On generalized Erdős–Ginzburg–Ziv constants of [formula omitted].

Abstract Let G be an additive finite abelian group with exponent exp (G) = n. For any positive integer k , the k th Erdős–Ginzburg–Ziv constant s k n (G) is defined as the smallest positive integer t such that every sequence S in G of length at least t has a zero-sum subsequence of length k n. It is easy to see that s k n (C n r) ≥ (k + r) n − r where n , r ∈ N. Kubertin conjectured that the equality holds for any k ≥ r. In this paper, we prove the following results: • [(1)] For every positive integer k ≥ 6 , we have s k n (C n 3) = (k + 3) n + O (n ln n). • [(2)] For every positive integer k ≥ 18 , we have s k n (C n 4) = (k + 4) n + O (n ln n). • [(3)] For n ∈ N , assume that the largest prime power divisor of n is p a for some a ∈ N. Forany fixed r ≥ 5 , if p t ≥ r for some t ∈ N , then for any k ∈ N we have s k p t n (C n r) ≤ (k p t + r) n + c r n ln n , where c r is a constant that depends on r. Our results verify the conjecture of Kubertin asymptotically in the above cases. [ABSTRACT FROM AUTHOR]