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

Edit December 16th 2014: I'd be interested in references about this topic as well.

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    $\begingroup$ 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? $\endgroup$ – Stefan Kohl Dec 16 '14 at 13:48
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    $\begingroup$ @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? $\endgroup$ – Johannes Hahn Dec 16 '14 at 14:07
  • $\begingroup$ @Johannes Hahn: you have perfectly understood what I meant. Thanks for the precise rephrasing though. $\endgroup$ – 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.

More generally, I don't recommend putting too much time into taking random conjectures in number theory and trying to figure out what they imply; there are a few important conjectures which have a lot of applications (e.g. RH, GUE Hypothesis, Elliott-Halberstam) but most conjectures in number theory are endpoint objectives, rather than stepping stones to other conjectures; they are useful for benchmarking the state of technical progress in the subject, but not directly applicable to other problems. This is particularly the case if the conjecture is referring directly to endpoint objects of study in number theory, such as the primes, as opposed to more technical but broadly useful objects of study (e.g. mean values of exponential sums or Dirichlet series).

Ultimately, analytic number theory is primarily a "second culture" subject, in which the techniques and general principles tend to be more important than the specific results proven (excepting some fundamentally important results, such as the prime number theorem, of course). See this essay by Gowers on the distinction between the two cultures of mathematics. Much of the value that arises when a long-standing conjecture in number theory is proved arises not from the logical consequences of the conjecture itself, but rather from the new techniques and methods (or perhaps the expansion of applicability of an older technique or method) that had to be introduced to resolve the conjecture, as these methods would then be likely to be applicable to many further problems, including those that had no direct relation to the original conjecture that was proved.

  • $\begingroup$ Thank you for your answer. I've been told that the considered asymptotics might imply the non existence of Landau-Siegel zeroes, can you please tell me more about this? $\endgroup$ – Sylvain JULIEN Dec 17 '14 at 10:27

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