I am studying about Selberg Sieve and I have a question regarding deriving an equation .
Consider S(x, z) = $\sum_{n \leq x} \delta ((gcd (n, P(z)) )$.
$P(z) = \prod_{p\leq z} p$, $\delta(m)$ = {1 if m =1 , 0 if m>1}
Given parameters {$\lambda_d$} such that (1) $\lambda_d$ is real (II) $\lambda_1 =1$.
then $\delta(m)$ = {1 if m =1 , 0 if m>1} $\leq (\sum_{d|m} {\lambda_d}^2)$.
It can be shown easily that $S(x, z)\leq \sum_{n \leq x} (\sum_{d|n , d|P(z) } \lambda_d)^2$.
The RHS of above equation was minimized by Selberg.
To do so we take $\lambda_d$ supported on integers d$\leq $z, ie assuming $\lambda_{d}=0$ if d>z.Then $S(x, z) \leq \sum_{n\leq x} (\sum_{d|n , d|P(z) }( \lambda_{d} )^2) $.
But I don't understand the next step where $\sum_{n\leq x} ( \sum_{d|n , d| P(z) } {\lambda_d}^2)$=
$\sum_{d_1 , d_2 \leq x, squarefree} \lambda_{d_1} \lambda_{d_2} $× number of elements in the set{ $n \leq x : d_1 | n, d_2 |n$}.
Can you please tell how to deduce the equality?