Zero-sum constants related to the Jacobi symbol

Santanu Mondal, Krishnendu Paul, Shameek Paul

Published 2021-11-29, updated 2022-12-01Version 2

For a weight-set $A\subseteq \mathbb Z_n$, the $A$-weighted Davenport constant $D_A(G)$ is defined to be the smallest natural number $k$ such that any sequence of $k$ elements in $\mathbb Z_n$ has an $A$-weighted zero-sum subsequence and the constant $C_A(n)$ is defined to be the smallest natural number $k$ such that any sequence of $k$ elements in $\mathbb Z_n$ has an $A$-weighted zero-sum subsequence of consecutive terms. We compute these constants for the weight set $S(n)=\big\{\,x\in U(n):\big(\frac{x}{n}\big)=1\,\big\}$ where the symbol $\big(\frac{x}{n}\big)$ is the Jacobi symbol. We also compute these constants for the weight-set $L(n;p)=\big\{\,x\in U(n):\big(\frac{x}{n}\big)=\big(\frac{x}{p}\big)\,\big\}$ where $p$ is a prime divisor of $n$.