On certain alternating series involving zeta and multiple zeta values
Résumé
In this article, we consider various generalizations of Euler's famous relation
\[
\gamma = \sum_{n\geq 2} (-1)^n \frac{\zeta(n)}{n}
\]
linking Euler's constant $\gamma$ to special values of the Riemann zeta function at positive integers. Among other things, we highlight the existence of a very similar relation for the Apostol-Vu harmonic zeta function which have never been noticed before.
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