Denote by $\lambda(n)$, the number of numbers between $0$ and $n$ which are the sum of two squares. Landau, and Ramanujan have proven independently, that $$\lambda(n) \sim \frac{n}{\sqrt{\ln(n)}}$$

And the constant of proportionality is the Landau-Ramanujan constant, which is a suspected transcendental.

Is there an exact formula for the function $\lambda(n)$? That is, is there an exact formula for the number of numbers between $0$ and $n$ which are the sum of two squares?

(On a lighter note, I wanted to ask another question which has been bothering me for quite a while. From what I have seen so far, (from the eyes of an absolute novice), in the analytic theory of numbers, I feel that there is more emphasis(or is there?)on finding out the asymptotic behavior of functions like $\lambda(n)$, rather than it's exact formula. Is this because of the difficulty of finding an exact formula? I must apologize if this question seems trivial. I just wanted some clarification and some advice on whether this was true.)

Thanks again in advance.