Reciprocity theorems for Bettin–Conrey sums

Juan S. Auli ¹   ; Abdelmejid Bayad ²   ; Matthias Beck ³

¹ Department of Mathematics Dartmouth College Hanover, NH 03755, U.S.A.
² Univ. Évry Université Paris-Sacly I.B.G.B.I. 23 Bd. de France 91025 Évry Cedex, France
³ Department of Mathematics San Francisco State University San Francisco, CA 94132, U.S.A.

Acta Arithmetica, Tome 181 (2017) no. 4, pp. 297-319

Résumé

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.

Détail

DOI : 10.4064/aa8580-8-2017

Keywords: recent work bettin conrey period functions eisenstein series naturally gave rise dedekind like sum biggl frac bigg sum k cot biggl frac bigg zeta biggl a frac bigg where mathbb positive coprime integers zeta denotes hurwitz zeta function derive reciprocity theorem these bettin conrey sums which odd negative integer explicitly given terms bernoulli numbers turn implies explicit formulas period functions appearing bettin conrey work study generalizations bettin conrey sums involving zeta derivatives multiple cotangent factors relate these special values estermann zeta function