A function related to the Mordell-Weil rank of elliptic curves (Q6628928)

For \(p\) a prime number, this paper defines the arithmetic function \(f_p(n)\) to be the number of pairs \((a, b) \in \mathbb{N}^2\) with \(ab = n\) and \(a + b\) a quadratic residue mod \(p\). This definition is motivated by comparison to the function \(f(n)\) which counts the number of pairs \((a, b) \in \mathbb{N}^2\) with \(ab = n\) and \(a + b\) a perfect square. The latter function is motivated by observing that \(\log_2 f(n)\) provides a lower bound for the rank of the elliptic curve with Weierstrass equation \(y^2 = x^3 - n x\) whenever \(n\) is squarefree. As a consequence, if \(f(n)\) is unbounded on squarefree integers, we may settle the question ``Are there elliptic curves of unbounded rank over \(\mathbb{Q}\)?'' in the affirmative.\N\NThis motivation dispensed with, the paper presents a straightforward but entertaining argument that for each prime \(p\), we have \(\sum_{\substack{n \leq x \\ n \ \text{squarefree}}} f_p(n) \gg x \log x\). This argument uses several standard tricks from analytic number theory, including interchanges of summation, Gauss sums, and a Tauberian theorem. One suspects that more careful follow-up work could prove more exact asymptotics for this partial sum.\N\NThe authors highlight that prior work shows that \(f(n)\) is unbounded if \(n\) is not required to be squarefree, but they do not emphasize that \(\sum_{n \leq x} f(n) \asymp x^{3/4}\) by the selfsame prior work. This discrepancy between the asymptotics of \(f(n)\) and \(f_p(n)\) means that there is little hope of extracting asymptotics about the average order of \(f(n)\) from their work on the average order of \(f_p(n)\), no matter how clever the sieve theory involved.