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I`m reading Melvin B. Nathanson's "Elementary Methods in Number Theory" and I can't think of a way of deducing Selberg's formula (9.3) from the prime number theorem. This is one of the tasks ...

I'm reading the elementary proof of prime number theorem (Selberg / Erdős, around 1949). One key step is to prove that, with $\vartheta(x) = \sum_{p\leq x} \log p$, $$(1) \qquad\qquad \vartheta(x) \...