Better than square-root cancellation for random multiplicative functions

Abstract

We investigate when the better than square-root cancellation phenomenon exists for ∑ n ≤ N a ( n ) f ( n ) \sum _{n\le N}a(n)f(n) , where a ( n ) ∈ C a(n)\in \mathbb {C} and f ( n ) f(n) is a random multiplicative function. We focus on the case where a ( n ) a(n) is the indicator function of R R rough numbers. We prove that log ⁡ log ⁡ R ≍ ( log ⁡ log ⁡ x ) 1 2 \log \log R \asymp (\log \log x)^{\frac {1}{2}} is the threshold for the better than square-root cancellation phenomenon to disappear.