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Counting lattice points in certain rational polytopes and generalized Dedekind sums. (English) Zbl 1384.05028

Summary: Let \(\mathcal{P} \subset \mathbb{R}^n\) be a rational convex polytope with vertices at the origin and on each positive coordinate axes. On the basis of the study on counting lattice points in \(t\mathcal{P}\) with positive integer \(t\), which is deeply connected with reciprocity laws for generalized Dedekind sums, we study the number of lattice points in the shifted polytope of \(t\mathcal{P}\) by a fixed rational point. Certain generalized multiple Dedekind sums appear naturally in the main result.

MSC:

05A15 Exact enumeration problems, generating functions
11F20 Dedekind eta function, Dedekind sums
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References:

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