  Let $D(G)$ be the Davenport constant of a finite Abelian group $G$. For a
positive integer $m$ (the case $m = 1$, is the classical one) let ${\mathsf
E}_m(G)$ (or $\eta_m(G)$, respectively) be the least positive integer $t$ such
that every sequence of length $t$ in $G$ contains $m$ disjoint zero-sum
sequences, each of length $|G|$ (or of length $\le exp(G)$ respectively). In
this paper, we prove that if $G$ is an~Abelian group, then ${\mathsf
E}_m(G)=D(G)-1+m|G|$, which generalizes Gao's relation. We investigate also the
non-Abelian case. Moreover, we examine the asymptotic behavior of the sequences
$({\mathsf E}_m(G))_{m\ge 1}$ and $(\eta_m(G))_{m\ge 1}.$ We prove
a~generalization of Kemnitz's conjecture. The paper also contains a result of
independent interest, which is a stronger version of a result by Ch. Delorme,
O. Ordaz, D. Quiroz. At the and we apply the Davenport constant to smooth
numbers and make a natural conjecture in the non-Abelian case.
