Legendre showed that an integer is the sum of three squares if and only if it is not of the form $4^n(8m + 7)$ for some nonnegative integers $n$ and $m$. However, I have been unable to find any information regarding the counting function.
Let $S(x)$ denote the number of positive integers $\leq x$ which are the sum of three squares. What is known regarding $L: =\displaystyle \lim_{x \to \infty} \frac{S(x)}{x}$? The above characterization of Legendre easily shows that $L \leq \frac{7}{8}$, but can we do better? Is an exact value of $L$ known to exist?
Answer from comments: The quantity $\displaystyle \lim_{x \to \infty}S(x)/x$ is trivially given by: $$ 1 - \frac{1}{8} \sum_{n=0}^{\infty} \frac{1}{4^n} = \frac{5}{6}. $$ Furthermore, P. Shiu has shown that $S(x) - \frac{5}{6} x = O( \log x)$.