Signed zero sum problems for metacyclic group

Let A be a nonempty subset of the integers and G a finite group written multiplicatively. The constant \(\eta _A(G)\) ${\eta }_A(G)$ ( \(s_A(G)\) $s_A(G)$ ) is defined to be the smallest positive integer t such that any sequence of length t of elements of G contains a nonempty A-weighted product one subsequence (that is, the terms of the subsequence could be ordered so that their A-weighted product is the multiplicative identity of the group G) of length at most \(\exp (G)\) $exp(G)$ (of length \(\exp (G)\) $exp(G)$ ). In this note, we shall calculate the value of \(\eta _\pm (G),D_\pm (G),E_\pm (G) \text{ and } s_\pm (G)\) ${\eta }_{\pm }(G),D_{\pm }(G),E_{\pm }(G)\text{and}{s}_{\pm }(G)$ for some metacyclic groups. In 2007, Gao et al. conjectured that \(s(G)=\eta (G)+\exp (G)-1\) $s(G)=\eta (G)+exp(G)-1$ holds for any finite abelian group G (see Gao et al. in Integers 7:A21, 2007). We will prove that this conjecture is true for some metacyclic groups. Furthermore, we study the Harborth constant \(\mathfrak {g}_\pm (G)\) ${\mathfrak{g}}_{\pm }(G)$ , where G is a group among one specific class of metacyclic groups.