The tangent function and power residues modulo primes

TL;DR

This paper evaluates products of tangent functions over mth power residues modulo primes, linking these products to prime representations and providing explicit formulas in special cases.

Contribution

It determines explicit values of tangent products over mth power residues modulo primes under specific conditions, extending known results and connecting to prime representations.

Findings

Explicit formulas for tangent products over mth power residues.

Connection between tangent products and prime representations like p=x^2+64y^2.

Special case formula when p=x^2+64y^2, involving powers of -2 and parity of y.

Abstract

Let $p$ be an odd prime, and let $a$ be an integer not divisible by $p$. When $m$ is a positive integer with $p\equiv1\pmod{2m}$ and $2$ is an $m$th power residue modulo $p$, we determine the value of the product $\prod_{k\in R_m(p)}\tan\pi\frac{ak}p$, where $$R_m(p)=\{0<k<p:\ k\in\mathbb Z\ \text{is an}\ m\text{th power reside modulo}\ p\}.$$ In particular, if $p=x^2+64y^2$ with $x,y\in\mathbb Z$, then $$\prod_{k\in R_4(p)}\left(1+\tan\pi\frac {ak}p\right)=(-1)^{y}(-2)^{(p-1)/8}.$$