Davenport's constant for groups with large exponent

Abstract

Let $G$ be a finite abelian group. We show that its Davenport constant $D(G)$ satisfies $D(G)\le \exp (G)+\frac{|G|}{\exp (G)}-1$, provided that $\exp (G)\ge \sqrt{|G|}$, and $D(G)\le 2\sqrt{|G|}-1$, if $\exp (G)<\sqrt{|G|}$. This proves a conjecture by Balasubramanian and the first named author.