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

The Prime Number Theorem asserts that the density of primes less than or equal to $N$ is asymptotically equal to $1\mathrm{/}\log N$. The density of prime triples in coprime triples in ${\mathbb{Z}}_{+}^3$ is determined to be $3\zeta (3)\mathrm{/}\log N$, where $\zeta$ is the Riemann zeta function. In this paper, we prove that the density of prime triples in coprime triples in the surface $S=\{z_0^2-z_1^2+z_2^2-z_0{z}_2=0\}$ is greater than $3\zeta (3)\mathrm{/}\log N$, meaning that $S$ meets primes more frequently. This surface is the eigensurface of the symmetric group ${\mathbf{S}}_3$ with respect to an irreducible representation.