On the number of zero sum subsequences

Let Zn be the cyclic group of order n. For a sequence S of elements in Zn, we use fn(S) to denote the number of subsequences, the sum of whose elements is zero. In this paper, we give a characterization on the sequences S of elements in Zn for which fn(S) < 2|S| − n + k − 1, under the restriction 1 ⩽ k ⩽ [n4] + 1. As consequences of this result, we obtain some further characterizations on the sequences S of n elements in Zn for which fn(S) is not large.