Abstract
TextIn this article, we study the nature of the forward shifted series σr=∑n>r|bn|n−r where r is a positive integer and bn 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. VideoFor a video summary of this paper, please visit https://youtu.be/uyLmgDh9JVs.
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