A Proof of a Conjecture of Zhi-Wei Sun on a Truncated Legendre-Symbol Determinant

Abstract

Let $p\ge 7$ be a prime with $p\equiv 3(\mathrm{mod}4)$ and let $\chi$ be the Legendre symbol modulo $p$. We prove that $\det[x+χ(j-k)]_{0\le j,k\le(p-7)/2}=\floor{(p-2)/3}^{2}x$ in $\mathbb{C}[x]$, which settles Conjecture~3.4 of Zhi-Wei Sun. The truncated matrix is a corner of Chapman's Legendre-symbol matrix $C$, and its determinant can be expressed through a handful of entries of $C^{-1}$ and of $C^{-1}\one$. Those entries are in turn read off from Vsemirnov's cyclotomic factorization of $C$ with the help of Schur's Pfaffian identity.