Secant Zeta Functions

Matilde Lalín; Francis Rodrigue; Mathew Rogers

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arXiv · 1304.3922

Secant Zeta Functions

Matilde Lalín·Francis Rodrigue·Mathew Rogers

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Abstract

We study the series $\psi_s(z):=\sum_{n=1}^{\infty} \sec(n\pi z)n^{-s}$, and prove that it converges under mild restrictions on $z$ and $s$. The function possesses a modular transformation property, which allows us to evaluate $\psi_{s}(z)$ explicitly at certain quadratic irrational values of $z$. This supports our conjecture that $\pi^{-k} \psi_{k}(\sqrt{j})\in\mathbb{Q}$ whenever $k$ and $j$ are positive integers with $k$ even. We conclude with some speculations on Bernoulli numbers.

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BibTeX RIS

Matilde Lalín, Francis Rodrigue, Mathew Rogers. 2013-04-14. Secant Zeta Functions. https://arxiv.org/abs/1304.3922

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