This question is moved from mathoverflow, there are several excellent answers at mathoverflow which improve my question greatly. For more information, please see the original question posted on mathoverflow, the link is :https://mathoverflow.net/questions/204733/distribution-of-composite-numbers
Motivation: I want to give an abstract formulation of Eratosthenes's Sieve and try to conject a property of Eratosthenes's Sieve. If this property are right, then I can use it to find a low bound of numbers of primes between any segment $[1,N]$ in the set of natural numbers, so this problem is connected to distribution of primes.
The following statement can be seen as conjectured properties of distribution of composite numbers.
* ** version 5 **: Let $N$ be a natural numbers. $d_i$ $(1\leq i\leq n)$ and $x$ are distinct prime numbers less than or equal to $\sqrt{N}$. $A_i=\{multiples\ of\ d_i\}\cap [1,N]$ $(1\leq i\leq n)$. If $x>3$, then the density of multiples of $x$ in $A_1\cup\cdots\cup A_n$ is less than or equal to $\frac{1}{x-2}$.*
This is false. Take $d_1=5$ say, and take $d_2,\dots,d_n$ to be primes near $\sqrt N$. Then $A_1$ has size about $N/5$, while $A_2\cup\dots\cup A_n$ has size about $(n-1)\sqrt N$. Now take $x=d_1$: the multiples of $x$ in $A_1\cup\dots\cup A_n$ is virtually all of it - it has density $1 > \frac1{5-2}$.