Explicit bounds for the graphicality of the prime gap sequence
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| Format: | Preprint |
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2025
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| author | Aggarwal, Keshav Frot, Robin Gou, Haozhe Wang, Hui |
| author_facet | Aggarwal, Keshav Frot, Robin Gou, Haozhe Wang, Hui |
| contents | We establish explicit unconditional results on the graphic properties of the prime gap sequence. Let $p_n$ denote the $n$-th prime number (with $p_0=1$) and $\mathrm{PD}_n = (p_\ell - p_{\ell-1})_{\ell=1}^n$ be the sequence of the first $n$ prime gaps. Building upon the recent work by Erdős \emph{et al}, which proved the graphic nature of $\mathrm{PD}_n$ for large $n$ unconditionally, and for all $n$ under RH, we provide the first explicit unconditional threshold such that: (1) For all $n \geq \exp\exp(30.5)$, $\mathrm{PD}_n$ is graphic. (2) For all $n \geq \exp\exp(34.5)$, every realization $G_n$ of $\mathrm{PD}_n$ satisfies that $(G_n, p_{n+1}-p_n)$ is DPG-graphic.
Our proofs utilize a more refined criterion for when a sequence is graphic, and better estimates for the first moment of large prime gaps proven through an explicit zero-free region and explicit zero-density estimate for the Riemann zeta function. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_24230 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Explicit bounds for the graphicality of the prime gap sequence Aggarwal, Keshav Frot, Robin Gou, Haozhe Wang, Hui Number Theory 11N05, 05C07, 11M26, 05C70 We establish explicit unconditional results on the graphic properties of the prime gap sequence. Let $p_n$ denote the $n$-th prime number (with $p_0=1$) and $\mathrm{PD}_n = (p_\ell - p_{\ell-1})_{\ell=1}^n$ be the sequence of the first $n$ prime gaps. Building upon the recent work by Erdős \emph{et al}, which proved the graphic nature of $\mathrm{PD}_n$ for large $n$ unconditionally, and for all $n$ under RH, we provide the first explicit unconditional threshold such that: (1) For all $n \geq \exp\exp(30.5)$, $\mathrm{PD}_n$ is graphic. (2) For all $n \geq \exp\exp(34.5)$, every realization $G_n$ of $\mathrm{PD}_n$ satisfies that $(G_n, p_{n+1}-p_n)$ is DPG-graphic. Our proofs utilize a more refined criterion for when a sequence is graphic, and better estimates for the first moment of large prime gaps proven through an explicit zero-free region and explicit zero-density estimate for the Riemann zeta function. |
| title | Explicit bounds for the graphicality of the prime gap sequence |
| topic | Number Theory 11N05, 05C07, 11M26, 05C70 |
| url | https://arxiv.org/abs/2512.24230 |