A shifted sum for the congruent number problem (Q2303443)

An integer is called \textit{congruent} if it is the area of some right triangle with rational side lengths. Determining which integers are congruent is called the \textit{congruent number problem}. Simple scaling arguments show that the congruent number problem will be solved once it is known which squarefree numbers appear as the squarefree part of the area of a primitive (i.e., all side lengths pairwise coprime) right triangle with integral sides. Let \(\mathcal{H}_t\) denote the set of hypotenuses of dissimilar primitive right triangles with squarefree part of their area equal to \(t\). In the present paper, the articles show that the sum \(C_t = \sum_{h \in \mathcal{H}_t} 1/h\) appears as the leading coefficient in the asymptotic formula for a certain shifted sum of square-detecting arithmetic functions. In particular: Let \(\tau(n) = 1\) if \(n\) is a square and \(0\) otherwise. For a squarefree number \(t\), let \(r_t\) denote the rank of the elliptic curve \(E_t : y^2 = x^3 - t^2x\) over \(\mathbb{Q}\). For \(X > 1\), define \[ S_t(X) = \sum_{m = 1}^X \sum_{n = 1}^X \tau(m+n)\tau(m-n)\tau(m)\tau(tn). \] The main theorem of the paper is the following asymptotic formula: \[ S_t(X) = C_tX^{1/2} + O_t((\log X)^{r_t/2}). \] The proof, which is highly readable and well-structured, begins by establishing the connection between primitive Pythagorean triples and arithmetic progressions of squares. This connnection provides the relationship between the sum \(S_t(X)\) and hypotenuses. To control the error term in the asymptotic formula above, the authors employ seminal work of Néron on counting rational points of bounded height on an elliptic curve.