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Math Help - Functional equation for Riemann zeta function

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    Functional equation for Riemann zeta function

    Let

    f(s)=\frac{1}{2}s(s-1)\pi^{-\frac{s}{2}}\Gamma(\frac{1}{2}s)\zeta(s).

    Prove that \underset{s\rightarrow o}{\lim}\, f(s)=\underset{s\rightarrow1}{\lim\,}f(s)=\frac{1}  {2}, and that f has no zeros outside the strip 0\leq\sigma\leq1.
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    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by Cairo View Post
    Let

    f(s)=\frac{1}{2}s(s-1)\pi^{-\frac{s}{2}}\Gamma(\frac{1}{2}s)\zeta(s).

    Prove that \underset{s\rightarrow o}{\lim}\, f(s)=\underset{s\rightarrow1}{\lim\,}f(s)=\frac{1}  {2}, and that f has no zeros outside the strip 0\leq\sigma\leq1.
    Remembering the 'Hadamard product'...

    \displaystyle \zeta(s)= \pi^{\frac{s}{2}}\ \frac{\prod_{p} (1-\frac{s}{p})}{2\ (s-1)\ \Gamma(1+\frac{s}{2})} (1)

    ... and taking into account that is...

    \displaystyle \Gamma(1+\frac{s}{2}) = \frac{s}{2}\ \Gamma(\frac{s}{2}) (2)

    ... You obtain...

    \displaystyle f(s) = \frac{1}{2}\ \prod_{p} (1-\frac{s}{p}) (3)

    ... where p are the 'non trivial zeroes' of \zeta (*) and they all have real part 0 \le \sigma \le 1. From (3) is evident that is...

    \displaystyle \lim_{s \rightarrow 0} f(s)=\frac{1}{2} (4)

    ... and because is f(s)=f(1-s) is also...

    \displaystyle \lim_{s \rightarrow 1} f(s)=\frac{1}{2} (5)

    Kind regards

    \chi \sigma
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