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

TL;DR

This paper investigates the distribution of primes on a specific algebraic surface related to the symmetric group S_3, demonstrating that primes occur more frequently there than in the general coprime triples setting.

Contribution

It proves that the density of prime triples on the eigensurface of S_3 exceeds the known density for coprime triples, revealing a higher prime occurrence rate on this surface.

Findings

Density of primes on the surface exceeds 3ζ(3)/log N

Primes are more frequent on the eigensurface of S_3

Surface meets primes more often than expected

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.