Prime values of \(f(a,b^2)\) and \(f(a,p^2)\), \(f\) quadratic (Q6610068)

The author of the paper considers the asymptotic formulas for primes of the shape \(f(a,b^2)\) with integers \(a,b\) and of the shape \(f(a,p^2)\) with integer \(a\) and prime \(p\). The following assertion is one of the main results proved in the paper.\N\NLet \(f(x,y)\in\mathbb{Z}[x,y]\) be an irreducible and primitive binary quadratic form with the property that \(f(x,1)\hspace{1mm}{\not}{\equiv}\ x(x+1)\pmod 2\). Then, for \(f\) positive definite, it holds that \N\[\N\sum_{\substack{m,l\in\mathbb{Z}\\\N0<f(m,l^2)\leqslant X}}\lambda(l)\lambda(f(m,l^2))=\frac{\nu_f\mathfrak{G}_f X^{3/4}}{(\log X)^2}\bigg(1+O\bigg(\frac{\log\log X}{\log X}\bigg)\bigg). \N\]\NHere \(\lambda\) is the prime indicator function,\N\begin{align*}\N\nu_f&=\prod_{p\, {\nmid} \, \Delta(f)}\bigg(1-\frac{\rho_f(p)}{p}\bigg)\bigg(1-\frac{1}{p}\bigg)^{-1}\prod_{p|\Delta(f)}\bigg(1-\frac{1}{p}\bigg)^{-1},\\\N\rho_f(p)&=\#\big\{x\,\pmod p: f(x,1)\equiv 0\,\pmod p\big\},\\\N\mathfrak{G}_f&= \mathrm{Area}\big\{(x,y)\in\mathbb{R}^2: f(x,y^2)\leqslant 1\big\},\N\end{align*}\Nand \(\Delta(f)\) is the discriminant of the quadratic form \(f\).