A note on some alternating series involving zeta and multiple zeta values

Abstract

In this article, we study a class of conditionally convergent alternating series including, as a special case, the famous series ∑n⩾2(−1)nζ(n)n which links Euler's constant γ to special values of the Riemann zeta function at positive integers. We give several new relations of the same kind. Among other things, we show the existence of a similar relation for the Apostol-Vu harmonic zeta function which have never been noticed before. We also highlight a deep connection with the Ramanujan summation of certain divergent series which originally motivated this work.