $ Re(s) > 1 $ are its corollaries. Subsequently we provide the asymptotic formulas for the Riemann zeta function and the Hurwitz zeta function on the half-plane $ Re(s) < 0 $. Finally, we study the asymptotic formulas of the inverse of the tails of the Dirichlet L-function for $ Re(s) > 1 $ and $ Re(s) < 0 $.

Keywords

• riemann zeta function
• hurwitz zeta function
• reciprocal sums
• asymptotic formulas
• dirichlet l-function