Timeline for Prime numbers in arithmetic progressions : uniformity with respect to the modulus
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| Nov 22, 2017 at 12:50 | answer | added | Olivier Ramaré | timeline score: 4 | |
| May 23, 2012 at 18:06 | vote | accept | js21 | ||
| Mar 9, 2012 at 20:22 | comment | added | Dimitris Koukoulopoulos | The proof of Linnik's theorem by Granville and Soundararajan can be found here dms.umontreal.ca/~andrew/PDF/OnePage.pdf (see also here dms.umontreal.ca/~andrew/Courses/MAT6627.H11.html). | |
| Mar 6, 2012 at 17:30 | history | edited | js21 | CC BY-SA 3.0 |
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| Mar 5, 2012 at 22:17 | comment | added | Anonymous | Granville--Sound's "pretentious perspective" is expounded upon in section 3 (Primes in Arithmetic Progressions, without L-functions) of this recent survey: dms.umontreal.ca/%7Eandrew/PDF/ItalySurvey.pdf | |
| Mar 4, 2012 at 9:46 | history | edited | js21 | CC BY-SA 3.0 |
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| Mar 3, 2012 at 22:45 | comment | added | Micah Milinovich | @Denis: Thank you for the clarification. | |
| Mar 3, 2012 at 20:17 | comment | added | Denis Chaperon de Lauzières | The proof of Friedlander and Iwaniec is in their sieve book, "Opera de cribro" (Chapter 24); however, it does use zeros of L-functions in addition to sieve methods. | |
| Mar 3, 2012 at 19:11 | comment | added | Micah Milinovich | I believe that there are now two elementary proofs of Linnik's theorem (in the sense that they avoid zeros of L-functions), one due to Friedlander-Iwaniec and another due to Granville-Soundararajan. I do not think either proof has appeared, but I heard Andrew Granville give a talk on the subject. | |
| Mar 3, 2012 at 15:25 | answer | added | Micah Milinovich | timeline score: 7 | |
| Mar 3, 2012 at 13:01 | comment | added | js21 | As far as I know, no elementary proof of Linnik's theorem is known (although easier theorems as Siegel's have elementary proofs). In an other article of A.Granville, it is suggested that Selberg's formula has been proved in the range $X>q^A$ (as in Linnik's theorem) by J. Friedlander in "Selberg’s formula and Siegel’s zero. Recent progress in analytic number theory, Academic Press, London—New York, 1981, 15–23" . If the proof relies only on a sieve argument, there's some chance that it's elementary ; but I haven't been able to find the aforedmentioned article, so I don't know. | |
| Mar 3, 2012 at 10:21 | comment | added | GH from MO | Note that $\epsilon=q^{-L}$ is admissible by Linnik's method, for $L>0$ a large effective constant, see Corollary 18.8 in Iwaniec-Kowalski: Analytic number theory. Of course it uses complex analysis very substantially. | |
| Mar 3, 2012 at 9:35 | comment | added | js21 | @Igor Rivin : as quid said, without complex analysis, but I wanted to skip the debate on the relevance of this concept. @Terry Tao : indeed A. Granville gives informations (obtained by elementary means) on the behaviour of $\Psi(X,q,a)$ when $X$ goes to infinity, but effectivity with respect to $q$ is lost at each ocurrence of the words "for $X$ sufficiently large" | |
| Mar 3, 2012 at 2:41 | comment | added | Terry Tao | You may find this paper of Granville relevant: ams.org/mathscinet-getitem?mr=1220462 (also available at dms.umontreal.ca/~andrew/PDF/PNTforaps.pdf ) | |
| Mar 3, 2012 at 0:54 | comment | added | user9072 | @Igor Rivin: Maybe I am missing something, but OP says "with the usual meaning of elementary in this context", so basically no complex analysis; which seems even more transparent as before complex analyis is mentioned expicitly as 'what not'. | |
| Mar 2, 2012 at 22:30 | comment | added | Igor Rivin | What DO you mean by elementary here? | |
| Mar 2, 2012 at 22:12 | history | asked | js21 | CC BY-SA 3.0 |