On an inequality connected with the approximation of the number of elements in a finite sequence of a special kind (Q2857903)

Let \(F\) be an irreducible polynomial with integer coefficients. Denote NEWLINE\[NEWLINE\begin{aligned} \mathcal{A}=&\{F(pq): p,q\;\text{primes}, p\neq q, (pq, F(0))=1, pq\leq x\},\\ \mathcal{A}_d=&\{ a\in \mathcal{A}: a\equiv 0\pmod d\}. \end{aligned}NEWLINE\]NEWLINENEWLINENEWLINEThe authors of the paper consider the approximation of \(\#\mathcal{A}_d\) by the expression \((\omega(d)X)/d\). Here \(\#\mathcal{A}_d\) denotes the number elements of the set \(\mathcal{A}_d\), \(\omega(d)\) is a multiplicative function, \(d\) is supposed to be square-free and NEWLINE\[NEWLINE X=\sum\limits_{pq\leq x, p>y, q>y}1 NEWLINE\]NEWLINE with \(y=y(x)=\exp\{(\log\log x)^3\}\).