On the distribution of a cotangent sum

Maier and Rassias computed the moments and proved a distribution result for the cotangent sum $c_0(a\mathrm{/}q):=-{\sum }_{m<q}\frac{m}{q}\cot (\frac{\pi ma}{q})$ on average over $1\mathrm{/}2<A_0\le a\mathrm{/}q<A_1<1$, as $q\to \mathrm{\infty }$. 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 $c_0$.