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Math Help - Help with proof.

  1. #1
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    Help with proof.

    How can you prove that

     \prod_{n=1}^{\infty}(1-x^{n})^{\mu(n)/n}=e^{-x}

    for |x|<1.

    Here, \mu(n) is the Mobius function.

    Thanks in advance.
    Last edited by CaptainBlack; July 18th 2011 at 11:37 PM. Reason: fix LaTeX
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Help with proof.

    Quote Originally Posted by Cairo View Post
    How can you prove that

     \overset{\infty}{\underset{n=1}{\prod}}(1-x^{n})^{\mu(n)/n}=e^{-x}

    for |x|<1.

    Here, \mu(n) is the Mobius function.

    Thanks in advance.
    What have you tried? Begin by taking the logarithm of both sides to get that, if F(x) is our function, \displaystyle \log(F(x))=\sum_{n=1}^{\infty}\frac{\mu(n)}{n}\log  \left(1-x^n\right)=-\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\frac{\mu(n)  }{n}\frac{x^{mn}}{m}. But show that the coefficient of x^n in this is \displaystyle \frac{-1}{n}\sum_{d\mid n}\mu(d) which I'm sure you know is -1 if n=1 and 0 otherwise. Thus, \log(F(x))=-x.
    Last edited by CaptainBlack; July 18th 2011 at 11:39 PM. Reason: fix quoted LaTeX
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  3. #3
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    Thumbs up Re: Help with proof.

    Thanks for this, Drexel.

    I was going down a very different road and trying to solve a first order differential equation by considering generating functions.

    Your approach looks much more easier to prove.

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