Publications de l’Institut Mathématique, Nouvelle Série, Vol. 102[116], pp. 155

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 (q, p) = \sum_{k = 1}^{p - 1} \{\frac{k q}{p}\} cot (\frac{π k}{p}) \cdot

First, we prove explicit formula for the symmetric sum $V (p, q) + V (q, p)$ 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 (p, q)

. Finally, by use of continued fraction theory, we give a formula for

V (p, q)

in terms of continued fraction of

\frac{p}{q}

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

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Electronic fulltext finalized on: 3 Nov 2017. This page was last modified: 29 Jan 2018.

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