1. ## Another phi proof

Let & denote the phi function.

Show that there are an infinite number of n such that &(n) / n < 1/4

Thanks for any help...

2. In...

http://www.mathhelpforum.com/math-he...-function.html

... it has been demonstrated that, given an $\displaystyle \varepsilon > 0$, it exists al least one $\displaystyle n$ for which is $\displaystyle \frac{\varphi (n)}{n} < \varepsilon$ ...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

3. One representation of the $\displaystyle \phi(n)$ function is:

$\displaystyle \phi(n)=n \Pi (1-\frac{1}{p})$

where the infinite product runs over all p's dividing n. So if at least the primes 2,3,5,7 divide n, then the fraction

$\displaystyle \frac {\phi(n)}{n} <= (1/2)(2/3)(4/5)(6/7) = 0.23$