A more intuitive proof of a sharp version of Hal\'asz's theorem

TL;DR

This paper presents a more intuitive proof of a sharp version of Halász's theorem for multiplicative functions, avoiding complex averaging techniques by using a circle method-inspired approach with Dirichlet convolutions and Perron's formula.

Contribution

It introduces a novel, more straightforward proof technique for Halász's theorem, bypassing traditional complex averaging and integration methods.

Findings

Proves a sharp version of Halász's theorem for multiplicative functions.

Provides a proof that is more intuitive and avoids complex averaging maneuvers.

Demonstrates the effectiveness of the circle method-inspired approach.

Abstract

We prove a sharp version of Hal\'asz's theorem on sums $\sum_{n \leq x} f(n)$ of multiplicative functions $f$ with $|f(n)|\le 1$. Our proof avoids the "average of averages" and "integration over $\alpha$" manoeuvres that are present in many of the existing arguments. Instead, motivated by the circle method we express $\sum_{n \leq x} f(n)$ as a triple Dirichlet convolution, and apply Perron's formula.