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Reciprocity theorems for Bettin–Conrey sums

Volume 181 / 2017

Juan S. Auli, Abdelmejid Bayad, Matthias Beck Acta Arithmetica 181 (2017), 297-319 MSC: Primary 11F20; Secondary 11L03, 11M35. DOI: 10.4064/aa8580-8-2017 Published online: 24 November 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.

Authors

  • Juan S. AuliDepartment of Mathematics
    Dartmouth College
    Hanover, NH 03755, U.S.A.
    e-mail
  • Abdelmejid BayadUniv. Évry
    Université Paris-Sacly
    I.B.G.B.I.
    23 Bd. de France
    91025 Évry Cedex, France
    e-mail
  • Matthias BeckDepartment of Mathematics
    San Francisco State University
    San Francisco, CA 94132, U.S.A.
    e-mail

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