On two conjectures involving quadratic residues

Fedor Petrov, Zhi-Wei Sun

Published 2019-07-30Version 1

Let $p$ be an odd prime with $p\equiv 1\pmod 4$. In this paper we confirm two conjectures of Sun involving quadratic residues modulo $p$. For example, we show that for any integer $a\not\equiv0\pmod p$ we have \begin{align*}&(-1)^{|\{1\le k<p/4:\ (\frac kp)=-1\}|}\prod_{1\le j<k\le (p-1)/2}(e^{2\pi iaj^2/p}+e^{2\pi iak^2/p}) \\=&\begin{cases}1&\text{if}\ p\equiv1\pmod 8,\\\left(\frac ap\right)\varepsilon_p^{-(\frac ap)h(p)}&\text{if}\ p\equiv5\pmod8,\end{cases} \end{align*} where $(\frac{\cdot}p)$ is the Legendre symbol, and $\varepsilon_p$ and $h(p)$ are the fundamental unit and the class number of the real quadratic field $\mathbb Q(\sqrt p)$.