Gaps Between Consecutive Primes and the Exponential Distribution

TL;DR

This paper investigates the statistical distribution of prime gaps, showing that their moments align with an exponential distribution and exploring implications for the size of the largest gaps, connecting to longstanding conjectures.

Contribution

It establishes a link between the moments of prime gaps and exponential distribution, providing new insights into the behavior of the largest prime gaps and their relation to conjectures.

Findings

Prime gaps' moments match those of an exponential distribution with mean log n.

Largest prime gaps are asymptotic to (log n)^2 under certain conjectures.

Numerical data suggests the limsup of scaled gaps may exceed predicted bounds.

Abstract

Based on the primes less than $4 \times 10^{18}$, Oliveira e Silva et al. (2014) conjectured an asymptotic formula for the sum of the $k$th power of the gaps between consecutive primes less than a large number $x$. We show that the conjecture of Oliveira e Silva holds if and only if the $k$th moment of the first $n$ gaps is asymptotic to the $k$th moment of an exponential distribution with mean $\log n$, though the distribution of gaps is not exponential. Asymptotically exponential moments imply that the gaps asymptotically obey Taylor's law of fluctuation scaling: variance of the first $n$ gaps $\sim$ (mean of the first $n$ gaps)$^2$. If the distribution of the first $n$ gaps is asymptotically exponential with mean $\log n$, then the expectation of the largest of the first $n$ gaps is asymptotic to $(\log n)^2$. The largest of the first $n$ gaps is asymptotic to $(\log n)^2$ if and only if the Cram\'er-Shanks conjecture holds. Numerical counts of gaps and the maximal gap $G_n$ among the first $n$ gaps test these results. While most values of $G_n$ are better approximated by $(\log n)^2$ than by other models, seven values of $n$ with $G_{n} >2 e^{-\gamma}(\log n)^2$ suggest that $\limsup_{n \to\infty} G_n/[2e^{-\gamma}(\log n)^2]$ may exceed 1.