Let $\tau$ be the divisor function, that is $$ \tau(n)=\sharp\{d \in \mathbb{N}, d|n\}. $$

I was wondering if anyone has ever proved an asymptotic estimate for the sum $$S(x):=\sum_{p,q\leq x}\tau(p^2+q^2),$$ where the summation is taken over pairs of primes.

One obviously expects $$S(x)\sim c\frac{x^2}{\log x}$$ as $x \to \infty,$ where $c$ is a positive constant which is an infinite product of Euler factors.

This is based on the heuristic that each of the $\pi(x)^2$ terms present in $S(x)$ is approximated by a constant multiple of $\log x$ on average.

Brun-Titchmarsch and Bombieri-Vinogradov can be used to prove the upper and the lower bound $$ c\frac{x^2}{\log x} (\frac{1}{2}+o(1)) \leq S(x) \leq c\frac{x^2}{\log x} (2+o(1)), $$ as $x\to \infty$ respectively.

But the question remains, $\textit{can we prove an asymptotic?}$

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