A MORE INTUITIVE PROOF OF A SHARP VERSION OF HALASZ'S THEOREM

We prove a sharp version of Halasz's theorem on sums Sigma(n <= x) f(n) of multiplicative functions f with vertical bar f(n)vertical bar <= 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 Sigma(n <= x) f(n) as a triple Dirichlet convolution and apply Perron's formula.