Secant Zeta Functions

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.

10 pages