Number of Natural Numbers Less than x which are Squares or Sums of Two Squares

Theorem
Let $x$ be a real number.

The number of natural numbers smaller than $x$ which are either square or the sum of $2$ squares is given by the expression:


 * $\map N x \approx \dfrac {k x} {\sqrt {\ln x} }$

where $k$ is given by:
 * $k = \sqrt {\dfrac 1 2 \displaystyle \prod_{\substack {r \mathop = 4 n \mathop + 3 \\ \text {$r$ prime} } } \paren {1 - \dfrac 1 {r^2} }^{-1} }$

The number $k$ is known as the Landau-Ramanujan constant: