Multiplicative functions in almost every short interval

Abstract

Estimating the size of sums of multiplicative functions over long intervals is a reasonably wellunderstood problem. It is accomplished by the prime number Theorem for the M¨obius function µ(n) and by Halasz’s Theorem for general multiplicative functions. Understanding these sums in short intervals is much harder, and it is usually hopeless unless we ”average” them and try to understand them only in almost every short interval. Matom¨aki and Radziwill recently published a result describing in incredible generality the size of these sums in almost every short interval, surpassing in several directions every previously known result. This paper is a survey on their result, starting from the case of the Liouville function λ(n) and interval size Xǫ which follows easily from their main ideas (even though it was previously open) building up close to the full generality of their main result.