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I'm working on understanding the following relation which was referenced in the Number Theory Forum some time ago:
[tex]x-\text{ln}(2\pi)-\sum_{\rho} \frac{x^{\rho}}{\rho}-\frac{1}{2}\text{ln}(1-\frac{1}{x^2})=
-\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{\zeta^{'}(z)}{\zeta(z)}\frac{x^z}{z}dz[/tex]
where:
[tex]\zeta(z)[/tex]
is the Reimann Zeta function.
(c is taken to be right of the critical strip although I'm barely qualified to even say that let alone evaluate it but I digress)
It's encountered in one of the proofs of the Prime Number Theorem. It's an interesting contour-integral to solve and involves lots of different concepts in math.
I'm having some problems applying Residue Integration to it, not to mention figuring out the limits for the residues but believe I can work them out with some time. Just thought I'd post the problem in case others are interested.
Oh yea: Here's the reference
:http://www.maths.ex.ac.uk/~mwatkins/zeta/pntproof.htm
[tex]x-\text{ln}(2\pi)-\sum_{\rho} \frac{x^{\rho}}{\rho}-\frac{1}{2}\text{ln}(1-\frac{1}{x^2})=
-\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{\zeta^{'}(z)}{\zeta(z)}\frac{x^z}{z}dz[/tex]
where:
[tex]\zeta(z)[/tex]
is the Reimann Zeta function.
(c is taken to be right of the critical strip although I'm barely qualified to even say that let alone evaluate it but I digress)
It's encountered in one of the proofs of the Prime Number Theorem. It's an interesting contour-integral to solve and involves lots of different concepts in math.
I'm having some problems applying Residue Integration to it, not to mention figuring out the limits for the residues but believe I can work them out with some time. Just thought I'd post the problem in case others are interested.
Oh yea: Here's the reference
:http://www.maths.ex.ac.uk/~mwatkins/zeta/pntproof.htm
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