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Journal Articles Journal of Number Theory Year : 2016

On shifted Mascheroni Series and hyperharmonic numbers

Paul Thomas Young
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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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Dates and versions

hal-01277623 , version 1 (22-02-2016)
hal-01277623 , version 2 (07-04-2016)

Identifiers

  • HAL Id : hal-01277623 , version 2

Cite

Marc-Antoine Coppo, Paul Thomas Young. On shifted Mascheroni Series and hyperharmonic numbers. Journal of Number Theory, 2016, 169, pp.1-20. ⟨hal-01277623v2⟩
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