Squarefree density of polynomials II

In this memoir we study the density of squarefree numbers in the set of numbers of the form \(\begin{aligned} \mathcal P(x,y) = bx^3+cy^k \end{aligned}\) $$\begin{array}{c}\hfill \mathcal{P}(x,y)=b{x}^3+c{y}^k\end{array}$$ where b, c are a non-zero integers with \((b,c)=1\) $(b,c)=1$ and \(k\ge 2\) $k\ge 2$ . Suppose that \(\theta \) $\theta$ is a constant with \(0 < \theta \le 3/k\) $0<\theta \le 3/k$ , that X is large, and \(Y\asymp X^{\theta }\) $Y\asymp X^{\theta }$ . Then take \(N_\mathcal {P}(X,Y)\) $N_{\mathcal{P}}(X,Y)$ to be the number of pairs of integers x and y with \(|x|\le X\) $|x|\le X$ , \(|y|\le Y\) $|y|\le Y$ such that \(\mathcal P(x,y)\) $\mathcal{P}(x,y)$ is squarefree. Suppose also that \(\rho _\mathcal {P}(d)\) ${\rho }_{\mathcal{P}}(d)$ is the number of solutions of \(\mathcal P(x,y)\equiv 0\hspace{3.33328pt}({\textrm{mod}}\,\,d)\) $\mathcal{P}(x,y)\equiv 0(\text{mod}d)$ . Then we show that \(\begin{aligned} N_\mathcal {P}(X,Y) = 4XY \mathfrak S_\mathcal {P} + O\left( XY(\log X)^{-1/2}\right) . \end{aligned}\) $$\begin{array}{c}\hfill N_{\mathcal{P}}(X,Y)=4XY{\mathfrak{S}}_{\mathcal{P}}+O\left(X,Y,{(logX)}^{-1/2}\right).\end{array}$$ where \(\mathfrak S_\mathcal {P}\) ${\mathfrak{S}}_{\mathcal{P}}$ is the anticipated density \(\begin{aligned} \mathfrak S_\mathcal {P} = \prod _p \left( 1-\frac{\rho _\mathcal {P}(p^2)}{p^4} \right) . \end{aligned}\) $$\begin{array}{c}\hfill {\mathfrak{S}}_{\mathcal{P}}=\prod _p\left(1,-,\frac{{\rho }_{\mathcal{P}}(p^2)}{p^4}\right).\end{array}$$ When \(k\ge 5\) $k\ge 5$ this is new.