Secant Zeta Functions

TL;DR

This paper investigates the secant zeta function, proving its convergence, exploring its modular properties, and providing explicit evaluations at quadratic irrationals, supporting conjectures about rationality involving Bernoulli numbers.

Contribution

It introduces the secant zeta function, proves its convergence, and derives explicit evaluations at quadratic irrationals, revealing new modular transformation properties.

Findings

Proves convergence of the series under certain conditions.

Establishes a modular transformation property for the function.

Provides explicit evaluations at quadratic irrational points.

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.