[proofplan]
We verify that the stated local-density data match the normalized Rosser-Iwaniec dimension-one fundamental lemma assumed in the theorem statement. The hypotheses give positivity of the Euler factors, squarefree support, level $D$, and the dimension-one product bound with constant $K$. The fundamental lemma then gives the upper and lower main terms in the same normalization of $V$, $F$, $f$, and the Rosser-Iwaniec weights, with constants depending only on $K$.
[/proofplan]
[step:Verify the local density and weight hypotheses]
The condition $0 \leq \omega(p) < p$ for every prime $p < z$ gives
\begin{align*}
0 < 1 - \frac{\omega(p)}{p} \leq 1
\end{align*}
for every prime $p < z$, so $V(z)$ is a positive finite Euler product. Since $\omega: \mathbb{N} \to [0,\infty)$ is multiplicative and supported on squarefree integers, the values entering the sums over $d \mid P(z)$ are exactly the squarefree local products determined by the prime densities $\omega(p)/p$. The functions $\lambda^+: \mathbb{N} \to [-1,1]$ and $\lambda^-: \mathbb{N} \to [-1,1]$ are, by hypothesis, the Rosser-Iwaniec upper and lower linear-sieve weights of level $D$ in the same normalization as the assumed fundamental lemma. In particular, they are supported on squarefree divisors of $P(z)$ and vanish for $d > D$.
[/step]
[step:Check the dimension-one sieve condition]
For all real $w$ and $y$ with $2 \leq w < y \leq z$, the theorem assumes
\begin{align*}
\prod_{w \leq p < y}\left(1 - \frac{\omega(p)}{p}\right)^{-1}
\leq \frac{\log y}{\log w}\left(1 + \frac{K}{\log w}\right).
\end{align*}
This is precisely the dimension-one upper Mertens-type condition required by the normalized Rosser-Iwaniec fundamental lemma assumed in the theorem statement, with dimension parameter $1$ and sieve constant $K$. The sieve ratio used by that lemma is
\begin{align*}
s = \frac{\log D}{\log z},
\end{align*}
which is the parameter appearing in the present statement.
[guided]
The point of this step is to match the abstract hypotheses of the linear sieve with the symbols in this theorem. The Euler factors are legitimate because $0 \leq \omega(p) < p$ implies
\begin{align*}
0 < 1 - \frac{\omega(p)}{p} \leq 1.
\end{align*}
Thus the reciprocal product in the dimension condition is well-defined. The assumed inequality
\begin{align*}
\prod_{w \leq p < y}\left(1 - \frac{\omega(p)}{p}\right)^{-1}
\leq \frac{\log y}{\log w}\left(1 + \frac{K}{\log w}\right)
\end{align*}
for every $2 \leq w < y \leq z$ is the dimension-one hypothesis: the product grows like one power of $\log y / \log w$, with the permitted error controlled by $K$. The level parameter in the Rosser-Iwaniec weights is $D$, and the sieving range is determined by primes $p < z$, so the corresponding sieve ratio is exactly
\begin{align*}
s = \frac{\log D}{\log z}.
\end{align*}
These verifications are the hypotheses needed before the normalized Rosser-Iwaniec fundamental lemma assumed in the theorem statement can be applied.
[/guided]
[/step]
[step:Apply the normalized Rosser-Iwaniec fundamental lemma]
The normalized Rosser-Iwaniec dimension-one fundamental lemma assumed in the theorem statement applies to the data verified above. Its hypotheses are the positivity of the local Euler factors, the squarefree multiplicative density $d \mapsto \omega(d)/d$ on divisors of $P(z)$, the level-$D$ Rosser-Iwaniec upper and lower weights, and the dimension-one product bound with constant $K$; these were verified in the preceding steps. It yields constants $C_+(K)>0$ and $C_-(K)>0$, depending only on $K$, such that the upper weights satisfy
\begin{align*}
\sum_{d \mid P(z)} \lambda^+(d) \frac{\omega(d)}{d}
\leq V(z)\left(F(s) + C_+(K)(\log D)^{-1/6}\right)
\end{align*}
and the lower weights satisfy
\begin{align*}
\sum_{d \mid P(z)} \lambda^-(d) \frac{\omega(d)}{d}
\geq V(z)\left(f(s) - C_-(K)(\log D)^{-1/6}\right).
\end{align*}
Renaming $C_+(K)$ and $C_-(K)$ as $C_+$ and $C_-$ gives the asserted constants and completes the proof.
[/step]
