Theory of RZF question

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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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Answers and Replies

  • #2
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I found an error, and the reason why i've been stuck. There is a typo in my book, maybe I should contact the publisher. I had to check Riemann's original handwritten manuscript in german to see that the second integral is from 0 to 1.

the second integral is not

[tex]\int_1^{\infty} \psi ({1 \over x}) x^{-1 \over 2} x^{{s \over 2}-1} dx[/tex]

but

[tex]\int_0^1 \psi ({1 \over x}) x^{-1 \over 2} x^{{s \over 2}-1} dx[/tex]
 
  • #3
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problem resolved. I verified and indeed

[tex]\pi^{-s \over 2} \Gamma ({s \over 2}) \zeta (s) = \int_1^{\infty} \psi (x) x^{{s \over 2}-1} dx + \int_0^1 \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]

=>

[tex]\zeta (s) = {\pi^{s \over 2} \over \Gamma (\frac{s}{2})} \left( \int_1^{\infty} \psi (x) x^{{s \over 2}-1} dx + \int_0^1 \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 \right)[/tex]

=>

[tex]\zeta (s) = {\pi^{s \over 2} \over \Gamma (\frac{s}{2})} \left( {1 \over s(s-1)} + \int_1^\infty \psi(x) \left( x^{{s \over 2}-1} + x^{-{s+1 \over 2}} \right) dx \right)[/tex]
 
  • #4
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The error Im talking about can be found in the middle of page 3 not counting the cover page.
http://www.claymath.org/millennium/Riemann_Hypothesis/1859_manuscript/EZeta.pdf [Broken]


Riemann's manuscript, bottom of page 2, where you can see the real version.
http://www.claymath.org/millennium/Riemann_Hypothesis/1859_manuscript/riemann1859.pdf [Broken]
 
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