In two important papers of Wirsing, namely "Das asymptotische Verhalten von Summen über multiplikative Funktionen" (1961) and its follow up (1967), several results on mean values of multiplicative functions are obtained. Specifically, Satz 1.1 and Satz 1.2.2 in the latter paper say the following. Suppose $f$ is a non-negative multiplicative function, satisfying $$\sum_{p \le x} \frac{\log p}{p} f(p) \sim \tau \log x$$ for some positive $\tau$ (here the sum is over primes). Suppose further there exists $C>0$ with $f(p)\le C$ for all primes $p$. Under some restrictions on the growth of $f(p^k)$ (which I shall not write down), one has $$(*) \sum_{n \le x} f(n) =\frac{e^{-\gamma \tau}}{\Gamma(\tau)} \frac{x}{\log x} \prod_{p \le x} ( \sum_{k \ge 0} f(p^k)/p^k) (1+o(1)),$$ where $\gamma$ is the Euler-Masscheroni constant. This is Satz 1.1. If furthermore we have a multiplicative, real function $g$ with $|g(n)|\le f(n)$ for all $n$, then Satz 1.2.2 says $$(**) \frac{\sum_{n \le x} g(n)}{\sum_{n \le x} f(n)} = \prod_{p} \frac{\sum_{ k\ge 0} g(p^k)/p^k}{\sum_{k \ge 0} f(p^k)/p^k} (1+o(1))$$
Q1: Is there any result making the dependence on $f$ and $g$ of the error term in $(**)$ explicit (perhaps under additional restrictions)?
For the applications I have in mind, I do not need a bound on the error term itself, but something weaker:
Q2: Is there a sense in which $(**)$ is uniform in $g$? Concretely, do we have $$\sup_{|g| \le f} \left| \frac{\sum_{n \le x} g(n)}{\sum_{n \le x} f(n)} / \prod_{p} \frac{\sum_{ k\ge 0} g(p^k)/p^k}{\sum_{k \ge 0} f(p^k)/p^k} - 1 \right| \to 0,$$ where the supremum is taken over all real multiplicative functions $g$ with $|g(n)| \le f(n)$? If not, is there some natural subset of these functions for which we do have uniformity?
