Timeline for better estimates than the prime number Theorem in Euclidean domains
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 21, 2022 at 23:47 | comment | added | Gerry Myerson | Perhaps it should be kept in mind that not every unique factorization domain is Euclidean. | |
| Jul 21, 2022 at 23:03 | comment | added | Pace Nielsen | @Wojowu Regarding the nonuniqueness of the norm; one could choose to use the minimal Euclidean norm, which is somewhat canonical. | |
| Jul 21, 2022 at 19:54 | comment | added | KConrad | You might like to examine Knopfmacher's book Abstract Analytic Number Theory, whose goal is to develop a framework for prime number theorem type results in different areas of math. My impression is that this book did not really have the kind of impact its author had hoped; I suspect many people familiar with the prime number theorem and Landau's prime ideal theorem will never have heard of it. For a review of the book online, see maa.org/press/maa-reviews/abstract-analytic-number-theory. | |
| Jul 21, 2022 at 19:50 | comment | added | KConrad | You write about "looking at possible candidates that can push the $n/\ln(n)$", but that is partly an artifact of how you choose to estimate. Even over the integers, the most classical case, the logarithmic integral ${\rm Li}(x)$ is understood to be a better estimate of $\pi(x)$ than $x/\log x$. | |
| S Jul 21, 2022 at 19:30 | history | suggested | J. W. Tanner | CC BY-SA 4.0 |
corrected typo.
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| Jul 21, 2022 at 18:54 | review | Suggested edits | |||
| S Jul 21, 2022 at 19:30 | |||||
| Jul 21, 2022 at 18:25 | comment | added | Johnny Cage | Unique..yes, sorry | |
| Jul 21, 2022 at 17:15 | comment | added | J. W. Tanner | Did you mean unique factorization domain when you typed unit factorization domain? | |
| Jul 21, 2022 at 11:59 | comment | added | Wojowu | In almost all cases (excluding number fields and function fields mentioned above) there will be infinitely many elements of norm less than a given bound. Furthermore the answer may depend on the choice of the Euclidean norm. | |
| Jul 21, 2022 at 11:56 | comment | added | Johnny Cage | @Wojowu: sure, I mean that using the euclidean norm you can always count the number of primes with norm less than $n$, compared with all elements with norm less than $n$ (or something similar) | |
| Jul 21, 2022 at 11:46 | comment | added | Wojowu | The question will necessarily run into the question of how one counts elements. There are only a few cases where there is a natural way to count them. In number fields you have an analogue of PNT called Landau's prime ideal theorem, and something similar should hold for function fields over finite fields. Over infinite fields matters look very different - e.g when measured by height, asymptotically 100% of polynomials in $\mathbb Q[x]$ are irreducible, but I could envision other ways to count which would give a different result. | |
| Jul 21, 2022 at 11:27 | history | asked | Johnny Cage | CC BY-SA 4.0 |