Note on the Davenport's 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 groups of rank three. Our main result is that: $$D(C_{n_1}\oplus C_{n_2}\oplus C_{n_3})\le (n_1-1)+(n_2-1)+(n_3-1)+1+ (a_3-3)(n_1-1),$$ where $1<n_1|n_2|n_3\in\mathbb{N}$ and $a_3\le 20369$ is a constant. Therefore $D(C_{n_1}\oplus C_{n_2}\oplus C_{n_3})$ 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.