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Reciprocity Theorems for Bettin--Conrey Sums

Research paper by Juan S. Auli, Abdelmejid Bayad, Matthias Beck

Indexed on: 25 Jan '16Published on: 25 Jan '16Published in: Mathematics - Number Theory



Abstract

Recent work of Bettin and Conrey on the period functions of Eisenstein series % and the second moments of the Riemann zeta function naturally gave rise to the Dedekind-like sum \[ c_{a}\left(\frac{h}{k}\right) \ = \ k^{a}\sum_{m=1}^{k-1}\cot\left(\frac{\pi mh}{k}\right)\zeta\left(-a,\frac{m}{k}\right), \] where $a \in \CC$, $h$ and $k$ are positive coprime integers, and $\zeta(a,x)$ denotes the Hurwitz zeta function. We derive a new reciprocity theorem for these \emph{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.