Central limit theorems for random multiplicative functions

Better than square-root cancellation for random multiplicative functions

“…one can take θ = π) by Soundararajan and the author [43], and also an average version of the result is proved by Benatar, Nishry and Rodgers [7]. The proof of the result in [43] is based on McLeish's martingale central limit theorem [35], and the method was pioneered by Harper in [23]. The proof reveals the connection between the existence of such a central limit theorem and a quantity called multiplicative energy of a := {a(n…”

“…It has been carried out in the concrete case where a(n) = e 2πinθ for some fixed real θ without too good Diophantine approximation properties (such θ has relative density 1 in R, e.g. one can take θ = π) by Soundararajan and the author [43], and also an average version of the result is proved by Benatar, Nishry and Rodgers [7]. The proof of the result in [43] is based on McLeish's martingale central limit theorem [35], and the method was pioneered by Harper in [23].…”

“…A special case of a(n) is an indicator function of a set A, and the quantity E × (A) is a popular object studied in additive combinatorics. It is now known [43] that a crucial condition for such a central limit theorem to hold for n≤N a(n)f (n) is that the set A has multiplicative energy ≤ (2 + )|A| 2 . See Section 9.1 for more discussions on a(n) being a "typical" choice.…”

“…See Section 9.1 for more discussions on a(n) being a "typical" choice. We refer readers who are interested in seeing more examples of when a central limit theorem holds for partial (restricted) sums of random multiplicative functions to [7,10,23,31,33,39,43].…”

“…However, to achieve such a bound on the fourth moment, one requires log R (log x) c for some constant c (by using the method in [43]), and thus this approach would not give the optimal range as in Theorem 1.3.…”