On the Distribution of a Cotangent Sum: Fig. 1.

Abstract

Maier and Rassias computed the moments and proved a distribution result for the cotangent sum <f>${c}_0(a/q):=-\\sum _{m\\lt q}\\frac mq\\cot ({\\pi ma}/{q})$</f> on average over <f>$\\frac {1}{2}\\lt A_0\\leq a/q\\lt A_1\\lt 1$</f>, as <f>$q\\rightarrow \\infty$</f>. We give a simple argument that recovers their results (with stronger error terms) and extends them to the full range <f>$1\\leq a\\lt q$</f>. Moreover, we give a density result for <f>${c}_0$</f> and answer a question posed by Maier and Rassias on the growth of the moments of <f>${c}_0$</f>.