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

Abstract

We prove a sharp version of Halász’s theorem on sums ∑ n ≤ x f ( n ) \sum _{n \leq x} f(n) of multiplicative functions f f with | f ( n ) | ≤ 1 |f(n)|\le 1 . Our proof avoids the “average of averages” and “integration over α \alpha ” manoeuvres that are present in many of the existing arguments. Instead, motivated by the circle method, we express ∑ n ≤ x f ( n ) \sum _{n \leq x} f(n) as a triple Dirichlet convolution and apply Perron’s formula.