Gaps Between Consecutive Primes and the Exponential Distribution

Based on the primes less than 4×1018, Oliveira e Silva et al. (Math. Comp., 83(288):2033–2060, 2014) conjectured an asymptotic formula for the sum of the kth 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 kth moment of the first n gaps is asymptotic to the kth 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 ∼ (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 Cramér-Shanks conjecture holds. Numerical counts of gaps and the maximal gap Gn among the first n gaps test these results. While most values of Gn are better approximated by ( log n)2 than by other models, seven exceptional values of n with Gn>2e−γ( log n)2 suggest that lim supn→∞Gn/2e−γ( log n)2 may exceed 1.