This seems so obvious that someone must have investigated it before, but I couldn't turn up anything substantive after a few Google searches.
Let $n$ be an odd natural number. It is a well-known fact that the Jacobi symbol $\left(\frac{a}{n}\right)=-1$ implies $a$ is not a QR modulo $n$, but that $\left(\frac{a}{n}\right)=1$ does not imply $a$ is a QR modulo $n$. I'm interested in all of the "natural questions" about the discrepancy $s_n=\frac{\#\{a\in(\mathbb{Z}/n\mathbb{Z})^\times\mid\left(\frac{a}{n}\right)=1\}-\#\{a\in(\mathbb{Z}/n\mathbb{Z})^\times\mid a\text{ is a QR}\}}{\phi(n)}.$ For example, can $s_n$ get arbitrarily close to 1? If not, what is $\sup(s_n)$? Given $r\in\mathbb{Q}\cap[0,1)$, is there always an $n$ such that $r=s_n$? Given $r\in\mathbb{Q}\cap[0,1)$, what is the asymptotic density of $\{n\in\mathbb{N}\mid s_n=r\}$? Are there $r$ such that this asymptotic density is positive?
Here is a graph I produced of $s_n$ for odd $1\leq n\leq 5000$:
The values that occurred were: $0, \frac{1}{4}, \frac{3}{8}, \frac{7}{16}, \frac{1}{2}, \frac{3}{4}$, which would certainly seem to point to there only being a few possible values of $s_n$; however, I didn't check any of the $n\geq 5000$, of which there are a substantial number...