Refined upper bounds for the linear Diophantine problem of Frobenius
Abstract
We study the Frobenius problem: given relatively prime positive integers a1,...,ad, find the largest value of t (the Frobenius number g(a1,...,ad)) such that m1 a1 + ... md ad = t has no solution in nonnegative integers m1,...,md. We introduce a method to compute upper bounds for g(a1,a2,a3), which seem to grow considerably slower than previously known bounds. Our computations are based on a formula for the restricted partition function, which involves Dedekind-Rademacher sums, and the reciprocity law for these sums.
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