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 ) ∑ n≤ Na(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 log log R (log log x)1/2 is the threshold for the better than square-root cancellation phenomenon to disappear.