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arXiv · 1901.10373

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

Atul Dixit·Rajat Gupta
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Abstract

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 $ζ(2m+1)$. The formula for $ζ^{2}(2m+1)$ is then generalized in two different directions, one, by considering the generalized divisor function $σ_z(n)$, 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 $ζ(1+ω)ζ(-1-ω)$, where $ω$ is a non-trivial zero of $ζ(z)$. We also evaluate a series involving the modified Bessel function of the second kind in the form of a rational linear combination of $ζ(4k-1)$ and $ζ(4k+1)$ for $k\in\mathbb{N}$.

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Atul Dixit, Rajat Gupta. 2019-01-29. A Ramanujan-type formula for $ζ^{2}(2m+1)$ and its generalizations. https://arxiv.org/abs/1901.10373

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