On the distribution of a cotangent sum

TL;DR

This paper provides a simplified proof of the distribution of a cotangent sum, extends the results to a broader range, and addresses questions about the growth of its moments, enhancing understanding of its probabilistic behavior.

Contribution

The authors offer a simpler proof of existing distribution results, extend the analysis to the entire range of parameters, and resolve an open question on the moments' growth.

Findings

Recovered distribution results with stronger error bounds

Extended the distribution analysis to the full parameter range

Answered a question on the growth of moments of the cotangent sum

Abstract

Maier and Rassias computed the moments and proved a distribution result for the cotangent sum $c_0(a/q):=-\sum_{m<q}\frac mq\cot(\frac{\pi ma}{q})$ on average over $1/2<A_0\leq a/q<A_1<1$, as $q\rightarrow \infty$. We give a simple argument that recovers their results (with stronger error terms) and extends them to the full range $1\leq a<q$. Moreover, we give a density result for $c_0$ and answer a question posed by Maier and Rassias on the growth of the moments of $c_0$.