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By the prime number theorem we know that the "average gap" between the first $n$ primes is $\ln p_n$. I would like to know the density of consecutive prime pairs whose gap is less than the average gap between the first $n$ primes? In other words, does the following limit exist and what is its value?

$$ \lim_{n \to \infty}\frac{1}{n}\#\Big\{m \le n : \frac{p_{m+1} - p_m}{\ln p_m} < 1\Big\} $$

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I'm not a number theorist, but I guess that for this sort of question, the primes in the interval $[0,x]$ can conjecturally be approximated by a rate $1/\log x$ Poisson point process, suggesting that the limit should be $1-1/e$. I don't know whether there is any hope of proving this. – Johan Wästlund Jun 27 '13 at 12:16
up vote 17 down vote accepted

It is conjectured (see e.g. Goldston-Pintz-Yildirim that

$$ \lim_{n \to \infty}\frac{1}{n}\#\Big\{m \le n : \alpha<\frac{p_{m+1} - p_m}{\ln p_m} < \beta \Big\} = \int_\alpha^\beta e^{-t} dt, $$ and thus in particular your value should be $1-e^{-1}$. (GPY proved that this is positive for any $0=\alpha<\beta$ when $\lim_{n \to \infty}$ is replaced by $\liminf_{n \to \infty}$). Gallagher proved this result under the Hardy-Littlewood $k$-prime conjecture, but since GPY also proved conditional results under the Elliot-Halberstam conjecture which are still weaker than this conjecture, it seems unlikely that the method used to prove Zhang's recent breakthrough result is sufficient to prove such a result (for reasons why Zhang's method can not give anything better than what can be proved under Elliot Halberstam, see this question Does Zhang's theorem generalize to $3$ or more primes in an interval of fixed length? )

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