Note on the index conjecture in zero-sum theory and its connection to a Dedekind-type sum

Let $S=(a_1)\cdots(a_k)$ be a minimal zero-sum sequence over a finite cyclic group $G$. The index conjecture states that if $k=4$ and $\gcd(|G|,6)=1$, then $S$ has index 1. In this note we study the index conjecture and connect it to a Dedekind-type sum. In particular we reprove a special case of the conjecture when $|G|$ is prime.