In two important papers of Wirsing, namely ["Das asymptotische Verhalten von Summen über multiplikative Funktionen"](https://link.springer.com/article/10.1007/BF01351892) (1961) and its [follow up](https://akademiai.com/doi/abs/10.1007/BF02280301?pubCode=mobile&journalCode=10473) (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
$$(\*\*) \frac{\sum\_{n \le x} g(n)}{\sum\_{n \le x} f(n)} = \prod\_{p} \frac{\sum\_{ k\ge 0} g(p^k)/p^k}{\sum\_{k \ge 0} f(p^k)/p^k} (1+o(1))$$



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> Q1: Is there any result making the dependence on $f$ and $g$ of the error term in $(\*\*)$ explicit (perhaps under additional restrictions)?
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For the applications I have in mind, I do not need a bound on the error term itself, but something weaker:



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> 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?
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Uniformity in Wirsing's Mean Value Theorems
Effective versions of Wirsing's theorems are worked out in a recent paper of G. Tenebaum: ``Moyennes effectives de fonctions multiplicatives complexes'', published in Ramanujan J. 44 (2017), no. 3, 641–701. Questions 1 and 2 are resolved by his Théorème 1.3. He shows that error term is uniform in the sense that it depends only on certain (simple) invariants of the functions at hand.


