Timeline for Does a positive binary quadratic form represent a set of primes possessing a natural density
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| Jun 16, 2010 at 20:05 | comment | added | Will Jagy | Thanks, Paul. That is the right way to do it in practice, of course. My first MO question was about the numbers integrally represented by $$ 2 x^2 + x y + 3 y^2 + z^3 - z, $$ with a complete answer by Kevin Buzzard. I would love to investigate similar constructions in class number five, thus $\Delta = -47$ and a monic irreducible quintic $f_{-47}(t) $ related to the principal form (as in the Cox book) and then an inhomogeneous polynomial, either $$ 2 x^2 + x y + 6 y^2 - f_{-47}(t) $$ or $$ 3 x^2 + x y + 4 y^2 - f_{-47}(t) . $$ | |
| Jun 16, 2010 at 19:48 | comment | added | paul Monsky | Will--I tend to do what people since Lagrange have done; find an odd b with b^2=-47 mod p, reduce the form px^2+bxy+cy^2 of discriminant -47, and see what I get. But this probably isn't what you want. | |
| Jun 16, 2010 at 19:18 | comment | added | paul Monsky | On page 195 of Hecke's paper in "Werke" he says he now can prove the existence of prime ideals in every ray class "ohne dass wir dazu die Existenz der zugehorigen Klassenkorper zu kennen brauchen" I gather from this and his comments on page 197 that the proof via the existence theorem only came 3 years later with Takagi. This seems surprising since Hilbert had quadratic reciprocity for number fields before 1900, and one only needs to construct the appropriate degree 2 extension, but Hecke is the expert. | |
| Jun 16, 2010 at 19:11 | comment | added | Will Jagy | Thanks, Paul. Do you have any opinion on the questions I posed to Victor Miller after his comment (directly under the initital question)? | |
| Jun 16, 2010 at 18:55 | comment | added | paul Monsky | Will--I was referring to Tate's thesis and the paper you mention. One more remark. Supose O is the ring of integers in K, [K:Q]=n, m is a modulus and S is a ray class for m. Hecke shows that the prime ideals of O in S have a Dirichlet density independent of S and a relative natural density independent of S (cf. page 197). The second result is no harder than the first once you know Hadamard's proof of PNT. For Dedekind's method shows the L-functions extend to Re s >1-(1/n). So the only problem is the real L at 1. Paul | |
| Jun 16, 2010 at 6:34 | comment | added | Robin Chapman | I presume Prof. Monsky is referring to Tate's PhD thesis which was published as an article in the classic collection Algebraic Number Theory edited by Cassels and Frohlich. | |
| Jun 16, 2010 at 4:03 | comment | added | Will Jagy | Dear Prof. Monsky, Could you please tell me where the Tate article appeared? Also, would you say that the Hecke results you mean appear in the piece Stevenhagen and Lenstra mention, Uber die L-Funktionen und den Dirichletschen Primzahlsatz fur einen beliebigen Zahlkorper, Nachr. Akad. Wiss. Gottingen Math.-Phys. (1917) pages 299-318 or Mathematische Werke, pages 178-197. Thanks in advance, Will. | |
| Jun 15, 2010 at 21:21 | comment | added | paul Monsky | There's an alternative to class-field theory here. For the most part the argument is like that used nowadays for showing primes in a progression have a relative density, but you have to use ray-class characters in the ring of integers of the quadratic field. The hard part is showing that the L-function attached to a real character doesn't vanish at 1. Once one can extend the L-function to all of C, one can give a Landau style argument. What Hecke did was show how to extend the L-function-- his argument is much like one of Riemann's; Tate later put it in the modern style. | |
| Jun 15, 2010 at 20:20 | comment | added | Will Jagy | I hope that I have managed to make genuine links by putting the string "http://" in front. On the Chebotarev density theorem proved for natural density, Stevenhagen and Lenstra give a specific 1917 Hecke article math.leidenuniv.nl/~hwl/papers/cheb.pdf Then Anatoly Preygel modular.fas.harvard.edu/129-05/final_papers/Anatoly-Preygel.pdf refers to a modern book, Larry Joel Goldstein , Analytic Number Theory (1971) while I already mentioned K. Prachar "Primzahlverteilung" (1957) so I am planning to go borrow more books from campus. | |
| Jun 15, 2010 at 20:06 | comment | added | Pete L. Clark | @Will: yes, that's what I say. I would be happy to have a precise reference though. | |
| Jun 15, 2010 at 19:02 | history | edited | Robin Chapman | CC BY-SA 2.5 |
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| Jun 15, 2010 at 18:27 | vote | accept | Will Jagy | ||
| Jun 15, 2010 at 18:24 | comment | added | Will Jagy | I used Google with "chebotarev density" "natural density". Our own Pete L. Clark, in some lecture notes, attributes the result for natural density to Hecke. www.math.uga.edu/~pete/8430notes5.pdf | |
| Jun 15, 2010 at 18:22 | history | edited | Robin Chapman | CC BY-SA 2.5 |
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| Jun 15, 2010 at 17:33 | history | answered | Robin Chapman | CC BY-SA 2.5 |