On zero-sum subsequences in a finite abelian group of length not exceeding a given number

Let G be an additive finite abelian group and let k∈ [exp(G),D(G)-1] be a positive integer. Denote by s≤ k(G) the smallest positive integer l∈ N∪ \+∞\ such that each sequence of length l over G has a non-empty zero-sum subsequence of length at most k. Let kG∈ [exp(G),D(G)-1] be the smallest positive integer such that s≤ D(G)-d(G)≤ D(G)+d for D(G)-d≥ kG. We conjecture that kG=D(G)+1/2 for finite abelian groups G with r(G)≥ 2 and D(G)=D^*(G). In this paper, we mainly study this conjecture for finite abelian p-groups and get some results to support this conjecture. We also prove that kG≤ D(G)-2 for all finite abelian groups G with r(G)≥ 2 except C₂³ and C₂⁴. In addition, we also get some lower bounds for the invariant s≤ k(G).