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We study the number of lattice points in integer dilates of the rational polytope P ¼ ðx 1 ; . . . ; x n Þ 2 R n 50 :where a 1 ; . . . ; a n are positive integers. This polytope is closely related to the linear Diophantine problem of Frobenius: given relatively prime positive integers a 1 ; . . . ; a n ; find the largest value of t (the Frobenius number) such that m 1 a 1 þ Á Á Á þ m n a n ¼ t has no solution in positive integers m 1 ; . . . ; m n : This is equivalent to the problem of finding the largest dilate tP such that the facet f P n k¼1 x k a k ¼ tg contains no lattice point. We present two methods for computing the Ehrhart quasipolynomials Lð % P P; tÞ :¼ #ðtP \ Z n Þ and LðP8; tÞ :¼ #ðtP8 \ Z n Þ: Within the computations a Dedekind-like finite Fourier sum appears. We obtain a reciprocity law for these sums, generalizing a theorem of Gessel. As a corollary of our formulas, we rederive the reciprocity law for Zagier's higher-dimensional Dedekind sums. Finally, we find bounds for the Fourier-Dedekind sums and use them to give new bounds for the Frobenius number. # 2002 Elsevier Science (USA)

We study the number of lattice points in integer dilates of the rational polytopewhere a1, . . . , an are positive integers. This polytope is closely related to the linear Diophantine problem of Frobenius: given relatively prime positive integers a1, . . . , an, find the largest value of t (the Frobenius number ) such that m1a1 + · · · + mnan = t has no solution in positive integers m1, . . . , mn. This is equivalent to the problem of finding the largest dilate tP such that the facet n k=1 x k a k = t contains no lattice point. We present two methods for computing the Ehrhart quasipolynomials L(P, t) := #(tP ∩ Z n ) and L(PWithin the computations a Dedekind-like finite Fourier sum appears. We obtain a reciprocity law for these sums, generalizing a theorem of 1

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