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

TL;DR

The paper investigates the density of prime triples among coprime triples on the eigensurface ${\mathcal{S}}$ of the symmetric group ${\bf S}_3$, defined by ${z_0^{2} - z_1^{2} + z_2^{2} - z_0z_2=0}$. It first establishes the baseline density ${\mathbb{P}}_+(N) = \frac{3\zeta(3)}{\log N}(1+o(1))$ for ${\mathbb{Z}_+^3}$ using Möbius inversion and prime counting results, then parameterizes integer points on ${\mathcal{S}}$ through coprime pairs $(m,n)$ to obtain ${\#}X_+^{\mathcal{S}}(N) = \frac{2N}{\log N} + o(\frac{N}{\log N})$. On the coprime surface side, it derives upper and lower bounds for ${\#}Y_+^{\mathcal{S}}(N)$ by analyzing coprime pairs in triangular regions with a modulo $3$ constraint, involving detailed estimates of sums of Euler's function and its ratio, leading to a quantified comparison of prime densities. Collectively, these results yield explicit liminf and limsup bounds for the ratio ${\mathbb{P}}_+^{\mathcal{S}}(N)/{\mathbb{P}}_+(N)$, demonstrating that the eigensurface ${\mathcal{S}}$ exhibits a higher asymptotic density of prime triples than the ambient space. The findings contribute to understanding how algebraic surfaces tied to group representations can influence arithmetic prime patterns and motivate further exploration of density phenomena on other eigensurfaces.

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 $3ζ(3)/\log N$, where $ζ$ 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 $3ζ(3)/\log N$, 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.

Figures (2)

• Figure 1: Diagram for the triangular regions (with $k_2<0$)
• Figure 2: Diagram for the elliptic region and its approximation

Theorems & Definitions (12)

• Theorem 1.1
• Theorem 2.1
• Theorem 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• Theorem 4.1
• Lemma 4.2
• Lemma 4.3