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?


$$
\lim\_{n \to \infty}\frac{1}{n}\#\Big\{m \le n : \frac{p\_{m+1} - p\_m}{\ln p\_m} < 1\Big\}
$$



Density of prime pairs whose gap is less than the average gap
It is conjectured (see e.g. Goldston-Pintz-Yildirim <http://arxiv.org/abs/1103.5886>) 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?](https://mathoverflow.net/questions/132731/does-zhangs-theorem-generalize-to-3-or-more-primes-in-an-interval-of-fixed-le) )


