An Euler-type formula for β(2n) and closed-form expressions for a class of zeta series

Abstract

In a recent work, Dancs and He found an Euler-type formula for ζ(2n+1), n being a positive integer, which contains a series they could not reduce to a finite closed form. This open problem reveals a greater complexity in comparison with ζ(2n), which is a rational multiple of π2n . For the Dirichlet beta function, the things are ‘inverse’: β(2n+1) is a rational multiple of π2n+1 and no closed-form expression is known for β(2n). Here in this work, I modify the Dancs–He approach in order to derive an Euler-type formula for β(2n), including β(2)=G, Catalan's constant. I also convert the resulting series into zeta series, which yields new exact closed-form expressions for a class of zeta series involving β(2n) and a finite number of odd zeta values. A closed-form expression for a certain zeta series is also conjectured.