We define $$a_N =\# \{ n \leq N, \exists (n_1,n_2) \in \mathbb{N}^2, n = n_1^2 + n_2^2 \}.$$ Can we have the exact value of $a_N$, or at least an asymptotic behavior such as $$ \alpha N \leq a_N \leq \beta N$$ for some constant $\alpha,\beta>0$? (Of course, the existence of $\beta$ is obvious since $a_N \leq N+1$. I am particularly interested by the existence of a constant $\alpha$.)
Thanks for your attention.
On
[Edit: I didn't read the post properly, so this answers a quite different though related question: what is the count of $(n_1,n_2)$ such that $n_1^2+n^2 \leq N$. Gerry Myerson's response answers the question that was actually asked.]
This is a nice question that was first studied thoroughly by Gauss and goes by the name 'Gauss's circle problem.'
We know that your $a_N$ is asymptotic to the area of a circle of radius $\sqrt{N}$; that is $a_N \sim \pi N$. A simple proof of this fact, with $r^2$ in place of your $N$ can be found in the first part of http://alpha.math.uga.edu/~pete/4400gausscircle.pdf
To obtain good bounds for $a_N-\pi N$ is still an open problem.
More information can be found at http://en.wikipedia.org/wiki/Gauss_circle_problem, or in classic books on number theory, like Hardy and Wright's 'An Introduction to the Theory of Numbers'.
E. Landau, "Über die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindestzahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate," Arch. Math. Phys. (3), v. 13, 1908, pp. 305-312 proved that the number of sums of two squares up to $x$ is asymptotic to $Bx/\sqrt{\log x}$, where $$B={1\over\sqrt2}\prod_{p\equiv3\bmod4}(1-p^{-2})^{1/2}$$ See also P. Shiu, Counting sums of two squares: the Meissel-Lehmer method, Math. Comp. volume 47, number 175, July 1986, pages 351-360, available at http://www.ams.org/journals/mcom/1986-47-175/S0025-5718-1986-0842141-1/S0025-5718-1986-0842141-1.pdf