# What would be the consequences of $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log k}$?

The question is in the title: what would be the number theoretic consequences if we managed to establish the conjectured asymptotic equality $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log k}$?

• What exactly do you mean by the expression $\lim\inf_{n\to\infty}p_{n+k}-p_{n} \sim k\log k$? -- For $k=1$ we get $\lim\inf_{n\to\infty}p_{n+1}-p_{n} \sim 0$, or how to read this? – Stefan Kohl Dec 16 '14 at 13:48
• @StefanKohl: would have guessed it's the obvious meaning that the function $k\mapsto \liminf_n p_{n+k}-p_n$ behaves asymptotically like $k\mapsto k \log k$, i.e. $\lim_{k\to\infty} \frac{\liminf_n p_{n+k}-p_n}{k \log k} = 1$. In particular there is no statement made for k=1. Why do you think that something non-obvious might be meant here? – Johannes Hahn Dec 16 '14 at 14:07
• @Johannes Hahn: you have perfectly understood what I meant. Thanks for the precise rephrasing though. – Sylvain JULIEN Dec 16 '14 at 14:12

The situation is similar to that in your previous question what would be the consequences on the distribution of primes of $\Lambda=\infty$? : by itself, not very much, because one only needs an arbitrarily sparse sequence of narrow prime clusters to establish the conjecture.
More substantial is the Hardy-Littlewood prime tuples conjecture, which implies the stated conjecture about $p_{n+k}-p_n$ (contingent on a further conjecture as to the narrowest admissible $k$-tuple, namely that the diameter of such a tuple is asymptotic to $k \log k$); this conjecture not only asserts the mere existence of prime tuples, but also predicts an asymptotic, and is broadly applicable to many further problems than the estimation of $\lim \inf p_{n+k}-p_n$, particularly if generalised to other patterns than tuples.