New upper bounds for the Davenport and for the Erdős–Ginzburg–Ziv constants

Abstract

Let G be a finite abelian group (written additively) of rank r with invariants n1, n2, . . . , nr, where nr is the exponent of G. In this paper, we prove an upper bound for the Davenport constant D(G) of G as follows; D(G) ≤ nr + nr-1 + (c(3) − 1)nr-2 + (c(4) − 1) nr-3 + · · · + (c(r) − 1)n1 + 1, where c(i) is the Alon–Dubiner constant, which depends only on the rank of the group \({{\mathbb Z}_{n_r}^i}\). Also, we shall give an application of Davenport’s constant to smooth numbers related to the Quadratic sieve.