Exact Rank of the Largest Good Lattice Point Matrix

Abstract

We prove that for every integer $n>2$ the largest good lattice point matrix $U_n=[u_h:h\in H_n]$, whose columns are generated by all integers $h<n$ coprime to $n$, has rank exactly $\phi (n)/2+1$. The upper bound follows from the pairing identity $u_h+u_{n-h}=(n,\dots ,n,2n{)}^{\top }$. For the lower bound the columns are centered and the resulting orbit under the multiplicative group $(\mathbb{Z}/n\mathbb{Z}{)}^{\times }$ is analysed by means of odd Dirichlet characters; the essential arithmetic input is the non-vanishing of ${\sum }_{a}a\psi (a)$ for odd primitive characters $\psi$, which is equivalent to Dirichlet's theorem $L(1,\psi )\ne 0$.