Timeline for Iterated logarithms in analytic number theory
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
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| Jun 18, 2022 at 9:25 | vote | accept | Jesse Elliott | ||
| Nov 18, 2021 at 1:39 | history | edited | Jesse Elliott | CC BY-SA 4.0 |
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| Nov 17, 2021 at 10:25 | comment | added | Jesse Elliott | The idea is that if Littlewood's lower bound on $\pi(x)-\operatorname{li}(x)$ were closest to the truth (much closer than any established prime number theorem), then one would expect that $\pi(x)-\operatorname{li}(x)$ were $O(\sqrt{x}(\log x)^{-1} (\log \log \log x)^{t})$ for all $t$ greater than some minimal $d \geq 1$, perhaps $d = 2$ if Montgomery's conjecture is correct. But outside a set of logarithmic measure $0$, it is probably more like $O(\sqrt{x}(\log x)^{-1} (\log \log x)^{t})$ for $t$ greater than some minimal $d$. With no more higher logs in either case. | |
| Nov 17, 2021 at 10:13 | comment | added | Jesse Elliott | Even so, geniuses like Tao and Maynard improving the power of $\log_3$ by one took a herculean effort. Many conjectures in number theory that extend the Riemann Hypothesis, including the one by Montgomery but also one by Gonek and Ng regarding the Mertens function, imply that the buck stops at $\log_3$. Of course they are much harder to establish than the Riemann Hypothesis. But they do have some supporters. I don't pretend to understand their heuristic argument. That's kind of the motivation behind my question... | |
| Nov 17, 2021 at 8:49 | comment | added | Wolfgang | In fact, does there exist a "natural" example where an exact order of magnitude contains a log iterated more than twice? I think that would be the revolutionary thing! (E.g. if Montgomery's conjecture turned out to be true.) | |
| Nov 17, 2021 at 8:43 | comment | added | Wolfgang | @JesseElliott Granted that there is a $\log \log \log \log$ that occurs. But this occurs just as part of an estimation, the statement being $\limsup\limits_{n\to\infty}\frac{p_{n+1}-p_n}{\log p_n}\cdot\frac{\left(\log\log\log p_n\right)^{2}}{ \log\log p_n \log\log\log\log p_n} = \infty$ (I suppose that is what you are referring to, from the page en.wikipedia.org/w/index.php?title=Cram%C3%A9r%27s_conjecture). So the numbers of iterated logs here are rather irrelevant, as no precise growth order is given. | |
| Nov 17, 2021 at 1:19 | comment | added | Tanmay Khale | (The most comprehensive account of sieve methods is Friedlander and Iwaniec's "Opera de Cribro." But, being written in 2010, it does not include the state-of-the art work on small and large gaps. These notes by Kevin Ford are also nice faculty.math.illinois.edu/~ford/sieve2020.pdf, and do cover small and large gaps.) | |
| Nov 17, 2021 at 1:09 | comment | added | Tanmay Khale | Regarding the final question in your edit: I would recommend the latter half of Dimitris Koukoulopoulos' "The Distribution of Prime Numbers" as a very accessible introduction to sieve theory. (It is probably my favorite mathematical text.) In particular, the last section (Part 6) covers the results on long and short gaps that you are interested in. | |
| Nov 16, 2021 at 23:37 | history | edited | LSpice | CC BY-SA 4.0 |
Capitalise title; delete spurious $x \to \infty$
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| Nov 16, 2021 at 23:25 | history | edited | Jesse Elliott | CC BY-SA 4.0 |
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| Nov 16, 2021 at 21:20 | history | edited | Jesse Elliott | CC BY-SA 4.0 |
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| Nov 16, 2021 at 16:16 | answer | added | TravorLZH | timeline score: 8 | |
| Nov 16, 2021 at 7:01 | history | became hot network question | |||
| Nov 16, 2021 at 7:00 | answer | added | 2734364041 | timeline score: 16 | |
| Nov 16, 2021 at 4:43 | answer | added | Pace Nielsen | timeline score: 39 | |
| Nov 16, 2021 at 1:18 | comment | added | Tanmay Khale | @Jesse Elliott You may find interesting Chapter 1 of Hall and Tenenbaum's "Divisors." | |
| Nov 15, 2021 at 21:26 | comment | added | Jesse Elliott | I never understood how that law applies, since it works only outside a set of measure zero. But in some cases in number theory, as in the study of prime gaps, there is a $\log \log \log \log$ that occurs. I don't think the law of the iterated logarithm is enough to explain this. And nothing I've seen in analytic number theory really uses the law. If you use the law in trying to study the Mertens function, for example, you probably get the wrong order of growth. (There are competing conjectures, and the one with a double log is less likely to be true than the one with the triple log.) | |
| Nov 15, 2021 at 20:58 | comment | added | Steve Huntsman | My guess is that apart from the PNT, a "metareason" for this would be the validity of probabilistic Ansätze (esp. en.wikipedia.org/wiki/Cram%C3%A9r%27s_conjecture). In such settings double logarithms are common, as in e.g. en.wikipedia.org/wiki/Law_of_the_iterated_logarithm and en.wikipedia.org/wiki/…. | |
| Nov 15, 2021 at 20:30 | history | asked | Jesse Elliott | CC BY-SA 4.0 |