Timeline for Primes and $x^2+2y^2+4z^2$
Current License: CC BY-SA 3.0
15 events
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| Jun 27, 2012 at 18:56 | comment | added | Will Jagy | Alright, so O+S 1997, if n > 0 and gcd(n, 10) = 1, where R_1(n) is the number of primitive representations by x^2 + y^2 + 10 z^2, and R_2(n) is the number of primitive representations by 2x^2 + 2y^2 + 3 z^2 - 2 zx, then we get $$ R_1(n)/2 + R_2(n) = h(-40n). $$ The only general formulation available is a combination of Dirichlet's class number formula with Siegel's weighted representation theorem. However, given your curiosity, you ought to find a copy of Gauss Disquisitiones and see what examples he gives. | |
| Jun 27, 2012 at 18:48 | comment | added | Will Jagy | @Joel, as Paul points out, the facts I used here are equivalent to representing $2p$ as the sum of three squares, due to Gauss and described in Grosswald's book and at mathoverflow.net/questions/3596/… A more realistic example in in Ono and Soundararajan, Ramanujan's ternary quadratic form (Inventiones 1997). | |
| Jun 27, 2012 at 16:02 | comment | added | paul Monsky | Joel--I seem to recall that the coefficients of Eisenstein series of half-integral weight can be expressed in terms of expressions involving class-numbers. So one should get generalizations of Gauss's result provided one takes a weighted sum of thetas over the forms in a genus. | |
| Jun 27, 2012 at 12:00 | comment | added | Joël | Thanks for this great answer. So, what is the general statement of the theorem you use (in your second paragraph), which compute the number of primitive representation by a ternary form of an integer $n$ by a certain class number of something related to $n$ ? I mean, can your method be applied in principle be applied to the same question to any ternary form ? | |
| Jun 27, 2012 at 11:55 | vote | accept | Joël | ||
| Nov 12, 2012 at 14:33 | |||||
| Jun 27, 2012 at 3:21 | comment | added | Will Jagy | Paul refers to his answer at mathoverflow.net/questions/74480/… | |
| Jun 27, 2012 at 2:42 | comment | added | paul Monsky | Will: As I remark in the last paragraph of my answer to my question, "More questions involving characteristic 2 theta series", the proof that (1/48)(the number of representations of 2p by x2+y2+z2) is odd or even, according as p is 7 or 15 mod 16, actually goes back to Hasse. | |
| Jun 27, 2012 at 2:05 | history | edited | Will Jagy | CC BY-SA 3.0 |
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| Jun 27, 2012 at 0:00 | history | edited | Will Jagy | CC BY-SA 3.0 |
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| Jun 26, 2012 at 21:04 | history | edited | Will Jagy | CC BY-SA 3.0 |
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| Jun 26, 2012 at 20:33 | history | edited | Will Jagy | CC BY-SA 3.0 |
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| Jun 26, 2012 at 19:44 | history | edited | Will Jagy | CC BY-SA 3.0 |
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| Jun 26, 2012 at 19:26 | comment | added | Will Jagy | @David, you are correct, I did not see that he had typed in the restriction that $x,y,z$ are odd. | |
| Jun 26, 2012 at 19:19 | comment | added | David E Speyer | Primes which are $1$, $3$, $5$ mod $8$ are not representable as $x^2+2y^2+4z^2$ with $(x,y,z)$ odd, so the number of representations is $0$ and is thus even. PS Nice answer! I learned a lot from it. | |
| Jun 26, 2012 at 19:15 | history | answered | Will Jagy | CC BY-SA 3.0 |