greetings . i have a couple of questions about the prime counting function .(adsbygoogle = window.adsbygoogle || []).push({});

when [itex]\pi _{0}(x)[/itex] changes by 1, then it's logical to assume that it should happen at a prime argument . meaning :

[tex]\lim_{\xi\rightarrow 0}\pi _{0}(x+\xi )-\pi _{0}(x-\xi )=1 [/tex]

implies that x is a prime .

is this a true assumption ?

according to the literature, we can expand [itex] \pi_{0}(x)[/itex] using the riemann R function .

[tex] R(x) = 1+\sum_{k=1}^{\infty}\frac{(ln x)^{k}}{k!k\zeta (k+1)}[/tex]

[tex] \pi_{0}(x)= R(x)-\sum_{\rho} R(x^{\rho })-\frac{1}{lnx}-\frac{1}{\pi}tan^{-1}\left( \frac{\pi}{lnx}\right) = \sum_{k=1}^{\infty}\frac{(lnx)^{k}[1-\sum \rho^{k} ]}{k!k\zeta(k+1)}-\frac{1}{lnx}-\frac{1}{\pi}tan^{-1}\left( \frac{\pi}{lnx}\right) [/tex][itex] \rho [/itex] being the nontrivial zeros of the zeta function .

is this correct !?!? i mean , is the expansion correct !?!?

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# Questions about the prime counting function

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