In the breakthrough work of the proof of weak twin prime conjecture, Goldstone, Pintz and Yildirim as well as Zhang use the following modified Selberg sieve:
$v=\lambda^2$
where $\lambda(n)$takes the form
$\lambda(n) = \sum_{d\in S_I:d|P(n)} \mu(d)a_d$
and choose $a_d := g\left(\frac{\log d}{\log R}\right)$
where g is A smooth function
I just ask a question that why is $a_d$ choosen in this form, what is the motivation for that? Thanks!