For natural numbers $a,b$, let $a$ be odd and let $f(a)$ be the number of $\,b<a\,$ such that $a^2+b^2$ is prime. In the diagram $f$ is blue, the function $g(n)=\frac{n}{\ln n}$ is red and $h(n)=\frac{1}{\pi}\cdot\frac{n}{\ln n}$ is green.
Since the blue curve is nearly limited by the red and the green curves it seems reasonable to make the conjecture:
There are real numbers $A$ and $B$ such that $A\cdot\frac{n} {\ln n}<f(n)<B\cdot\frac{n}{\ln n}$, where $n$ is odd.
What would be the intuition behind this?