Gaps Between Consecutive Primes and the Exponential Distribution

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.