Let \(A \subseteq \mathbb{Z}_n\) be a subset. A sequence \(S = (x_1, \ldots, x_k)\) is said to be an \(A\)-weighted zero-sum sequence if there exist \(a_1, \ldots, a_k \in A\) such that \(a_1x_1 + \cdots + a_kx_k = 0\). The \(A\)-weighted Davenport constant \(D_A\) is defined to be the smallest natural number \(k\) such that every sequence of \(k\) elements in \(\mathbb{Z}_n\) has an \(A\)-weighted zero-sum subsequence. The constant \(C_A\) is defined to be the smallest natural number \(k\) such that every sequence of \(k\) elements in \(\mathbb{Z}_n\) has an \(A\)-weighted zero-sum subsequence having consecutive terms. Let \(\Omega(n)\) be the total number of prime factors of \(n\), counting multiplicities. When \(n\) is odd, let \(S(n)\) be the set of all units in \(\mathbb{Z}_n\) whose Jacobi symbol with respect to \(n\) is 1. The authors study the constants \(C_{S(n)}\) and \(D_{S(n)}\).
Theorem 23. Let \(n\) be squarefree. If \(n\) is prime we have \(D_{S(n)} = 3\). If \(n\) is not a prime and every prime divisor of \(n\) is at least seven, we have \(D_{S(n)} = \Omega(n) + 1\).
Theorem 24. Let \(n\) be squarefree. If \(n\) is a prime, then \(C_{S(n)} = 3\). If \(n\) is not a prime and every prime divisor of \(n\) is at least seven, then \(C_{S(n)} = 2^{\Omega(n)}\).
For a prime divisor \(p\) of \(n\), the authors also compute these constants for a related weight-set \(L(n;p)\), which consists of all units \(x\) in \(\mathbb{Z}_n\) such that the Jacobi symbol of \(x\) with respect to \(n\) equals the Legendre symbol of \(x\) with respect to \(p\). They show that even though these weight-sets \(A\) may have half the size of \(U(n)\) (which is the set of units of \(\mathbb{Z}_n\)), the corresponding \(A\)-weighted constants are the same as those for the weight-set \(U(n)\).