Timeline for Probability of finding a prime number between $x-\ln(x)$ and $x+\ln(x)$

8 events

when toggle format	what		by	license	comment
May 4 at 18:15	comment	added	Andreas Blass		@GHfromMO Thanks for the correction, which fortunately also makes the next sentence correct.
May 4 at 18:12	comment	added	GH from MO		@AndreasBlass The density is not $x/(\ln x)$ but $1/\ln x$.
May 4 at 17:29	comment	added	Andreas Blass		@YCor A very simple observation that helped me see what's going on: The prime number theorem says that, near $x$, the density of primes is approximately $x/(\ln x)$. So in your interval of length $2c\ln x$, we can expect approximately $2c$ primes. That "explains" the factor $2$ in the comments by Lucia and Wojowu. That the distribution should be Poisson (in Lucia's comment) seems to be another instance of the distribution of primes being random unless one can find a reason against randomness.
May 4 at 8:24	comment	added	Wojowu		@YCor The introduction of that paper answers that too. It is known that the probability is positive for all large enough $N$, and is conjectured to tend to $1-e^{-2c}$.
May 4 at 7:46	comment	added	YCor		Is this specific to this interval $[x-\ln x,x+\ln x]$ or is there something similar for any interval, say, $[x-c\ln x,x+c\ln x]$ for $c>0$?
May 4 at 5:28	history	edited	GH from MO	CC BY-SA 4.0	added 17 characters in body
May 4 at 4:10	history	edited	GH from MO	CC BY-SA 4.0	added 60 characters in body
May 4 at 4:04	history	answered	GH from MO	CC BY-SA 4.0