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Math Help - Riemann Zeta Function

  1. #1
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    Riemann Zeta Function

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

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

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

    Kind regards

    \chi \sigma

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

    Last edited by chisigma; September 10th 2009 at 12:07 PM. Reason: added graphics...
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