On shifted Mascheroni series and hyperharmonic numbers

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• 作者：Marc-Antoine Coppo^a ; ^{coppo@unice.fr" class="auth_mail" title="E-mail the corresponding author} ; Paul Thomas Young^b ; ^{paul@math.cofc.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Mascheroni series ; Euler&ndash ; Mascheroni constant ; Riemann zeta function ; Ramanujan summation ; Hyperharmonic numbers ; Wolstenholme primes
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：169
• 期：Complete
• 页码：1-20
• 全文大小：341 K

文摘

In this article, we study the nature of the forward shifted series ${\sigma }_r={\sum }_{n>r}\frac{|b_n|}{n-r}$ where r   is a positive integer and b_n$b_n$ are Bernoulli numbers of the second kind, expressing them in terms of the derivatives ζ^′(−k)${\zeta }^{\prime }(-k)$ 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)$\zeta (2k+1)/\zeta (2k)$. 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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