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Math Help - Another phi proof

  1. #1
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    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...
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  2. #2
    MHF Contributor chisigma's Avatar
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    In...

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

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

    Kind regards

    \chi \sigma
    Last edited by chisigma; April 8th 2010 at 11:49 PM. Reason: simple orthographic error...
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  3. #3
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    One representation of the \phi(n) function is:

    \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

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