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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Sign up to join this communityDenote 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 $n$?
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.
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: