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.
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$\begingroup$ It all depends on the shape of the polynomial. For example, oddly enough, there are always solutions for the curves of triangular numbers. math.stackexchange.com/questions/794510/… If there are variables in the polynomial 2, then I think there will always be solutions. $\endgroup$– individCommented Jan 18, 2022 at 5:56
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This problem has been addressed in a paper of Friedlander and Iwaniec in Acta Mathematica 1978, called Quadratic polynomials and quadratic forms. 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.
