The distribution of small gaps between successive primes

Abstract:

For $r \geqq 1$ and large N, a well-known conjecture of Hardy and Littlewood implies that the number of primes $p \leqq N$ such that $p + 2r$ is the least prime greater than p is asymptotic to \[ \int _2^N {\left ( {\sum \limits _{k = 1}^r {\frac {{{A_{r,k}}}}{{{{(\log x)}^{k + 1}}}}} } \right )} \;dx,\] where the ${A_{r,k}}$ are certain constants. We describe a method for computing these constants. Related constants are given to 10D for $r = 1(1)40$, and some empirical evidence supporting the conjecture is mentioned.