Results 1 to 3 of 3

Thread: Help with proof.

  1. #1
    Member
    Joined
    May 2008
    Posts
    140

    Help with proof.

    How can you prove that

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

    for $\displaystyle |x|<1$.

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

    Thanks in advance.
    Last edited by CaptainBlack; Jul 18th 2011 at 10:37 PM. Reason: fix LaTeX
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    22

    Re: Help with proof.

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

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

    for $\displaystyle |x|<1$.

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

    Thanks in advance.
    What have you tried? Begin by taking the logarithm of both sides to get that, if $\displaystyle F(x)$ is our function, $\displaystyle \displaystyle \log(F(x))=\sum_{n=1}^{\infty}\frac{\mu(n)}{n}\log \left(1-x^n\right)=-\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\frac{\mu(n) }{n}\frac{x^{mn}}{m}$. But show that the coefficient of $\displaystyle x^n$ in this is $\displaystyle \displaystyle \frac{-1}{n}\sum_{d\mid n}\mu(d)$ which I'm sure you know is $\displaystyle -1$ if $\displaystyle n=1$ and $\displaystyle 0$ otherwise. Thus, $\displaystyle \log(F(x))=-x$.
    Last edited by CaptainBlack; Jul 18th 2011 at 10:39 PM. Reason: fix quoted LaTeX
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    May 2008
    Posts
    140

    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.

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 15
    Last Post: Jun 8th 2011, 11:13 AM
  2. Replies: 5
    Last Post: Oct 19th 2010, 10:50 AM
  3. [SOLVED] direct proof and proof by contradiction
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: Feb 27th 2010, 10:07 PM
  4. Proof with algebra, and proof by induction (problems)
    Posted in the Discrete Math Forum
    Replies: 8
    Last Post: Jun 8th 2008, 01:20 PM
  5. proof that the proof that .999_ = 1 is not a proof (version)
    Posted in the Advanced Applied Math Forum
    Replies: 4
    Last Post: Apr 14th 2008, 04:07 PM

Search Tags


/mathhelpforum @mathhelpforum