I'm stuck on the one (if not many) step of the proof of Corollary 6.2 (page 58) from the book "Opera de Cribro" by Friedlander and Iwaniec.
The statement is a corrollary of Brun's pure sieve as follow:
Suppose $|r_d|\leq g(d)d$ where $d\mid P(z)$. Then, $$|S(\mathcal{A},z)-XV(z)|\leq2V(z)X(\log X)^{-1}+X^\frac{3}{4},$$ for $X\geq4$ and $$4\leq z\leq X^{1/c\log(V(z)^{-1}\log X)}.$$
It is proved in the book that $$|S(\mathcal{A},z)-XV(z)|\leq\left(\frac{eG}{r}\right)^r(X+z^r)+e^r,$$ where $$G=\sum_{p\mid P(z)}g(p)\leq-\log V(z).$$ It is clear that $$z^r\leq X \quad \text{and} \quad e^r\leq X^\frac{1}{\log4}\leq X^\frac{3}{4}.$$ Thus, it remains to show that $$\left(\frac{eG}{r}\right)^r\leq\frac{V(z)}{\log X}.$$ This is where I'm stuck. Any help would be appreciated.
P.S. I'm not providing definitions for everything here due to there are a lot. My hope is that people who read this is familiar with the notations of Friedlander and Iwaniec, or similar sieve notations. However, if neeeded, I can offer more explainations.