Evaluation of Series Involving the Product of the Tail of ${\zeta(k)}$ and ${\zeta(k+1)}$

Abstract

We aim at evaluating the following class of series involving the product of the tail of two consecutive zeta function values

$$\sum\limits_{n=1}^{\infty}\left(\zeta(k)-1-\frac{1}{2^k}-\cdots-\frac{1}{n^k}\right)\left(\zeta(k+1)-1-\frac{1}{2^{k+1}}-\cdots-\frac{1}{n^{k+1}}\right),$$

where ${k\geq 2}$ is an integer. We show that the series can be expressed in terms of Riemann zeta function values and a special integral involving a polylogarithm function.