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Math Help - Boundedness of Riemann Zeta function

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    Boundedness of Riemann Zeta function

    Show that \frac{\zeta '(s)}{\zeta (s)} = -\sum_{n=1}^{\infty}\frac{\Lambda (n)}{n^s} is bounded in the right half plane Re s > 2.

    ( \Lambda (n) is the Mangoldt Function)
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by EinStone View Post
    Show that \frac{\zeta '(s)}{\zeta (s)} = -\sum_{n=1}^{\infty}\frac{\Lambda (n)}{n^s} is bounded in the right half plane Re s > 2.

    ( \Lambda (n) is the Mangoldt Function)
    Note that  \Lambda(n)\leq \log(n)\leq n .

    Let  s=\sigma+it

    So  \left|\frac{\zeta '(s)}{\zeta (s)}\right| \leq \sum_{n=1}^\infty \left|\frac{\Lambda(n)}{n^s} \right| \leq \sum_{n=1}^\infty \left|\frac{n}{n^s} \right| = \sum_{n=1}^\infty \left|\frac{1}{n^{s-1}} \right| = \sum_{n=1}^\infty \frac{1}{n^{\sigma-1}} = \zeta(\sigma-1) .

    Since  \zeta(\sigma-1) is bounded for  \sigma = \Re(s)>2 , we are done.
    Last edited by chiph588@; May 29th 2010 at 07:22 PM.
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