Problems and results on determinants involving Legendre symbols

Abstract

In this paper we investigate determinants whose entries are linear combinations of Legendre symbols. We deduce some new results in this direction; for example, we prove that for any prime $p\equiv3\pmod4$ we have $$\det\left[x+\left(\frac{j-k}p\right)+\left(\frac jp\right)-\left(\frac kp\right)\right]_{0\le j,k\le(p-1)/2}=4,$$ where $(\frac{\cdot}p)$ is the Legendre symbol. We also pose many conjectures for further research. For example, for any prime $p>3$ we conjecture that \begin{align*}&\ \det\left[\left(\frac{j+k}p\right)+\left(\frac{j-k}p\right)+\left(\frac{jk}p\right)\right]_{1\le j,k\le(p-1)/2} \\=&\ \begin{cases}(\frac 2p)p^{(p-5)/4}&\text{if}\ p\equiv1\pmod4, \\(-1)^{(h(-p)-1)/2}(1-(2-(\frac 2p))h(-p))p^{(p-3)/4}&\text{if}\ p\equiv3\pmod4, \end{cases}\end{align*} where $h(-p)$ is the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-p})$.