Refined upper bounds for the linear Diophantine problem of Frobenius

Abstract

We study the Frobenius problem: given relatively prime positive integers a 1,…, a d , find the largest value of t (the Frobenius number g( a 1,…, a d )) such that ∑ k=1 d m k a k = t has no solution in nonnegative integers m 1,…, m d . We introduce a method to compute upper bounds for g( a 1, a 2, a 3), 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.