Some mathematical remarks on the polynomial selection in NFS (Q2826687)

The authors of the paper consider the proportion of friable values of a binary quadratic form \(F\in\mathbb{Z}[x_1,x_2]\) on sets of certain special construction. An integer \(n\) is said to be \(y\)-friable if its greatest prime factor \(P(n)\) satisfies the inequality \(P(n)\leqslant y\). Let \(x\mathcal{K}\) denote the set obtained from the compact set \(\mathcal{K}\) rescaling it by the factor \(x\). Denote \(u=\log x/\log y\) and NEWLINE\[NEWLINE\begin{aligned} \Psi_F(x\mathcal{K}, y)=&\left|\left\{(n_1,n_2)\in x\mathcal{K}\cap\mathbb{Z}^2: \text{gcd}(n_1,n_2)=1, P(F(n_1,n_2))|\leqslant y\right\}\right|,\\ \Psi(x,y)=& \left|\left\{1\leqslant n\leqslant x: P(n)\leqslant y\right\}\right|.\end{aligned}NEWLINE\]NEWLINE The following assertion is one of the main results of the paper.NEWLINENEWLINELet \(F\in\mathbb{Z}[x_1,x_2]\) be an irreducible quadratic form such that discriminant is negative and fundamental and let NEWLINE\[NEWLINE \mathcal{K}_F=\{(x_1,x_2)\in \mathbb{R}^2: |F(x_1,x_2)|\leqslant 1\}. NEWLINE\]NEWLINE Then, for some positive \(\kappa\) and an arbitrary positive \(\varepsilon\) one has that NEWLINE\[NEWLINE \frac{\Psi_F(x\mathcal{K}_F, y)}{\Psi_F(x\mathcal{K}_F, \infty)}=\frac{\Psi(x^2\text{e}^{\alpha},y)}{\Psi(x^2\text{e}^{\alpha},\infty)}\left(1+O\left(\frac{\log^2(u+1)}{\log^2y}\right)\right) NEWLINE\]NEWLINE uniformly in the domain NEWLINE\[NEWLINE x\geqslant 3,\;\;\exp\left\{(\log\log x)^{5/3+\varepsilon}\leqslant y\leqslant x^2(\log x)^{-\kappa}\right\}, NEWLINE\]NEWLINE where \(\alpha\) denotes the quantity related with Murphy's \(\alpha\) function.NEWLINENEWLINEThe proof of the above assertion is based on a extremely deep analysis of properties of Murphy's \(\alpha\) function.