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> > **Question:** Are the properties as follows holds?
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**Version 1:** the answer by Bjørn Kjos-Hanssen


Let $P$ be a positive integers. We written: $P=$ $a\_1^{x\_1}a\_2^{x\_2}...a\_n^{x\_n}$ $=b\_1^{y\_1}b\_2^{y\_2}...b\_k^{y\_k}$ where $a\_i, b\_j$ are integers greater than $1$.


if $\prod\_{i=1}^n a\_i \le \prod\_{j=1}^k b\_j $ then


1) $$\prod\_{i=1}^n (1-\frac{1}{a\_i}) \le \prod\_{j=1}^k (1-\frac{1}{b\_j}) $$


2) $$\prod\_{i=1}^n (1-\frac{1}{a\_i})^{x\_i} \le \prod\_{j=1}^k (1-\frac{1}{b\_j})^{y\_j} $$


3) $$\frac{\varphi(P)}{P} \le \prod\_{i=1}^n (1-\frac{1}{a\_i})$$


Where $\varphi(P)$ is the [Euler's totient function](https://en.wikipedia.org/wiki/Euler%27s_totient_function) of $P$


**Comment:** The answer of Bjørn Kjos-Hanssen is: 



> 
> No, let $b\_1,b\_2,a\_1,a\_2$ be $2,6,3,4$. Then $$ (1-1/2)(1-1/6)=5/12
>  \not\ge 1/2=(1-1/3)(1-1/4) $$
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But I think the asnswer is not counter example. Because $2.6 \le 3.4$ we can write 


$$ (1-1/2)(1-1/6)=5/12
\le 1/2=(1-1/3)(1-1/4) $$.


To clearly I changes the question as follows:


**Version 2:**


Let $P$ be a positive integers. We written: $P=$ $a\_1^{x\_1}a\_2^{x\_2}...a\_n^{x\_n}$ $=b\_1^{y\_1}b\_2^{y\_2}...b\_k^{y\_k}$ where $a\_i, b\_j$ are integers greater than $1$.



> 
> if $\prod\_{i=1}^n a\_i < \prod\_{j=1}^k b\_j $ and $a\_i \le b\_j$ then
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1) $$\prod\_{i=1}^n (1-\frac{1}{a\_i}) \le \prod\_{j=1}^k (1-\frac{1}{b\_j}) $$


2) $$\prod\_{i=1}^n (1-\frac{1}{a\_i})^{x\_i} \le \prod\_{j=1}^k (1-\frac{1}{b\_j})^{y\_j} $$


3) $$\frac{\varphi(P)}{P} \le \prod\_{i=1}^n (1-\frac{1}{a\_i})$$


Where $\varphi(P)$ is the [Euler's totient function](https://en.wikipedia.org/wiki/Euler%27s_totient_function) of $P$



The inequality $\Pi (1-\frac{1}{a_i})^{x_i} \le \Pi (1-\frac{1}{b_j})^{y_j} $ hold?
$2^46^118^1=3^34^116^1(=2^63^3)$. 


$2\cdot6\cdot18=216>192=3\cdot4\cdot16$. 


$(1/2)(5/6)(17/18)=85/216<15/32=(2/3)(3/4)(15/16)$. 


Concerning the question about the phi-function, for $$n=2^{28}3^{16}=4\times6\times8\times9\times12\times16\times18\times24\times27\times32\times36\times48\times54$$ we have $${\phi(n)\over n}={1\over3}=.333\dots>.32956\dots={3\over4}\cdot{5\over6}\cdot\cdots\cdot{53\over54}$$


