In the "Sieve methods" notes of Dimitris Koukoulopoulos (see http://www.dms.umontreal.ca/~koukoulo/documents/notes/sievemethods.pdf), the following useful result can be found:

**Theorem 0.4.1.** *Let $g$ be a multiplicative arithmetic function with nonnegative values and such that $g(p) \leq C_5 / p$ for all prime numbers $p$, and
$$-L + k \log z \leq \sum_{p \leq z} g(p) \log p \leq C_6 + k \log z ,$$
for some constants $k, L, C_5, C_6 > 0$.
Then
$$\sum_{n \leq z} \mu^2(n) g(n) = \frac{\mathfrak{S}(z)}{\Gamma(k + 1)} \cdot (\log z)^k \cdot \left\{1 + O_{k, C_5, C_6}\!\left(\frac{L}{\log z}\right)\right\}$$,
for $z \geq 2$, where
$$\mathfrak{S}(z) := \prod_{p \leq z}(1 + g(p))\left(1 - \frac1{p}\right)^k.$$*
I am looking for a standard reference (a book or a published article) containing Theorem 0.4.1, or a result from which Theorem 0.4.1 follows quickly.

Thank you for any suggestion.

Sieve Methods, Chapter 5, specifically Lemma 5.4. $\endgroup$ – so-called friend Don Jan 30 '17 at 20:01