The Density of Primes in the Eigensurface of ${\bf S}_3$
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| Format: | Preprint |
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2025
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| _version_ | 1866918263776083968 |
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| author | Geng, Liang He, Wei Yang, Rongwei |
| author_facet | Geng, Liang He, Wei Yang, Rongwei |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_21488 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Density of Primes in the Eigensurface of ${\bf S}_3$ Geng, Liang He, Wei Yang, Rongwei Number Theory 11A41, 11D45, 47A13, 20C30 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. |
| title | The Density of Primes in the Eigensurface of ${\bf S}_3$ |
| topic | Number Theory 11A41, 11D45, 47A13, 20C30 |
| url | https://arxiv.org/abs/2512.21488 |