Explicit bounds for the graphicality of the prime gap sequence

Abstract

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 Erd\H{o}s \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.