Ramanujan’s formula for odd zeta values: a proof by Mittag-Leffler expansion and applications

Abstract

Using Mittag-Leffler expansion a novel proof of Ramanujan’s famous formula for zeta (2m-1) is presented. Here, the formula can be derived by Taylor series of a generating function and its Mittag-Leffler expansion. Furthermore, generalized formulas for the fast-converging series of Plouffe are offered. But Ramanujan’s equation not only produces series for odd zeta values, surprisingly it is shown that in the limit to its singularity it provides the identities for even zeta values, namely Euler’s classical formula. Finally, a new triangle identity of Ramanujan’s formula is presented, hinting that properties and symmetries of the equation are far from all uncovered. For this work, a new representation of the equations is introduced, that significantly simplifies analyses and applications.