Timeline for Number of products of distinct primes lying in an interval

9 events

when toggle format	what		by	license	comment
Jun 20, 2014 at 15:56	vote	accept	Stanley Yao Xiao
Jun 20, 2014 at 15:45	answer	added	Emil Jeřábek		timeline score: 5
Jun 20, 2014 at 15:18	comment	added	Stanley Yao Xiao		I phrased thr question in terms of counting, which is hopefully more clear.
Jun 20, 2014 at 15:16	history	edited	Stanley Yao Xiao	CC BY-SA 3.0	added 47 characters in body; edited title
Jun 20, 2014 at 15:05	comment	added	Robert Israel		Can you clarify? Do you mean you are taking the product of a fixed number $n$ of primes randomly chosen from all the primes less than $Y$? Obviously if at least two of those primes are greater than $Y/k$ the product will be greater than $Y^2/k^2$...
Jun 20, 2014 at 14:38	comment	added	Emil Jeřábek		Sorry, a factor of $X$ is missing in the last but one expression (but it is swamped in the final $o(1)$, so it doesn’t matter).
Jun 20, 2014 at 14:28	comment	added	Emil Jeřábek		If I understand the question correctly, it is $2^{-\pi(Y)}$ times the number of square-free $Y$-smooth numbers below $X$. If $X=CY$ for fixed $C$, a $6/\pi^2$ fraction of numbers below $X$ are square-free, and all but an $O(1/\log X)$ fraction are $Y$-smooth, so the result is ~ $2^{-\pi(Y)}6/\pi^2=2^{-(1+o(1))Y/\log Y}$.
Jun 20, 2014 at 13:57	history	edited	Stanley Yao Xiao	CC BY-SA 3.0	added 158 characters in body
Jun 20, 2014 at 13:43	history	asked	Stanley Yao Xiao	CC BY-SA 3.0