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

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.