i have a question about the relation between the riemann zeta function and the prime counting function . one starts with the formal definition of zeta :(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \zeta (s)=\prod_{p}\frac{1}{1-p^{-s}} [/tex]

then :

[tex] ln(\zeta (s))= -\sum_{p}ln(1-p^{-s})=\sum_{p}\sum_{n=1}^{\infty}\frac{p^{-sn}}{n}[/tex]

using the trick :

[tex] p^{-sn}=s\int_{p^{n}}^{\infty}x^{-s-1}dx [/tex]

then :

[tex] \frac{ln\zeta (s)}{s} = \sum_{p}\sum_{n=1}^{\infty}\int_{p^{n}}^{\infty}x^{-s-1}dx[/tex]

up until now, things make perfect sense , but the following line is mysterious to me :

[tex] \frac{\ln\zeta(s)}{s}=\int_{0}^{\infty}f(x)x^{-s-1}dx[/tex]

where [itex] f(x) [/itex] is the weighted-prime counting function .

how is this formula derived !?!?

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# Zeta function the the orime counting function

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