On the distribution of a cotangent sum

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 $c_0$.