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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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Contributor : Marc-Antoine Coppo <>
Submitted on : Thursday, April 7, 2016 - 4:04:48 PM
Last modification on : Thursday, December 10, 2020 - 12:16:10 PM
Long-term archiving on: : Monday, November 14, 2016 - 9:48:37 PM


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  • HAL Id : hal-01277623, version 2



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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