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camilus
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theory of Riemann zeta function question
analytically continuing the Riemann zeta function (RZF) using the gamma function leads to this identify:
[tex]n^{-s} \pi^{-s \over 2} \Gamma ({s \over 2}) = \int_0^{\infty} e^{-n^2 \pi x} x^{{s \over 2}-1} dx[/tex] ________(1)
and from that we can build a similar expression incorporating the RZF:
[tex]\pi^{-s \over 2} \Gamma ({s \over 2}) \zeta (s) = \int_0^{\infty} \psi (x) x^{{s \over 2}-1} dx[/tex] ________(2)
where
[tex]\psi (x) = \sum_{n=1}^{\infty} e^{-n^2 \pi x}[/tex]
is the Jacobi theta function.
Then Riemann proceeds to use the functional equation for the theta function:
[tex]2\psi (x) +1 = x^{-1 \over 2} (2 \psi ({1 \over x})+1)[/tex]
to equate (2) with:
[tex]\pi^{-s \over 2} \Gamma ({s \over 2}) \zeta (s) = \int_1^{\infty} \psi (x) x^{{s \over 2}-1} dx + \int_1^{\infty} \psi ({1 \over x}) x^{-1 \over 2} x^{{s \over 2}-1} dx+{1 \over 2} \int_0^{1} (x^{-1 \over 2} x^{{s \over 2}-1} - x^{{s \over 2}-1}) dx[/tex]
This is the step Im stuck on, Im trying to figure out what he did to get that last equation. Any help would be greatly appreciated.
analytically continuing the Riemann zeta function (RZF) using the gamma function leads to this identify:
[tex]n^{-s} \pi^{-s \over 2} \Gamma ({s \over 2}) = \int_0^{\infty} e^{-n^2 \pi x} x^{{s \over 2}-1} dx[/tex] ________(1)
and from that we can build a similar expression incorporating the RZF:
[tex]\pi^{-s \over 2} \Gamma ({s \over 2}) \zeta (s) = \int_0^{\infty} \psi (x) x^{{s \over 2}-1} dx[/tex] ________(2)
where
[tex]\psi (x) = \sum_{n=1}^{\infty} e^{-n^2 \pi x}[/tex]
is the Jacobi theta function.
Then Riemann proceeds to use the functional equation for the theta function:
[tex]2\psi (x) +1 = x^{-1 \over 2} (2 \psi ({1 \over x})+1)[/tex]
to equate (2) with:
[tex]\pi^{-s \over 2} \Gamma ({s \over 2}) \zeta (s) = \int_1^{\infty} \psi (x) x^{{s \over 2}-1} dx + \int_1^{\infty} \psi ({1 \over x}) x^{-1 \over 2} x^{{s \over 2}-1} dx+{1 \over 2} \int_0^{1} (x^{-1 \over 2} x^{{s \over 2}-1} - x^{{s \over 2}-1}) dx[/tex]
This is the step Im stuck on, Im trying to figure out what he did to get that last equation. Any help would be greatly appreciated.
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