NOTE ON THE DAVENPORT CONSTANT FOR FINITE ABELIAN GROUPS WITH RANK THREE

Let G be a finite abelian group and D(G) denote the Davenport constant of G. We derive new upper bound for the Davenport constant for all finite abelian groups of rank three. Our main result is that D(C-n1 circle plus C-n2 circle plus C-n3) <= (n(1) - 1) + (n(2) - 1) + (n(3) - 1) + 1 + (a(3) - 3)(n(1) - 1), where 1 < n(1)vertical bar n(2)vertical bar n(3) is an element of N and a(3) <= 20369 is a constant. Therefore, D(C-n1 circle plus C-n2 circle plus C-n3) grows linearly with the variables n(1), n(2), n(3). The new result is the given upper bound for a(3). Finally, we give an application of the Davenport constant to smooth numbers.