On shifted Mascheroni Series and hyperharmonic numbers

Abstract : In this article, we study the nature of the forward shifted series σ r = n>r |bn| n−r where r is a positive integer and b n are Bernoulli numbers of the second kind, expressing them in terms of the derivatives ζ (−k) of zeta at the negative integers and Euler's constant γ. These expressions may be inverted to produce new series expansions for the quotient ζ(2k + 1)/ζ(2k). Motivated by a theoretical interpretation of these series in terms of Ramanujan summation, we give an explicit formula for the Ramanujan sum of hyperharmonic numbers as an application of our results.
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Marc-Antoine Coppo, Paul Thomas Young. On shifted Mascheroni Series and hyperharmonic numbers. Journal of Number Theory, Elsevier, 2016, 169, pp.1-20. ⟨hal-01277623v2⟩

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