Let $r_2(n)$ denote the number of ways in which a positive integer $n$ is expressed as the sum of two squares (integers). I would like to know if there is a result that gives the exact behavior of (as $X\to\infty$) $$ \sum_{n\leq X}r_2(n^2+d^2) $$ for a fixed $d\in\mathbb{N}$. I think that it is expected to have the behaviour $A(d)X \log X + B(d) X + ERROR$. I am interested in the constants $A(d)$ and $B(d)$. Do you know if some paper establishes this result?
Thank you very much
Just to expand a bit on Lucia's comment: It is well known that $$r_2(n) = 4(\tau_1(n)-\tau_3(n)) = 4\tau(n) - 8\tau_3(n)$$ Where $d\tau_1(n)$ is the number of divisors of $n$ that are $1\mod4$, $\tau_3(n)$ likewise for divisors that are $3\mod4$, and $\tau(n)$ is the ordinary divisor function.
The paper referenced in Lucia's comment shows that $$\sum_{n\leq X}{\tau(n^2+d^2)} = A(d)X\log X + B(d)X + O(x^{8/9}\log^3(x))$$ So it will suffice to consider the sum $$\sum_{n\leq X}{\tau_3(n^2+d^2)}$$ However, notice that $\tau_3(n^2+d^2) = \tau_3(\gcd(n,d)^2)$, because $\tau_3$ is multiplicative and no prime number $q=3\mod4$ divides a sum of two coprime squares. So $\tau_3(n^2+d^2)$ is periodic$\mod n$, and therefore our sum is approximately linear (with an error of $O(1)$) as requested.
