[proofplan]
We use the general Rosser-Iwaniec beta-sieve theorem as an external prerequisite. First we state the needed form of that theorem with arbitrary sieve dimension $\kappa$. Then we verify that the present hypotheses are exactly its $\kappa=1$ hypotheses. Finally we identify the general sifting functions with the standard dimension-one functions $F$ and $f$, and use their exponential normalization to obtain the final $1+O(e^{-s})$ consequence.
[/proofplan]
[step:State the external beta-sieve input]
We use the following standard Rosser-Iwaniec beta-sieve theorem as a prerequisite. For a fixed dimension parameter $\kappa>0$, suppose the local densities satisfy $0\le \omega(p)<p$, a uniform local bound $\omega(p)/p\le 1-1/L$, and the dimension-$\kappa$ product estimate
\begin{align*}
\prod_{\substack{w \le p < z,\ p \in \mathcal P}}\left(1-\frac{\omega(p)}{p}\right)^{-1} \le A\left(\frac{\log z}{\log w}\right)^\kappa
\end{align*}
for all $2\le w<z$. If $s=\log D/\log z$ satisfies the usual beta-sieve level restriction
\begin{align*}
s\le \frac{\log D}{(\log\log D)^2}
\end{align*}
for sufficiently large $D$, then the theorem constructs Rosser-Iwaniec beta-sieve weights of level $D$ and functions $F_\kappa$ and $f_\kappa$ such that
\begin{align*}
\sum_{d\mid P(z)}\lambda_d^+\frac{\omega(d)}{d}=V(z)\{F_\kappa(s)+O(e^{-s})\}
\end{align*}
for $s\ge 1$, with the implicit constant depending only on $L$, $A$, and $\kappa$, and
\begin{align*}
\sum_{d\mid P(z)}\lambda_d^-\frac{\omega(d)}{d}=V(z)\{f_\kappa(s)+O_\eta(e^{-s})\}
\end{align*}
for each fixed $\eta>0$ and $s\ge 2+\eta$, with the lower implicit constant also allowed to depend on $\eta$. This external theorem is broader than the present statement: it treats arbitrary sieve dimension $\kappa$, while the present theorem records only the linear-sieve case $\kappa=1$.
[/step]
[step:Specialize the beta-sieve theorem to dimension one]
The hypotheses in the statement are precisely the preceding theorem's hypotheses with $\kappa=1$. The product condition becomes
\begin{align*}
\prod_{\substack{w \le p < z,\ p \in \mathcal P}}\left(1-\frac{\omega(p)}{p}\right)^{-1} \le A\frac{\log z}{\log w},
\end{align*}
which is exactly the displayed dimension-one regularity assumption. The local bounds $0\le \omega(p)<p$ and $\omega(p)/p\le 1-1/L$ are also stated explicitly. The level parameter in the external theorem is the same
\begin{align*}
s=\frac{\log D}{\log z},
\end{align*}
and the required restriction on $s$ is the displayed hypothesis
\begin{align*}
s\le \frac{\log D}{(\log\log D)^2}
\end{align*}
for all sufficiently large $D$. Therefore the upper part of the beta-sieve theorem gives
\begin{align*}
\sum_{d\mid P(z)}\lambda_d^+\frac{\omega(d)}{d}=V(z)\{F_1(s)+O(e^{-s})\}
\end{align*}
for $s\ge 1$, and the lower part gives
\begin{align*}
\sum_{d\mid P(z)}\lambda_d^-\frac{\omega(d)}{d}=V(z)\{f_1(s)+O_\eta(e^{-s})\}
\end{align*}
for each fixed $\eta>0$ and $s\ge 2+\eta$. By definition of the standard dimension-one linear-sieve functions used in the statement, $F_1=F$ and $f_1=f$. This proves the two displayed estimates.
[/step]
[step:Derive the final asymptotic form]
It remains only to justify the last consequence. The statement normalizes the dimension-one sifting functions by
\begin{align*}
F(s)=1+O(e^{-s})
\end{align*}
and
\begin{align*}
f(s)=1+O(e^{-s})
\end{align*}
as $s\to\infty$. Substituting the first relation into the upper estimate gives
\begin{align*}
\sum_{d\mid P(z)}\lambda_d^+\frac{\omega(d)}{d}=V(z)\{1+O(e^{-s})\}
\end{align*}
in the upper admissible range. Substituting the second relation into the lower estimate gives the same form for $\lambda_d^-$ in the lower admissible range, with the constant allowed to depend on the fixed $\eta$. This is the asserted final estimate.
[guided]
The point of using the beta-sieve theorem is that it is a more general prerequisite than the result being proved here. It treats arbitrary dimension $\kappa$, and the present theorem is obtained by setting $\kappa=1$. With this specialization, the beta-sieve product estimate is exactly
\begin{align*}
\prod_{\substack{w \le p < z,\ p \in \mathcal P}}\left(1-\frac{\omega(p)}{p}\right)^{-1} \le A\frac{\log z}{\log w}
\end{align*}
for every $2\le w<z$. The level relation is also the same:
\begin{align*}
s=\frac{\log D}{\log z} \le \frac{\log D}{(\log\log D)^2}
\end{align*}
for all sufficiently large $D$.
The upper beta-sieve conclusion at $\kappa=1$ is
\begin{align*}
\sum_{d\mid P(z)}\lambda_d^+\frac{\omega(d)}{d}=V(z)\{F(s)+O(e^{-s})\},
\end{align*}
for $s\ge 1$. The lower beta-sieve conclusion is
\begin{align*}
\sum_{d\mid P(z)}\lambda_d^-\frac{\omega(d)}{d}=V(z)\{f(s)+O_\eta(e^{-s})\},
\end{align*}
for every fixed $\eta>0$ and $s\ge 2+\eta$.
The final simplification uses the exponential normalizations
\begin{align*}
F(s)=1+O(e^{-s})
\end{align*}
and
\begin{align*}
f(s)=1+O(e^{-s})
\end{align*}
as $s \to \infty$. Combining these exponential estimates with the two Rosser-Iwaniec asymptotics gives
\begin{align*}
\sum_{d\mid P(z)}\lambda_d^\pm\frac{\omega(d)}{d}=V(z)\{1+O(e^{-s})\}
\end{align*}
in the corresponding admissible range, with the lower-sieve constant allowed to depend on the fixed $\eta$.
[/guided]
[/step]
