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Math Help - Euler's Phi Function

  1. #1
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    Euler's Phi Function

    σ(n)Φ(n)≥n^2(1-1/p1^2)(1-1/p2^2)...(1-1/pr^2)

    I can show that

    σ(n)Φ(n)≥n^2(1-1/p1)(1-1/p2)...(1-1/pr)

    but am unsure of how to do it with the pi^2.
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  2. #2
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    Quote Originally Posted by bluedevilgirl7 View Post
    σ(n)Φ(n)≥n^2(1-1/p1^2)(1-1/p2^2)...(1-1/pr^2)

    I can show that

    σ(n)Φ(n)≥n^2(1-1/p1)(1-1/p2)...(1-1/pr)

    but am unsure of how to do it with the pi^2.

    I think it'd be a good idea if you defined what you use: what's that sigma function: the sum of all the divisors of the number n? And those p_i's are the prime divisors of n? If so the claim is false:

    \phi(6)=2\Longrightarrow 6\phi(6)=6\cdot 2=12\,,\,\,but\,\,6^2\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)=36\cdot\frac{3}{4}\cdot \frac{8}{9}=24
    By the way, without the squares we get equality with n = 6.

    Tonio
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