Timeline for Proving Mertens' theorem using the prime number theorem
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 27, 2022 at 11:46 | comment | added | Daniel Loughran | Yes this was the original approach I tried and got stuck at exactly showing convergence of this integral. Still it is nice to see that the PNT with the weak error term $O(x/ (\log x)^2 )$ is sufficient to prove Merten's Theorem using this approach. | |
| Dec 24, 2022 at 21:36 | comment | added | GH from MO | Yes. Currently I don't see an easier way than the one outlined in Montgomery-Vaughan's book. On the other hand, Axer's theorem is not obvious at all. | |
| Dec 24, 2022 at 19:38 | comment | added | Greg Martin | Ah, is that the detail that your method gets around while this method doesn't? | |
| Dec 24, 2022 at 19:33 | comment | added | GH from MO | How do you justify the convergence of $\int_2^\infty \frac{\pi(t)-\mathop{\rm li}(t)}{t^2} \,dt$? The PNT states that $\pi(t)-\mathop{\rm li}(t)=o(t/\log t)$, but substituting this directly does not guarantee convergence as $\int_2^\infty \frac{1}{t\log t\log\log t} \,dt$ diverges (say). | |
| Dec 24, 2022 at 18:06 | history | answered | Greg Martin | CC BY-SA 4.0 |