On some determinants involving the tangent function

Zhi-Wei Sun

Published 2019-01-15Version 1

Let $p$ be an odd prime and let $a,b\in\mathbb Z$ with $p\nmid ab$. In this paper we mainly evaluate $$T_p^{(\delta)}(a,b):=\det\left[\tan\pi\frac{aj^2+bk^2}p\right]_{\delta\le j,k\le (p-1)/2}\ \ (\delta=0,1).$$ For example, in the case $p\equiv3\pmod4$ we show that $T_p^{(1)}(a,b)=0$ and $$T_p^{(0)}(a,b)=\begin{cases} 2^{(p-1)/2}p^{(p+1)/4}&\text{if}\ (\frac{ab}p)=1, \\p^{(p+1)/4}&\text{if}\ (\frac{ab}p)=-1.\end{cases}$$ When $(\frac{-ab}p)=-1$, we also evaluate the determinant $\det[\cot\pi\frac{aj^2+bk^2}p]_{1\le j,k\le(p-1)/2}.$ We also pose several conjectures one of which states that the class number of the quadratic field $\mathbb Q(\sqrt{p^*})$ with $p^*=(-1)^{(p-1)/2}p$ is equal to $$\left(\frac{-2}p\right)2^{-(p-3)/2}p^{-(p-5)/4}\det\left[\cot\pi\frac{jk}p\right]_{1\ls j,k\ls (p-1)/2}.$$