EMIS ELibM Electronic Journals 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)= k=1 p-1 kq pcotπk p·

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 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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