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Thread: Riemann Zeta Function

  1. #1
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    Riemann Zeta Function

    Show that $\displaystyle \zeta(s) < 0 $ for $\displaystyle 0 < s < 1 $.
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  2. #2
    MHF Contributor chisigma's Avatar
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    The Laurent expansion of the $\displaystyle \zeta(*)$ around $\displaystyle s=1$ is...

    $\displaystyle \zeta(s)= \frac{1}{s-1} + \sum_{n=0}^{\infty} \frac{(-1)^{n}}{n!}\cdot \gamma_{n}\cdot (s-1)^{n}$ (1)

    ... where the $\displaystyle \gamma_{n}$ are the so called 'Stieltjes constants'. For $\displaystyle s$ 'not too far' from 1 the term $\displaystyle \frac{1}{s-1}$ is dominant, so that $\displaystyle \zeta(*)$ is negative in $\displaystyle 0 < s < 1$. The nearest zero on the real axis is at $\displaystyle s=-2$...

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$

    P.S. That's a graphich I've made about a year ago using the (1)...

    Last edited by chisigma; Sep 10th 2009 at 11:07 AM. Reason: added graphics...
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