I have moved this question to math.stackexchange.com. People who are interested in this question can discuss at :https://math.stackexchange.com/questions/1272431/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 statements can be seen as **conjectured properties** of distribution of composite numbers.

**Version 1**:

Let $A_1, \cdots, A_n$ be finite sets of natural numbers and $x>1$ is a natural number.

If the following conditions are satisfied:

$(1)$ each $A_i$ $(1\leq i\leq n)$ is an arithmetic sequence with common difference $d_i$ being prime number and each term being a multiple of $d_i$,

$(2)$ the common differences $d_i$ $(1\leq i\leq n)$ are coprime to each other and also they are coprime to $x$,

$(3)$ the density of multiples of $x$ in each $A_i$ $(1\leq i\leq n)$ is less than or equal to $1/d$ $(d\geq 1)$,

$(4)$ for any subset $J\subset [1,\cdots,n]$, the density of multiples of $x$ in $\bigcap_{j\in J}A_j$ is also less than or equal to $1/d$,

then the density of multiples of $x$ in $A_1\cup\cdots\cup A_n$ is also less than or equal to $1/d$.*

**Version 2**:

Let $A_1, \cdots, A_n$ be finite sets of natural numbers and $x$ is a prime number.

If the following conditions are satisfied:

$(1)$ each $A_i$ $(1\leq i\leq n)$ is an arithmetic sequence with common difference $d_i$ being prime number and each term being a multiple of $d_i$,

$(2)$ the common differences $d_i$ $(1\leq i\leq n)$ and $x$ are distinct from each other,

$(3)$ the density of multiples of $x$ in each $A_i$ $(1\leq i\leq n)$ is less than or equal to $1/d$ $(d\geq 1)$,

$(4)$ for any subset $J\subset [1,\cdots,n]$, the density of multiples of $x$ in $\bigcap_{j\in J}A_j$ is also less than or equal to $1/d$,

then the density of multiples of $x$ in $A_1\cup\cdots\cup A_n$ is also less than or equal to $1/d$.*

Is this question easy for experts? I think the result should be right, but I can not prove it. I hope someone can give some hints to prove this conjecture or some counterexamples to improve this conjecture.

I am not very sure that the $(4)$ is implied by the conditions $(1),(2), (3)$. But indeed this condition is satisfied by the Eratosthenes's Sieve in which $d_i$s and $x$ are distinct primes. In order for insurance purposes, I add the $(4)$ condition.

## -------------------------------------------------------------------------------

An example shows that the $(1)$ condition (especially, the first term of $A_i$ should be a multiple of $d_i$) is necessary:

## $A=\{1,4,7,10,13,16\}$ with common difference $3$, $B=\{3,8,13,18,23,28\}$ with common difference $5$, $d=3$, $x=4$. $A\cup B=\{1,3, 4,7,8,10,13,16,18,23, 28\}$. Both densities of multiples of $x=4$ in $A$ and $B$ are $1/3$, the density of multiple of $x=4$ in $A\cup B$ is $4/11$ which is large than $1/3$.

The example of The Masked Avenger shows that it is better to suppose $d_i$ and $x$ to be prime numbers. His example: $A_i=3^i\times \{1,2,3,4\}$, $d_i=3^i$, $x=4$.

## -------------------------------------------------------------------------------------

Since The Masked Avenger gives his ingenious answer and disproves the two statements above, I give the last version which is my original motivation.

- **Last version **:

Let $M,N$ be natural numbers, $d_i$ $(1\leq i\leq n)$ and $x$ are distinct prime numbers less than or equal to $\sqrt{M+N}$, $A_i=\{multiples\ of\ d_i\}\cap [N, N+M]$ $(1\leq i\leq n)$.

If the density of multiples of $x$ in each $A_i$ $(1\leq i\leq n)$ is less than or equal to $1/d$ $(d\geq 1)$,

then the density of multiples of $x$ in $A_1\cup\cdots\cup A_n$ is also less than or equal to $1/d$.*

# ---------------------------------------------------------------------------------------

I am particularly grateful to The Masked Avenger and Włodzimierz Holsztyński for their ingenious counterexamples.

Now I give a weaker version about distribution of composite numbers. May be theire examples can also disprove this conjecture. But I need time to study their examples.

*** **weak version **: Let $K,L,N$ be natural numbers and $K+L\leq N$. $d_i$ $(1\leq i\leq n)$ and $x$ are distinct prime numbers less than or equal to $\sqrt{N}$. $A_i=\{multiples\ of\ d_i\}\cap [K, K+L]$ $(1\leq i\leq n)$.

If $x>2， L> \sqrt{N}$, then the density of multiples of $x$ in $A_1\cup\cdots\cup A_n$ is less than or equal to $\frac{1}{x-2}$.***

*** ** 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 $\frac{1}{x-2}$.***

19more comments