Dedekind sums involving Jacobi modular forms and special values of Barnes zeta functions
Annales de l'Institut Fourier, Volume 61 (2011) no. 5, pp. 1977-1993.

In this paper we study three new shifted sums of Apostol-Dedekind-Rademacher type. The first sums are written in terms of Jacobi modular forms, and the second sums in terms of cotangent and the third sums are expressed in terms of special values of the Barnes multiple zeta functions. These sums generalize the classical Dedekind-Rademacher sums. The main aim of this paper is to state and prove the Dedekind reciprocity laws satisfied by these new sums. As an application of our Dedekind reciprocity law we show how to derive all the well-known results on Dedekind reciprocity law studied by Hall-Wilson-Zagier, Beck-Berndt-Dieter, Katayama and Nagasaka-Ota-Sekine.

Dans ce papier nous étudions trois types nouveaux de sommes shiftées de Dedekind-Apostol-Rademacher. Les premières sommes sont écrites à l’aide des formes modulaires de Jacobi et les deuxièmes sont écrites en termes de valeurs de fonctions cotangentes et les troisièmes sont exprimées à l’aide de valeurs spéciales de fonctions zêta multiples de Barnes. Le résultat principal de cet article est de montrer une loi de réciprocité de Dedekind satisfaites par ces nouvelles sommes. Nos résultats recouvrent ceux de Hall-Wilson-Zagier sur les sommes classiques de Dedekind-Rademacher, ceux de Beck-Berndt-Dieter sur les sommes cotangentes et d’autres résultats obtenus par Ota et Nagasaka sur les sommes de Dedekind, attachées aux dérivées premières de fonctions zêta de Barnes.

DOI: 10.5802/aif.2663
Classification: 11F20,  11F50,  11F66,  11F67,  11M41
Keywords: Elliptic Dedekind sums, modular forms, theta functions, ellpitic functions, Bernoulli functions, Jacobi modular forms
Bayad, Abdelmejid 1; Simsek, Yilmaz 2

1 Université d’Evry Val d’Essonne Département de mathématiques Bd. F. Mitterrand, 91025 Evry Cedex (France)
2 University of Akdeniz Faculty of Arts and Science Department of Mathematics 07058 Antalya (Turkey)
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Bayad, Abdelmejid; Simsek, Yilmaz. Dedekind sums involving Jacobi modular forms and special values of Barnes zeta functions. Annales de l'Institut Fourier, Volume 61 (2011) no. 5, pp. 1977-1993. doi : 10.5802/aif.2663. http://archive.numdam.org/articles/10.5802/aif.2663/

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