# Sum of multiplicative arithmetic function over squarefree numbers

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.

• Results of this type are often put under the umbrella of "Wirsing's theorem". Unfortunately I can't provide a precise reference like you want, but hopefully this will help you isolate one. – Daniel Loughran Jan 30 '17 at 17:59
• Have a look at Halberstam and Richert's Sieve Methods, Chapter 5, specifically Lemma 5.4. – so-called friend Don Jan 30 '17 at 20:01