Better than square-root cancellation for random multiplicative functions

We investigate when the better than square-root cancellation phenomenon exists for ∑ n ≤ N a (n) f (n) ₍ ₍a (n) f (n), where a (n) ∈ C a (n) 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 R (x) ^ {12} is the threshold for the better than square-root cancellation phenomenon to disappear.