On the distribution of a cotangent sum

Abstract

Maier and Rassias computed the moments and proved a distribution result for the cotangent sum $c_0(a/q):=-{\sum }_{m<q}\frac{m}{q}\cot (\frac{\pi ma}{q})$ on average over $1/2<A_0\le a/q<A_1<1$, as $q\to \infty$. We give a simple argument that recovers their results (with stronger error terms) and extends them to the full range $1\le a<q$. 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 $c_0$.

10 pages, 2 figures

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• Asymptotics for moments of certain cotangent sums for arbitrary exponents
• Sur certaines équations fonctionnelles approchées, liées à la transformation de Gauss
• The error for the second moment of cotangent sums related to the Riemann Hypothesis
• Series representation of a cotangent sum related to the Estermann zeta function
• Linear correlations of the divisor function