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Exact Rank of the Largest Good Lattice Point Matrix
Abstract
We prove that for every integer n>2 the largest good lattice point matrix Un=[uh:h∈ Hn], whose columns are generated by all integers h<n coprime to n, has rank exactly φ(n)/2+1. The upper bound follows from the pairing identity uh+un−h=(n,…,n,2n)⊤. For the lower bound the columns are centered and the resulting orbit under the multiplicative group (ℤ/nℤ)× is analysed by means of odd Dirichlet characters; the essential arithmetic input is the non-vanishing of ∑aa ψ(a) for odd primitive characters ψ, which is equivalent to Dirichlet's theorem L(1,ψ)≠ 0.
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BibTeX
@misc{xu2026exact,
title = {{Exact Rank of the Largest Good Lattice Point Matrix}},
author = {Long-Hao Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15042},
eprint = {2610.15042},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15042},
}
RIS
TY - UNPB
TI - Exact Rank of the Largest Good Lattice Point Matrix
AU - Long-Hao Xu
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.15042
UR - https://arxiv.org/abs/2610.15042
AB - We prove that for every integer n>2 the largest good lattice point matrix U_n=[u_h:h∈ H_n], whose columns are generated by all integers h<n coprime to n, has rank exactly φ(n)/2+1. The upper bound follows from the pairing identity u_h+u_n−h=(n,…,n,2n)^⊤. For the lower bound the columns are centered and the resulting orbit under the multiplicative group (ℤ/nℤ)^× is analysed by means of odd Dirichlet characters; the essential arithmetic input is the non-vanishing of ∑_aa ψ(a) for odd primitive characters ψ, which is equivalent to Dirichlet's theorem L(1,ψ)≠ 0.
ER -