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 $n$?



Number of prime numbers in a range
Theorem 3.2 in [Maynard](http://arxiv.org/pdf/1405.2593.pdf) 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.


