Analytically Continue RZF Using Gamma Function: Step By Step Guide

[camilus]

theory of Riemann zeta function question

analytically continuing the Riemann zeta function (RZF) using the gamma function leads to this identify:

$$n^{-s} \pi^{-s \over 2} \Gamma ({s \over 2}) = \int_0^{\infty} e^{-n^2 \pi x} x^{{s \over 2}-1} dx$$ ________(1)

and from that we can build a similar expression incorporating the RZF:

$$\pi^{-s \over 2} \Gamma ({s \over 2}) \zeta (s) = \int_0^{\infty} \psi (x) x^{{s \over 2}-1} dx$$ ________(2)

where

$$\psi (x) = \sum_{n=1}^{\infty} e^{-n^2 \pi x}$$

is the Jacobi theta function.

Then Riemann proceeds to use the functional equation for the theta function:

$$2\psi (x) +1 = x^{-1 \over 2} (2 \psi ({1 \over x})+1)$$

to equate (2) with:

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

This is the step I am stuck on, I am trying to figure out what he did to get that last equation. Any help would be greatly appreciated.

 

[camilus]

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

$$\int_1^{\infty} \psi ({1 \over x}) x^{-1 \over 2} x^{{s \over 2}-1} dx$$

but

$$\int_0^1 \psi ({1 \over x}) x^{-1 \over 2} x^{{s \over 2}-1} dx$$

 

[camilus]

problem resolved. I verified and indeed

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

=>

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

=>

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

 

[camilus]

The error I am 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


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