Rapidly convergent zeta series for $ζ{(2n+1)}$ and $β{(2n)}$ and their generalization

Details Rapidly convergent zeta series for $ζ{(2n+1)}$ and $β{(2n)}$ and their generalization Rapidly convergent zeta series for $ζ{(2n+1)}$ and $β{(2n)}$ and their generalization

2012-03-26

arxiv.org/abs/1203.5660v1

Mathematics

Number Theory

9 pages, no figures. Submitted to Int. Transf. Special Functions (03/06/2012)

Scientific paper

In a very recent work on Euler-type formulae for even Dirichlet beta values, i.e. $\beta{(2n)}$, $n$ being a positive integer, I have derived an exact closed-form expression for a family of zeta series. From this family, the analytical results I have found for certain related series led me to conjecture exact expressions for two other families of zeta series. Here in this work, I make use of a classical Wilton's formula to prove the simpler conjecture. The other conjecture is proved in two independent forms. The comparison of this second result with a well-known theorem by Srivastava yields a new identity relating $\beta{(2n)}$ to the first derivatives of the Riemann zeta and the Hurwitz zeta functions. Indeed, the generalization of the mentioned results for zeta series has led me to detect an error in a theorem by Katsurada. The corrected version of this theorem is presented here.

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