Directed cycles with zero weight in Zpk

For a finite abelian group A, define f(A) to be the minimum integer such that for every complete digraph Γ on f vertices and every map w:E(Γ)→A, there exists a directed cycle C in Γ such that ∑e∈E(C)w(e)=0. The study of f(A) was initiated by Alon and Krivelevich (2021). In this article, we prove that f(Zpk)=O(pk(log⁡k)2), where p is prime, with an improved bound of O(klog⁡k) when p=2. These bounds are tight up to a factor which is polylogarithmic in k.