A note on certain 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 $\sum_{n\geq 2}(−1)^n \frac{ζ(n}{n}$ which links Euler’s constant $\gamma$ 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.
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Submitted on : Monday, October 28, 2019 - 4:57:29 PM
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Marc-Antoine Coppo. A note on certain alternating series involving zeta and multiple zeta values. Journal of Mathematical Analysis and Applications, Elsevier, 2019, 475, pp.1831-1841. ⟨10.1016/j.jmaa.2019.03.057⟩. ⟨hal-01735381v5⟩

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