Publications de l’Institut Mathématique, Nouvelle Série Vol. 102[116], pp. 155–174 (2017)

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## Explicit and asymptotic formulae for Vasyunin-cotangent sums

### Mouloud Goubi, Abdelmejid Bayad, Mohand Ouamar Hernane

Laboratoire d’Algèbre et Théorie des Nombres, Department of Mathematics, University of UMMTO, Tizi-ouzou, Algeria; Département de mathématiques, Université d’Evry Val d’Essonne, Evry Cedex, France; Département d’Algèbre et Théorie des Nombres, Faculté de Mathématiques, Université des Sciences et de la technologie, Houari-Boumediène (USTHB), Alger, Algérie

Keywords: Vasyunin-cotangent sum, Estermann zeta function, fractional part function, Riemann hypothesis

@abstract: For coprime numbers $p$ and $q$, we consider the Vasyunin-cotangent sum

$V\left(q,p\right)=\sum _{k=1}^{p-1}\left\{\frac{kq}{p}\right\}cot\left(\frac{\pi k}{p}\right)·$

First, we prove explicit formula for the symmetric sum $V\left(p,q\right)+V\left(q,p\right)$ which is a new reciprocity law for the sumsabove. This formula can be seen as a complement to the Bettin–Conrey result Theorem 1.

Second, we establish an asymptotic formula for $V\left(p,q\right)$. Finally, by use of continued fraction theory, we give a formula for $V\left(p,q\right)$ in terms of continued fraction of $\frac{p}{q}$.

Classification (MSC2000): 11B99, 11F67, 11E45; 11M26, 11B68

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