Timeline for $\{(\log n)^\alpha\}$ not equidistributed if $0<\alpha\leq 1$, so how is it distributed?
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Oct 29, 2021 at 18:10 | history | edited | Vincent Granville | CC BY-SA 4.0 |
There was a missing term ($-n$) in the definition of $g_n(x)$. Now corrected.
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| Oct 29, 2021 at 0:34 | history | edited | Vincent Granville | CC BY-SA 4.0 |
added 144 characters in body
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| Oct 28, 2021 at 20:48 | history | edited | Vincent Granville | CC BY-SA 4.0 |
See update at the bottom
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| Oct 28, 2021 at 17:22 | vote | accept | Vincent Granville | ||
| Oct 28, 2021 at 4:40 | answer | added | Iosif Pinelis | timeline score: 8 | |
| Oct 28, 2021 at 3:41 | comment | added | Vincent Granville | @Terry: Thank you very much for your answer and very interesting paper. I read more and more of what you write in number theory, but still has a very long way to go to understand even 30% of it. I am very honored that you commented on my question. | |
| Oct 28, 2021 at 2:56 | comment | added | Vincent Granville | @Conrad: yes, and my previous comment (about an answer to an MO question) actually is based on the Fejer criteria, which I was not aware of. What I offer here is a method to find what the limiting distribution might be, and if uniform it means equidistribution. Not sure if it is somehow related to Fejer's result, but it is an effective method that will tell you what that distribution is, either way. | |
| Oct 28, 2021 at 2:41 | comment | added | Conrad | Feijer criterion trivially implies the equidistribution of $(\log n)^a, a>1$ modulo $1$ ($f'(x) \to 0, x|f'(x)| \to \infty$ for $x \to \infty$ and $f'$ monotonic implies $f(n)$ is uniformly distributed modulo $1$ | |
| Oct 28, 2021 at 2:41 | comment | added | Terry Tao | Many years ago, an undergraduate student did an honours project on this topic: math.ucla.edu/~tao/preprints/ThesisSyd.pdf | |
| Oct 28, 2021 at 2:15 | comment | added | Vincent Granville | As I am doing more research, I've found this MO answer to a question closely related to mine: mathoverflow.net/questions/107195/…. It confirms $\{(\log n)^{1+\epsilon}\}$ is equidistributed. So the originality of my question is in the case where equidistribution fails. | |
| Oct 28, 2021 at 2:10 | history | edited | user44143 | CC BY-SA 4.0 |
restated questions as questions
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| Oct 28, 2021 at 1:35 | history | edited | Vincent Granville | CC BY-SA 4.0 |
added 10 characters in body
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| Oct 28, 2021 at 0:17 | comment | added | Vincent Granville | It is "easy" to work on the case $\{(\log_b n)^\alpha\}$ using my methodology. Replace the number $2$ in my last formula, by $b$. | |
| Oct 27, 2021 at 23:58 | history | asked | Vincent Granville | CC BY-SA 4.0 |