Reciprocity Theorems for Bettin--Conrey Sums

TL;DR

This paper establishes new reciprocity theorems for Bettin--Conrey sums, connecting them to Bernoulli numbers and special zeta function values, thereby advancing understanding of period functions related to Eisenstein series.

Contribution

It introduces a novel reciprocity theorem for Bettin--Conrey sums and explores their generalizations involving zeta derivatives and multiple cotangent factors.

Findings

Derived explicit reciprocity formulas for Bettin--Conrey sums.

Connected Bettin--Conrey sums to Bernoulli numbers for odd negative integers.

Linked generalized sums to special values of the Estermann zeta function.

Abstract

Recent work of Bettin and Conrey on the period functions of Eisenstein series 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 $a\in\mathbb{C}$, $h$ 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.