A Ramanujan-type formula for $ζ^{2}(2m+1)$ and its generalizations

Atul Dixit, Rajat Gupta

Published 2019-01-29Version 1

A Ramanujan-type formula involving the squares of odd zeta values is obtained. The crucial part in obtaining such a result is to conceive the correct analogue of the Eisenstein series involved in Ramanujan's formula for $\zeta(2m+1)$. The formula for $\zeta^{2}(2m+1)$ is then generalized in two different directions, one, by considering the generalized divisor function $\sigma_z(n)$, and the other, by studying a more general analogue of the aforementioned Eisenstein series, consisting of one more parameter $N$. A number of important special cases are derived from the first generalization. For example, we obtain a series representation for $\zeta(1+\omega)\zeta(-1-\omega)$, where $\omega$ is a non-trivial zero of $\zeta(z)$. We also evaluate a series involving the modified Bessel function of the second kind in the form of a rational linear combination of $\zeta(4k-1)$ and $\zeta(4k+1)$ for $k\in\mathbb{N}$.