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

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.