# On the zero-sum constant, the Davenport constant and their analogues

## TL;DR

This paper explores the properties of zero-sum constants in finite Abelian groups, generalizes known relations, investigates asymptotic behaviors, and extends some results to non-Abelian groups, with applications to smooth numbers.

## Contribution

It proves a general formula for ${
m E}_m(G)$ in Abelian groups, extends the relation to non-Abelian groups, and examines asymptotic behaviors of related sequences.

## Key findings

- ${
m E}_m(G)=D(G)-1+m|G|$ for Abelian groups
- Generalization of Kemnitz's conjecture
- Stronger version of a result by Delorme, Ordaz, Quiroz

## Abstract

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.