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 Erdős 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 $n\ge \exp \exp (30.32)$, ${\mathrm{PD}}_n$ is graphic. (2) For all $n\ge \exp \exp (34.33)$, 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.

15 pages, incorporates the referees' comments; to appear in Analysis Mathematica. Comments welcome!