Timeline for The number of integral solutions to $x^2+y^2-az^2=0$
Current License: CC BY-SA 3.0
5 events
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| Nov 16, 2015 at 15:34 | history | edited | GH from MO |
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| Nov 16, 2015 at 15:22 | comment | added | Lucia | The answer below gives a nice reference. But this particular case is pretty easy. Clearly one can assume that $a$ is an integer and a sum of two squares. Then you are asking for $\sum_{z \le T} r(az^2)$ where $r(n)$ is the number of ways of writing $n$ as a sum of two squares. This is essentially summing a multiplicative function, which is $3$ on primes $p$ that are $1\pmod 4$ (and not dividing $a$), and $1$ on the primes $p$ that are $3\pmod 4$. The usual argument via Dirichlet series now gives the asymptotic. | |
| Nov 16, 2015 at 13:40 | vote | accept | Keivan Karai | ||
| Nov 16, 2015 at 10:47 | answer | added | Daniel Loughran | timeline score: 8 | |
| Nov 16, 2015 at 10:23 | history | asked | Keivan Karai | CC BY-SA 3.0 |