It is well known that $50$ % of the primes are of the form $x^2 + y^2$.
Many variants exists where a rational amount of primes is of some integer polynomial form.
But I wonder ; are there integer symmetric polynomials $P$ such that $Q$ % of the primes are of the form $P$ where $Q$ is an irrational number ?
If so , does $Q$ need to be an algebraic number ?