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So we have $$\sum_{n \leq x} \frac{\Lambda (n)}{n}=\log{x}+C+o(1)$$ where $C$ is a constant, its partial summation is $$\sum_{n \leq x} \frac{\Lambda (n)}{n}=\frac{\psi(x)}{x}+\int_1^x \frac{\psi (t)}{t^2} dt$$ How should I go from here to prove that $\psi(x) \sim x$, which is a equivalent form of PNT.

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I don't know. What makes you think it is possible to go from those two equations to $\psi(x)\sim x$? –  Gerry Myerson Oct 16 '11 at 11:57
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up vote 3 down vote accepted

Do you mean $O(1)$ or $o(1)$ in your first equation? If the former, then (a) it doesn't make sense to include the $C$ term, since it can be absorbed into the error and (b) I don't think that estimate is strong enough to prove PNT.

If you mean $o(1)$, then see my blog post, especially part 3.

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FYI part 4. there includes $1/\log1$ summands. –  anon Oct 16 '11 at 21:37
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