= Solution
Let $L=\sqrt D$. Regroup the finite <divisor sum> using the <least common multiple>:
$$
\sum_{d\mid n}\lambda_d
=\sum_{[u,v]\mid n}\rho_u\rho_v
=\left(\sum_{d\mid n}\rho_d\right)^2.
$$
This is nonnegative. If $n$ has no <prime factor> in $\mathcal P$, only $d=1$ can contribute to the inner <sum>, so it equals $\rho_1=1$. This proves the required pointwise <upper-bound sieve> inequality. Whenever a term defining $\lambda_d$ is nonzero, $u,v\leq L$ and their <prime factors> lie in $\mathcal P$. Hence $d=[u,v]\leq uv\leq L^2=D$, and every <prime factor> of $d$ lies in $\mathcal P$. Thus these are \b[upper-bound sieve weights of level $D$], as described by the <Selberg least-common-multiple weights>.
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