A Strengthening of Theorems of Hal\'asz and Wirsing

TL;DR

This paper extends classical results on multiplicative functions by providing upper bounds and asymptotic formulas for the ratio of partial sums, with applications to explicit estimates and generalizations of Wirsing's theorem.

Contribution

It offers new bounds and asymptotic formulas for sums of multiplicative functions, extending Halász's and Wirsing's theorems with explicit estimates and broader applicability.

Findings

Derived upper bounds for the ratio |M_g(x)|/M_{|g|}(x)

Proved an asymptotic formula when arg(g(p)) is small

Provided explicit estimates for ratios involving multiplicative functions

Abstract

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 in several ways by proving upper bounds for the ratio of $\frac{|M_g(x)|}{M_{|g|}(x)}$, for any strongly multiplicative, complex-valued function $g(n)$ under certain assumptions on the sequence $\{g(p)\}_p$. We further prove an asymptotic formula for this ratio in the case that $|\text{arg}(g(p))|$ is sufficiently small uniformly in $p$. In so doing, we recover a new proof of an explicit lower mean value estimate for $M_{f}(x)$ for any 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 main theorem 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 multiplicative functions 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 extends recent work due to Elliott.