Number of quadratic residues for $p=4k+3$ and $p=4k+1$ less than $p/2$?

Question

I know the number of quadratic residues less than $p/2$ for $p=4k+1$ is $p+1/4$, would it be the same result for $p=4k+3$?

Answer 1

For $p=4k+1$, the number of quadratic residues less than $\frac{p}{2}$ is $\frac{p-1}{4}$. In other words there are equal numbers of them less than $\frac{p}{2}$ and more than $\frac{p}{2}$. This follows from the fact that $-1$ is a quadratic residue and therefore each quadratic residue $i$ is paired with a quadratic residue $p-i$.

There is no simple result for $p=4k+3$. However, if you check various primes, you will find that for these primes more than half the residues are less than $\frac{p}{2}$.