Reciprocity theorems for Bettin–Conrey sums
Volume 181 / 2017
Abstract
Recent work of Bettin and Conrey on the period functions of Eisenstein series naturally gave rise to the Dedekind-like sum \[ c_{a}\biggl(\frac{h}{k}\bigg) = k^{a}\sum_{m=1}^{k-1}\cot\biggl(\frac{\pi mh}{k}\bigg)\zeta\biggl(-a,\frac{m}{k}\bigg), \] where $a \in \mathbb C$, $h$ and $k$ are positive coprime integers, and $\zeta(a,x)$ denotes the Hurwitz zeta function. We derive a new reciprocity theorem for these Bettin–Conrey sums , which in the case of an odd negative integer $a$ can be explicitly given in terms of Bernoulli numbers. This in turn implies explicit formulas for the period functions appearing in Bettin–Conrey’s work. We study generalizations of Bettin–Conrey sums involving zeta derivatives and multiple cotangent factors and relate these to special values of the Estermann zeta function.
