Explicit bounds for the graphicality of the prime gap sequence

TL;DR

This paper provides explicit unconditional bounds on when the prime gap sequence is graphic, improving understanding of the graphical properties of prime gaps with precise thresholds based on advanced number theory techniques.

Contribution

It establishes the first explicit unconditional thresholds for the graphicality of prime gap sequences, refining criteria and estimates using zero-free regions and zero-density estimates of the Riemann zeta function.

Findings

Prime gap sequence is graphic for all n ≥ exp exp(30.5).

Every realization of the prime gap sequence satisfies DPG-graphic for n ≥ exp exp(34.5).

Utilizes refined graphic criteria and explicit bounds from analytic number theory.

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.