= Solution
The remainders in the <sieve distribution> are defined by
$$
r(d)=A_d-Xg(d),\qquad A_d=\sum_{d\mid n}a_n.
$$
Multiply the pointwise inequality from part (a) by $a_n\geq0$ and sum. Finite rearrangement gives
$$
S\leq\sum_d\lambda_dA_d
=X\sum_d\lambda_dg(d)+\sum_{d\leq D}\lambda_dr(d).
$$
To identify the main quadratic form, put $h(t)=\prod_{p\mid t}(g(p)^{-1}-1)$. For <squarefree integers> $u,v$, <multiplicativity> gives
$$
\sum_{t\mid(u,v)}h(t)=\prod_{p\mid(u,v)}g(p)^{-1},\qquad
g([u,v])=g(u)g(v)\sum_{t\mid(u,v)}h(t).
$$
The second identity is also valid if $u$ or $v$ is not <squarefree>: both sides then vanish. Therefore the <Selberg sieve diagonalization> gives
$$
\sum_d\lambda_dg(d)
=\sum_{u,v\leq L}\rho_u\rho_vg([u,v])
=\sum_{t\leq L}h(t)\left(\sum_{\substack{m\leq L\\t\mid m}}g(m)\rho_m\right)^2
=\Sigma.
$$
This proves $S\leq X\Sigma+\sum_{d\leq D}\lambda_dr(d)$. Notice that the hypotheses do not force $\rho_m$ to vanish on non-<squarefree integers>; those coefficients simply make no contribution to this main quadratic form because $g(m)=0$.
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