DEDEKIND SUMS INVOLVING JACOBI MODULAR FORMS AND SPECIAL VALUES OF BARNES ZETA FUNCTIONS

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Bayad A., ŞİMŞEK Y.

ANNALES DE L INSTITUT FOURIER, vol.61, no.5, pp.1977-1993, 2011 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 61 Issue: 5
  • Publication Date: 2011
  • Doi Number: 10.5802/aif.2663
  • Journal Name: ANNALES DE L INSTITUT FOURIER
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.1977-1993
  • Keywords: Elliptic Dedekind sums, modular forms, theta functions, ellpitic functions, Bernoulli functions, Jacobi modular forms, COTANGENT SUMS
  • Akdeniz University Affiliated: Yes

Abstract

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.