Density of forms x^2+y^3, x^2+y^4,

1. Mar 4, 2008

CRGreathouse

Consider all numbers of a given form x^2 + y^4 for any nonnegative integers x and y. Their density seems to vary roughly as Kn^(3/4), though perhaps there's a logarithmic term in there. I was trying to refine, generalize, and estimate with this.

(Note: I can't get \tilde{O} or even \stackrel{\~}{O} to work, so I'm using O*(x) for O(x log^k x) for some unspecified/unknown k.)

1. After numerical experiments, it seems reasonable to conjecture that the density of $x^2+y^a$ is $O*(n^{1/2+1/a})$. If true, I imagine this is well-known. Does anyone have a name for this result (as theorem or conjecture), a reference, or know a reference? Is it perhaps too basic for that?
2. It seems reasonable to generalize to $x^a+y^b$ is $O*(n^{1/a+1/b})$ for suitable a and b. Anything known about this?
3. Does anyone know about the logarithmic factors I've hidden in my soft-Os? I know that the density of solutions for $x^2+y^2$ is
$$\frac{Kn}{\sqrt{\log n}}$$
with K = 0.76422... the Landau-Ramanujan constant. Generalizing much too far it would be easy to guess something like
$$\mathcal{O}(\frac{n^{1/a+1/b}}{(\log n)^k})$$
for some kind of k(a, b)... but my numerical evidence doesn't really cover small factors like logarithms. (I had enough trouble distinguishing the exponents, thank you very much. The sequences are slow to converge to their limits.)
4. Along those lines: any ideas for speeding up calculations of relevant constants?