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Journal Articles Journal of Mathematical Analysis and Applications Year : 2019

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 $\sum_{n\geq 2}(−1)^n \frac{ζ(n}{n}$ 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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Dates and versions

hal-01735381 , version 1 (15-03-2018)
hal-01735381 , version 2 (09-05-2018)
hal-01735381 , version 3 (06-09-2018)
hal-01735381 , version 4 (29-10-2018)
hal-01735381 , version 5 (28-10-2019)
hal-01735381 , version 6 (22-03-2021)
hal-01735381 , version 7 (06-10-2021)

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Cite

Marc-Antoine Coppo. A note on some alternating series involving zeta and multiple zeta values. Journal of Mathematical Analysis and Applications, 2019, 475, pp.1831-1841. ⟨10.1016/j.jmaa.2019.03.057⟩. ⟨hal-01735381v7⟩
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