Arithmetic Upper and Lower Bounds for the Denumerant Function

Abstract

We use an elementary arithmetic argument, going back to Blom and Fr{ö}berg, to obtain upper and lower bounds for the denumerant function of Sylvester for arbitrary positive integers $a_1,\dots ,a_k$. The bounds hold along the multiples of $\gcd (a_1,\dots ,a_k)$, share the leading term of the denumerant, and yield its asymptotic behaviour with an explicit error term; for large $n$ and under a mild condition, a polynomial refinement of the lower bound improves on the lower bound of Agnarsson.

arXiv comment16 pages. Apart from minor corrections, the results are those of the previous versions; only the presentation has changed: lighter notation, statements for arbitrary gcd(a_1,...,a_k) (which reduces to the coprime case), self-contained and fully elementary proofs (no Popoviciu formula, no differentiation), a few complementary remarks, and a comparison with later work of Gajdzica