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Given an arithmetic function $g(n)$ write $M_g(x) := \sum_{n \leq x} g(n)$.We extend and strengthen the results of a fundamental paper of Hal\'{a}sz inseveral ways by proving upper bounds for the ratio of$\frac{|M_g(x)|}{M_{|g|}(x)}$, for any strongly multiplicative, complex-valuedfunction $g(n)$ under certain assumptions on the sequence $\{g(p)\}_p$. Wefurther prove an asymptotic formula for this ratio in the case that$|\text{arg}(g(p))|$ is sufficiently small uniformly in $p$. In so doing, werecover a new proof of an explicit lower mean value estimate for $M_{f}(x)$ forany non-negative, multiplicative function satisfying $c_1 \leq |f(p)| \leq c_2$for $c_2 \geq c_1 > 0$, by relating it to $\frac{x}{\log x}\prod_{p \leq x}\left(1+\frac{f(p)}{p}\right)$. As an application, we generalize our maintheorem in such a way as to give explicit estimates for the ratio$\frac{|M_g(x)|}{M_{f}(x)}$, whenever $f: \mathbb{N} \rightarrow (0,\infty)$and $g: \mathbb{N} \rightarrow \mathbb{C}$ are strongly multiplicativefunctions that are uniformly bounded on primes and satisfy $|g(n)| \leq f(n)$for every $n \in \mathbb{N}$. This generalizes a theorem of Wirsing and extendsrecent work due to Elliott.

A Strengthening of Theorems of Hal\'asz and Wirsing