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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.

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