Timeline for What is the high-concept explanation on why real numbers are useful in number theory?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 15, 2019 at 9:44 | history | made wiki | Post Made Community Wiki by Todd Trimble | ||
| Jun 15, 2012 at 8:42 | comment | added | GH from MO | @David: It depends on what you call "analytic". The statement captures a deep property of the integers using the notion of "limit". I would not call it "analytic" just because the notion of "limit" is used in it. Similarly, I would not call a theorem from algebraic number theory "algebraic" just because it uses notions from algebra (e.g. group, ring, field, prime ideal etc.) | |
| Jun 14, 2012 at 3:00 | comment | added | David Corwin | Isn't the theorem itself an analytic statement? | |
| Apr 15, 2011 at 22:36 | comment | added | GH from MO | @Junkie: Pintz confirmed that he had indeed proved Siegel's theorem in an elementary way: matwbn.icm.edu.pl/ksiazki/aa/aa24/aa2458.pdf His paper combines rather simple facts from real analysis in very clever way. It also seems that Linnik (1950) found the first elementary proof. | |
| Apr 15, 2011 at 10:55 | comment | added | Franz Lemmermeyer | Already Gauss in his Disquisitiones observed connections between the class number and what later were realized to be quaternionic effects. Venkov used quaternions and Gauss's approach for proving the class number formula in most cases; for details, see Rehm's article in "Ternary Quadratic Forms and Norms" edited by Taussky. This was taken up in the last few years by a number of people. | |
| Apr 15, 2011 at 9:23 | comment | added | GH from MO | @Junkie: I will ask Pintz. | |
| Apr 15, 2011 at 6:58 | comment | added | Junkie | Didn't Pintz have a long series of Acta Arithmetica papers on Elementary methods in the theory of L-functions? I think you can get D-H w/o complex analysis via that. | |
| Apr 14, 2011 at 21:31 | comment | added | GH from MO | @Pete: The proofs known to me are closely tied with the Deuring-Heilbronn phenomenon of $L$-functions. @David: I highly recommend front.math.ucdavis.edu/1001.0897 | |
| Apr 14, 2011 at 18:17 | comment | added | David E Speyer | @Pete I'd be very interested to see you try. This is essentially equivalent to showing that the class numbers of quadratic imaginary fields go to infinity (see eqn. 35 at mathworld.wolfram.com/SumofSquaresFunction.html .) I don't think there should be any easy way to prove that! But I've never understood how the quaternion picture ties in with the class number picture, so I'd love for you to turn out one of your notes explaining it. | |
| Apr 14, 2011 at 17:54 | comment | added | Pete L. Clark | Hmm. My feeling is that it should be possible to prove this using quaternion arithmetic, although I am not prepared to back this up with details at the moment. | |
| Apr 14, 2011 at 14:42 | history | answered | GH from MO | CC BY-SA 3.0 |