Period functions and cotangent sums

TL;DR

This paper studies the period function related to a divisor sum, extends its analytic properties, and introduces cotangent sums that generalize Dedekind sums, providing new proofs and formulas in analytic number theory.

Contribution

It offers a new analytic continuation of the period function, a simple proof of the Voronoi formula, and introduces a family of cotangent sums with reciprocity properties.

Findings

Analytic continuation of the period function to |rac{z}|<rac{ ext{pi}}{ ext{arg} z}

A new proof of the Voronoi formula

A reciprocity formula for Vasyunin sums

Abstract

We investigate the period function of $\sum_{n=1}^\infty\sigma_a(n)\e{nz}$, showing it can be analytically continued to $|\arg z|<\pi$ and studying its Taylor series. We use these results to give a simple proof of the Voronoi formula and to prove an exact formula for the second moments of the Riemann zeta function. Moreover, we introduce a family of cotangent sums, functions defined over the rationals, that generalize the Dedekind sum and share with it the property of satisfying a reciprocity formula. In particular, we find a reciprocity formula for the Vasyunin sum.