I am reading the book "Opera de Cribro - John B. Friedlander, Henryk Iwaniec" and in pages 5,6 I do not understand why and how they chose $X$, $A(x)$, $A_d(x)$, $g(p)$ and $r_d(x)$.
any hints will be appreciated.
I am reading the book "Opera de Cribro - John B. Friedlander, Henryk Iwaniec" and in pages 5,6 I do not understand why and how they chose $X$, $A(x)$, $A_d(x)$, $g(p)$ and $r_d(x)$.
any hints will be appreciated.
Your questions are answered by (1.2) and the preceding display in the book along with the surrounding text. In the specific examples that you ask about, $A(x)$ is the number of elements of $\mathcal{A}$ up to $x$, while $A_d(x)$ is the number of elements of $\mathcal{A}$ up to $x$ that are divisible by $d$. Once you know $A_d(x)$, you need to approximate it reasonably with an expression of the form $g(d)X$, where $g(d)$ is a multiplicative function independent of $x$. In particular, $X\approx A(x)$, since $g(1)=1$. Finally, $r_d(x)$ is the error of your approximation, so that $A(x)=g(d)X+r_d(x)$ as (1.2) states. Apriori, you can choose $g(d)$ and $X$ arbitrarily, but you need some bounds on $r_d(x)$ in order to sieve $\mathcal{A}$ efficiently.