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The proof can be found on this site.

Newman's "Natural proof"(Analytic) of Prime Number Theorem (1980)

I do not understand well for part(VI), why do we consider such integral would imply $\theta(x)$~$x$ ?

Any one can elaborate a bit for me? Thank you.

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dave
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  • Could you be more precise about your question " why do we consider such integral would imply $\theta(x)$~$x$ ?" Answer: Because it proves PNT? – Dietrich Burde Mar 15 '17 at 16:21
  • @DietrichBurde To be more precise, I cannot see the link between this integral and the $\theta(x)$~$x$. – dave Mar 15 '17 at 16:28
  • The link is $(V)$. If the integral is not convergent, this contradicts $(V)$. Hence we must have $\theta(x)\sim x$, which is the PNT. – Dietrich Burde Mar 15 '17 at 16:33
  • Am voting to close since this problem is entirely subsumed by the linked question unless the OP still does not understand. – daniel Mar 06 '19 at 19:45

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