Elliptic curves with square-free \(\Delta\) (Q2800156)

Let \(A,B\in \mathbb Z\) and \(E_{A,B}\) be the elliptic curve defined by the equation: \(y^2=x^3+Ax+B\). The discriminant of \(E_{A,B}\) is \(-16\Delta_{A,B}\) with \(\Delta_{A,B}=4A^3+27B^2\). It is an important but unsolved question whether \(\Delta_{A,B}\) is prime infinitely often. For the density of square free \(\Delta_{A,B}\), in [J. Reine Angew. Math. 680, 69--151 (2013; Zbl 1295.11058)], the author and \textit{T. D. Browning} showed the following estimation: NEWLINE\[NEWLINE \sum_{H(A,B)\leq X}\mu^2(\Delta_{A,B})=4X^{10}\prod_{p\text{ prime}}\left(1-\frac{\sigma(p^2)}{p^4}\right)+O(X^{7+\varepsilon}), NEWLINE\]NEWLINE for any positive number \(\varepsilon\), where \(H(A,B)\) is the exponential height of \(E_{A,B}\), \(\sigma(p^2)=\sharp\{\alpha,\beta:\Delta_{\alpha,\beta}\equiv 0 \mod p^2\}\) and \(\mu(n)\) is the Möbius function with the convention that \(\mu(0)=0\) and \(\mu(-n)=\mu(n)\).NEWLINENEWLINEIn this article, under the Riemann hypothesis for Dirichlet \(L\)-functions, by introducing a Schwartz class function \(\Gamma\) on \(\mathbb R\) such that \(\hat \Gamma(0)=1\) where \(\hat \Gamma\) is the Fourier transformation of \(\Gamma\), the author presents a smoothed version of the above result with an improved error term: NEWLINE\[NEWLINE \sum_{A,B}\Gamma(A/X^4)\Gamma(B/X^6)\mu^2(\Delta_{A,B})=\frac{X^{10}}3\prod_{p>3}\left(1-\frac{2p-1}{p^3}\right)+O(X^{7-5/27+\varepsilon}). NEWLINE\]NEWLINE The improvement is obtained from evaluating explicitly the complete exponential sums to square moduli that appear in the estimation instead of bounding them, and from saving a factor by averaging the moduli.