On Davenport constant of finite abelian groups

$G$ be an additive finite abelian group. The Davenport constant $\mathsf D(G)$ is the smallest integer $t$ such that every sequence (multiset) $S$ over $G$ of length $|S|\ge t$ has a non-empty zero-sum subsequence. Recently, B. Girard proved that for every fixed integer $r > 1$ the Davenport constant $\mathsf D(C_n^r)$ is asymptotic to $rn$ when $n$ tends to infinity. In this paper, for every fixed positive integer $r$, we prove that $$\mathsf D(C_n^r)=rn+O(\frac{n}{\ln n}).$$ This is an explicit version of the above result of B. Girard. Furthermore, we can get better estimates of the error term for some $n$ of special types. Finally, we get an asymptotic result for some finite abelian groups of special types. Our proof combines a classical argument in the zero-sum theory together with some basic tools and results from analytic number theory.