Denote by $A_n$ the number of prime numbers between $n$ and $n + \log n$.

Is it true that $A_n < const$?

UPD: Is it true that $A_n > \log \log n$ (or something another) for infinite number $n$?

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    $\begingroup$ No, thanks to the recent breakthroughs in bounded gaps between primes. $\endgroup$ – Lucia Jul 1 '15 at 21:06
  • $\begingroup$ @Lucia It's been over 2 years, can we still call it recent? :P $\endgroup$ – Wojowu Jul 1 '15 at 21:09
  • $\begingroup$ I don't understand: Are you mean results about twin primes? $\endgroup$ – Alexey Milovanov Jul 1 '15 at 21:49
  • $\begingroup$ If the prime k-tuples conjecture holds, it may be that for infinitely many n that A_n is larger than (log n) /(log log n). Gerhard "Still Too Early To Tell" Paseman, 2015.07.01 $\endgroup$ – Gerhard Paseman Jul 1 '15 at 22:06

Theorem 3.2 in Maynard shows that there are many values $x$ for which the interval $[x,x+\log x]$ contains $\gg \log \log x$ primes. This is a quantification of his earlier breakthrough work where he showed that there are intervals of length $\ll e^{(4+o(1))m}$ containing $m$ primes.

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    $\begingroup$ Excellent! For some reason I did not remember this refinement. $\endgroup$ – GH from MO Jul 1 '15 at 22:48

Based on the work of James Maynard and independently by Terry Tao, it is known that for any positive integer $k$, there exists a positive number $h(k)$ such that there exist infinitely many $k$-tuples of consecutive primes $p_1 < \cdots < p_k$ such that $p_k - p_1 < h(k)$. Applying this result to the current situation, for each positive integer $k$ there exist infinitely many $n$ such that between $n$ and $n + \log n$ there are more than $k$ primes (it is necessary of course to have $\log n > h(k)$, hence $A_n$ is not bounded.

See: http://annals.math.princeton.edu/2015/181-1/p07

and for the arxiv version:


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