Timeline for How does one prove that the density of unusual numbers is $\ln 2$?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 12, 2020 at 14:21 | vote | accept | Andrei Sipoș | ||
| Apr 11, 2020 at 6:21 | answer | added | Gerry Myerson | timeline score: 3 | |
| Apr 11, 2020 at 6:18 | comment | added | Gerry Myerson | Finch gives a reference to Greene & Knuth, Mathematics for the Analysis of Algorithms, 3rd ed., pages 95-98. It appears those pages are freely accessible at link.springer.com/content/pdf/bbm%3A978-0-8176-4729-2%2F1.pdf | |
| Apr 11, 2020 at 4:33 | comment | added | GH from MO | @NoamD.Elkies: $\sum_{p \leq y} 1/p$ equals $\log\log y + M + o(1)$, where $M$ is the Meissel–Mertens constant. Of course the presence of $M$ does not matter for your argument. | |
| Apr 11, 2020 at 3:07 | comment | added | Noam D. Elkies | Thanks. If the OP [original proposer] is willing to accept it as an answer I can post it as such. The question as stated asks for a reference, not a proof, and I didn't give a reference. | |
| Apr 11, 2020 at 3:00 | comment | added | Steven Landsburg | @NoamD.Elkies : That should really be an answer, not a comment! | |
| Apr 11, 2020 at 2:40 | comment | added | Andrei Sipoș | Thank you for the answer! | |
| Apr 11, 2020 at 2:18 | comment | added | Noam D. Elkies | If $n$ is "unusual" then its large prime factor $p$ is unique. To count "unusual" $n \leq x$, sum over prime $p \leq x$ the number of "unusual" $n$ that are multiples of $p$. The count is $p-1$ if $p \leq \sqrt x$, and $\lfloor x/p \rfloor$ if $\sqrt x < p \leq x$. By the Prime Number Theorem (or even Chebyshev), $\sum_{p \leq \sqrt x} (p-1) \ll x/\log x$. This leaves essentially $x \sum_{\sqrt x < p \leq x} 1/p + O(x/\log x)$. Now use $\sum_{p \leq y} 1/p = \log\log y + o(1)$ for $y=\sqrt x$ and $y=x$ to get $x (\log\log x - \log\log x^{1/2} + o(1)) = x \log 2 + o(x)$, QED. | |
| Apr 11, 2020 at 1:41 | history | asked | Andrei Sipoș | CC BY-SA 4.0 |