Let $b:\mathbb N\to \{0,1\}$ be the indicator function of integers that are a sum of two non-zero integer squares. Let $f(t)\in \mathbb Z[t]$ be an irreducible polynomial of degree $2$ with positive leading coefficient and not of the form $(at+b)^2+c^2$ for some rationals $a,b,c$. Then standard sieve heuristics would suggest that $$ \frac{x}{\sqrt{\log x} }\ll \sum\_{n\leq x } b(|f(t)|) \ll \frac{x}{\sqrt{\log x}}.$$ My question is: has this problem been studied before and are there any results confirming the lower bound, even for at least one polynomial $f$? The upper bounds are directly proved by the Brun or the Selberg sieve.



How often is the value of a quadratic polynomial equal to a sum of two integer squares?
This problem has been addressed in a paper of Friedlander and Iwaniec in Acta Mathematica 1978, called [Quadratic polynomials and quadratic forms](https://projecteuclid.org/journals/acta-mathematica/volume-141/issue-none/Quadratic-polynomials-and-quadratic-forms/10.1007/BF02545740.full). Under general conditions they count the number of integers $n\le x$ for which the values $g(n) = an^2+ bn+c$ (with $a>0$, and $a$, $b$, $c$ integers) may be represented by a given quadratic form $\phi(u,v) = Au^2+Buv+Cv^2$. The main result gives a lower bound of order $x/\sqrt{\log x}$ as may be expected, provided a (necessary) local condition is met.


