On the asymptotics of a cotangent sum related to the Estermann zeta function

TL;DR

This paper derives an extended asymptotic expansion for a cotangent sum linked to the Estermann zeta function by applying the Poisson summation formula, adding three more terms to previous results.

Contribution

It provides a more detailed asymptotic expansion of the cotangent sum related to the Estermann zeta function, extending prior work with additional terms.

Findings

Derived three additional terms in the asymptotic expansion

Extended understanding of the cotangent sum's behavior for large k

Applied Poisson summation to finite sums in this context

Abstract

The sum $c_0(1/k)=-\sum_{m=1}^{k-1}(m/k)\cot(m{\pi}/k)$ is related to the Estermann zeta function. A recent paper computes the first two terms of the large-$k$ asymptotic expansion of $c_0(1/k)$. Using the Poisson summation formula for finite sums, we find three additional terms.