Timeline for Jacobi's theorem on sums of two squares (reference request)
Current License: CC BY-SA 4.0
14 events
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| Mar 14, 2019 at 7:23 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 23, 2016 at 17:39 | history | edited | Carlo Beenakker | CC BY-SA 3.0 |
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| Jun 23, 2016 at 17:20 | history | edited | Carlo Beenakker | CC BY-SA 3.0 |
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| Jun 23, 2016 at 17:15 | history | edited | Carlo Beenakker | CC BY-SA 3.0 |
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| Jun 23, 2016 at 17:02 | history | edited | Carlo Beenakker | CC BY-SA 3.0 |
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| Jun 23, 2016 at 16:21 | history | edited | Carlo Beenakker | CC BY-SA 3.0 |
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| Jun 22, 2016 at 23:50 | comment | added | GH from MO | I agree with Jeremy Rouse. The quoted formula (13) expresses in analytic form the simple fact that $\sum_{m+n=t}r_{A,B}(m)r_{A,B}(n)=\sum_{kA+lB=t}r(k)r(l)$. Indeed, both sides count the number of quadruples $(u,v,x,y)\in\mathbb{Z}^4$ such that $Au^2+Bv^2+Ax^2+By^2=t$. In contrast, Jacobi's formula has to do with arithmetic, e.g. the fundamental theorem of arithmetic in $\mathbb{Z}[i]$. | |
| Jun 22, 2016 at 20:37 | comment | added | Carlo Beenakker | the formula immediately generalizes to more than two squares, $n=\sum_{k=1}^N A_k x_k^2$, see equation 15 of the cited paper; I find it actually more memorable than unwieldy. | |
| Jun 22, 2016 at 19:50 | comment | added | Jeremy Rouse | I don't consider this formula to be "similar" to Jacobi's formula - it is way too unwieldy. In cases that the binary form $Q = x^2 + ay^2$ has class number $1$, there should be a formula equating the theta series of $Q$ with an Eisenstein series, and be very similar to Jacobi's formula. This would occur only for $a = 1, 2, 3$ and $7$, though. | |
| Jun 22, 2016 at 18:34 | history | edited | Carlo Beenakker | CC BY-SA 3.0 |
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| Jun 22, 2016 at 18:28 | history | edited | Carlo Beenakker | CC BY-SA 3.0 |
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| Jun 22, 2016 at 18:20 | history | edited | Carlo Beenakker | CC BY-SA 3.0 |
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| Jun 22, 2016 at 17:17 | history | edited | Carlo Beenakker | CC BY-SA 3.0 |
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| Jun 22, 2016 at 17:11 | history | answered | Carlo Beenakker | CC BY-SA 3.0 |