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.
Tell me more
×
Mathematics Stack Exchange is a question and answer site for
people studying math at any level and professionals in related fields. It's 100% free, no registration required.
|
|
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. |
|||
|
|
