The inverses of tails of the generalized Riemann zeta function within the range of integers

Abstract

In recent years, many mathematicians researched infinite reciprocal sums of various sequences and evaluated their value by the asymptotic formulas. We study the asymptotic formulas of the infinite reciprocal sums formed as $ \left(\sum^{\infty}_{k = n} \frac{1}{k^r(k+t)^s} \right)^{-1} $ for $ r, s, t \in \mathbb{N^+} $, where the asymptotic formulas are polynomials.

Keywords

• riemann zeta function
• hurwitz zeta function
• reciprocal sums
• asymptotic formulas
• infinite sums