Disproof of a Conjectured Upper Bound for the Davenport Constant

Abstract

Let $G= C_{n_1}\oplus\cdots\oplus C_{n_r}$ be a finite abelian group with $1<n_1\mid\cdots\mid n_r$, and let $\rr(G)=r$ denote its rank. The Davenport constant $\DD(G)$ is the least integer $\ell$ such that every sequence of $\ell$ elements of $G$ contains a nonempty zero-sum subsequence, and $\DD^*(G)=1+\sum_{i=1}^r(n_i-1)$ is its classical lower bound. A long-standing conjecture \cite[Conjecture 3.7]{GG06} asserts that $\DD(G)\le\DD^*(G)+\rr(G)-1$. In this paper, we disprove this conjecture. More strongly, we prove that $\sup_{\rr(G)=r}\bigl(\DD(G)-\DD^*(G)\bigr)=\infty$ for every fixed $r\ge8$. Thus the classical lower bound does not approximate the Davenport constant within an additive error depending only on the rank. Our result also disproves the Narkiewicz--\'{S}liwa conjecture of 1982 \cite{NS82} on the Narkiewicz constant, arising in algebraic number theory from the quantitative study of algebraic integers with unique factorization. The same amplification of the Davenport excess yields counterexamples to Girard's conjecture \cite[Conjecture 1.2]{Girard08} on the cross numbers of long zero-sum-free sequences. As a further main result, we establish the uniform upper bound $$\DD(G)\le\frac{16}{5}\rr(G)\exp(G)$$ for every nontrivial finite abelian group $G$. The classical estimate of van Emde Boas and Kruyswijk \cite{vEBK69} gives $\DD(G)\le\exp(G)\left(1+\log\frac{|G|}{\exp(G)}\right) \le\exp(G)\bigl(1+(\rr(G)-1)\log\exp(G)\bigr)$. Our bound removes the logarithmic factor $\log\exp(G)$ from this classical estimate, and replaces it with the absolute constant $16/5$.