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

Let S=(a1)⋯(ak) 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.