Prime number theorem is equivalent to the (asymptotic) equality:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \Psi \sim x [/tex]

Where the "Psi" is Tsebycheff (can,t be a more complicate surname in science?.... ) function equal to complex integral:

[tex] \int_{c-i\infty}^{c+i\infty}dsx^{s}\frac{\zeta'(s)}{\zeta(s) s} [/tex]

but a factor [tex] 2i\pi [/tex] then the "proof" should be easy...get an asymptotic expansion using "saddle point2 or "steepest descent" method and check that keeping the first term the integral is asymptotic to "x".

It seems me a too much easy proof ..it,s strange that Hadamard or other didn,t use that trick to proof PNT or that this theorem is so difficult to prove.

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# A fast and easy proof of PNT?

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