Displaying similar documents to “Riemann sums over polytopes”

Dedekind sums involving Jacobi modular forms and special values of Barnes zeta functions

Abdelmejid Bayad, Yilmaz Simsek (2011)

Annales de l’institut Fourier

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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...

The number of vertices of a Fano polytope

Cinzia Casagrande (2006)

Annales de l’institut Fourier

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Let X be a Gorenstein, ℚ -factorial, toric Fano variety. We prove two conjectures on the maximal Picard number of X in terms of its dimension and its pseudo-index, and characterize the boundary cases. Equivalently, we determine the maximal number of vertices of a simplicial reflexive polytope.

A generalization of the reciprocity law of multiple Dedekind sums

Masahiro Asano (2007)

Annales de l’institut Fourier

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Various multiple Dedekind sums were introduced by B.C.Berndt, L.Carlitz, S.Egami, D.Zagier and A.Bayad. In this paper, noticing the Jacobi form in Bayad [4], the cotangent function in Zagier [23], Egami’s result on cotangent functions [14] and their reciprocity laws, we study a special case of the Jacobi forms in Bayad [4] and deduce a generalization of Egami’s result on cotangent functions and a generalization of Zagier’s result. Further, we consider their reciprocity laws. ...