For a pair of positive integers $m$ and $n$, put

$$\displaystyle \Psi(m,n) = 2^{-\omega(m)} \sum_{a | m} \left(\frac{n}{a}\right),$$ where $\left(\frac{\cdot}{\cdot}\right)$ is the Jacobi symbol. Let $\sum^\ast$ denote a sum over square-free numbers. Let $X$ be a positive number. Can one evalute the sum


$$\displaystyle \sideset{}{^\ast}\sum_{\substack{D \leq X \\ d_1 d_2 d_3 = D \\ d_1 - d_2 + d_3 = 0}} \Psi(d_1, d_2) \Psi(d_2,d_3) \Psi(d_3,d_1)?$$

A similar, but more straightforward question, was addressed by Fouvry and Kluners in this paper, where they evaluated the sum

$$\displaystyle \sideset{}{^\ast} \sum_{\substack{D \leq X \\ d_1 d_2 = D}} \Psi(d_1, d_2)\Psi(d_2,d_1).$$

The cyclic nature of the sum (1) leads me to believe that a similar argument should work, but the presence of the linear relation between the divisors $d_1, d_2, d_3$ complicates things.


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