A more intuitive proof of a sharp version of Halász's theorem

Abstract

We prove a sharp version of Halász's theorem on sums ${\sum }_{n\le x}f(n)$ 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 ${\sum }_{n\le x}f(n)$ as a triple Dirichlet convolution, and apply Perron's formula.

A short proof of Halász's theorem using our new approach

Cited by in corpus (2)

• A new proof of Halász's Theorem, and its consequences
• Zeros of partial sums of $L$-functions