On shifted Mascheroni series and hyperharmonic numbers

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 bnbn are Bernoulli numbers of the second kind, expressing them in terms of the derivatives ζ′(−k)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)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.