The Density of Primes in the Eigensurface of ${\bf S}_3$

Abstract

The Prime Number Theorem asserts that the density of primes less than or equal to $N$ is asymptotically equal to $1/\log N$. The density of prime triples in coprime triples in $\mathbb{Z}^3_+$ is determined to be $3\zeta (3)/\log N$, where $\zeta$ is the Riemann zeta function. In this paper, we prove that the density of prime triples in coprime triples in the surface $S=\{z_0^{2} - z_1^{2} + z_2^{2} - z_0z_2=0\}$ is greater than $3\zeta (3)/\log N$, meaning that $S$ meets primes more frequently. This surface is the eigensurface of the symmetric group ${\bf S}_3$ with respect to an irreducible representation.