Timeline for Frobenius density theorem
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
|
|
| Mar 12, 2014 at 8:21 | comment | added | ACL | @DavidSpeyer. Indeed. What one needs for the natural density is the absence of zero on the real line $\mathop{\rm Re}(s)=1$. The tauberian theorem of Ikehara shows that this condition is also sufficient. Newman's tauberian theorem is a bit weaker and requires some zero-free region; however, the standard (Hadamard) proof of non-vanishing implies the required estimate. For a more precise asymptotic expansion, larger zero-free regions are needed, with upper-bounds at infinity. The notion of analytic density is weaker and for that, one only needs to understand the pole at $s=1$. | |
| Mar 12, 2014 at 2:53 | comment | added | Igor Rivin | @DavidSpeyer yes, I came to the same conclusion, but maybe ACL means that under some conditions (Euler product?) one gets nonvanishing on the line for free. A little hard to believe, but what do I know... | |
| Mar 12, 2014 at 2:42 | comment | added | David E Speyer | @ACL Your notes show that you do need more analytic information. Propositions 1.4.7 and 1.4.8 are additional analytic information that is not needed in the Dirichlet density proof; for Dirichlet density you only need to prove these in a neighborhood of $s=1$. | |
| Mar 6, 2014 at 13:28 | comment | added | Igor Rivin | @ACL Thanks! One lives and learns... | |
| Mar 6, 2014 at 12:59 | comment | added | ACL | I say exactly that. I wrote the details in a mid-graduate course on number theory (4th year univ. in France). See the text on math.u-psud.fr/~chambert/enseignement/2007-08/h4/coursh4.pdf | |
| Mar 6, 2014 at 12:50 | comment | added | Igor Rivin | @ACL So are you saying (for example) that to get from Dirichlet's theorem on primes in progressions to a natural density statement you just need a Tauberian argument? I thought (just as Speyer does, apparently) that you needed more analytic information on the $L$-functions involved. | |
| Mar 6, 2014 at 2:12 | answer | added | David E Speyer | timeline score: 8 | |
| Mar 6, 2014 at 1:41 | comment | added | ACL | @IgorRivin: To pass from Dirichlet density to natural density, you just need powerful enough tauberian theorems. I would bet that Newman's "easy tauberian theorem" (see Zagier's AMM paper) is enough for Frobenius. | |
| Mar 6, 2014 at 0:26 | history | edited | Igor Rivin | CC BY-SA 3.0 |
there seems to be a subtlety about what the FDT actually is...
|
| Jul 8, 2013 at 0:28 | vote | accept | Igor Rivin | ||
| Jul 7, 2013 at 19:40 | comment | added | user9072 | I just changed the link, since it pointed neither to the question nor the answer you mention, but to an other answer. | |
| Jul 7, 2013 at 19:37 | history | edited | user9072 | CC BY-SA 3.0 |
changed link to point to Zieve's answer instead of Rivin's answer
|
| Jul 7, 2013 at 19:31 | answer | added | abz | timeline score: 16 | |
| Jul 7, 2013 at 18:29 | history | asked | Igor Rivin | CC BY-SA 3.0 |