Timeline for Proving Mertens' theorem using the prime number theorem
Current License: CC BY-SA 4.0
9 events
when toggle format | what | by | license | comment | |
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Feb 7, 2023 at 14:01 | comment | added | GH from MO | @KConrad Thank you. My impression is that Dimitrov's argument is essentially the same as what the proof of Axer's theorem would give in this instance. I might be wrong of course. | |
Feb 7, 2023 at 8:39 | comment | added | KConrad | An approach to showing PNT implies $\sum_{p \leq x} (\log p)/p = \log x + C + o(1)$, or more precisely $\sum_{n \leq x} \Lambda(n)/n = \log x - \gamma + o(1)$, without using Axer's theorem is indicated in the second half of Dimitrov's answer at mathoverflow.net/questions/95743/…. | |
Dec 27, 2022 at 11:43 | vote | accept | Daniel Loughran | ||
Dec 23, 2022 at 22:49 | comment | added | GH from MO | @DanielLoughran I don't, but I have not thought about this very hard. I just recall your original question was asked from me before (in private), and I found it rather nontrivial: I ended up looking it up in Montgomery-Vaughan. BTW if you assume the PNT with a goodish error term, it is easier to derive a version of $(\ast)$ with a goodish error term. So whatever you are looking for, working with effective error terms (rather than just $o(1)$ type error terms) might be helpful in achieving your goal. | |
Dec 23, 2022 at 17:46 | comment | added | Daniel Loughran | Many thanks. Do you know any more direct approach which avoids using logarithmic weights? | |
Dec 23, 2022 at 12:03 | comment | added | GH from MO | @VladimirDotsenko Corrected, thanks. I copied the display from the original post, hence the mistake. (Now the original post is corrected as well.) | |
Dec 23, 2022 at 12:03 | history | edited | GH from MO | CC BY-SA 4.0 |
added 21 characters in body
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Dec 23, 2022 at 11:59 | comment | added | Vladimir Dotsenko | I sincerely hope that you and the OP meant something like $o(1/\log x)$. In particular, $O(\log(x))$ makes the presence of $\log\log x+M$ in (*) completely pointless? | |
Dec 23, 2022 at 11:35 | history | answered | GH from MO | CC BY-SA 4.0 |