A few notes on the asymptotic behavior of Rademacher random multiplicative functions

Let Xₚ, p be a sequence of independent random variables s. t. (Xₚ= 1) =1/2. Let ⱼ=₏|₉Xₚ if j is square free and ⱼ=0 otherwise. Denote Sₙ==₁ⁿ_. The from this point of view proving limit theorems for Sₙ is natural problem, since Sₙ mimics the behavior of e^ (β). It is a natural guiding conjecture that Sₙ/ n obeys the central limit theorem (CLT). However, S. Chatterjee conjectured (as expressed in 25) that the CLT should not hold. Chatterjee's conjecture was proved by Harper 17, and by now it is a direct consequence of a more recent breakthrough by Harper Har20 that Sₙbₙ 0 in L¹, where bₙ= (n^1/2 ( ( (n) ) ) ^-1/4) uₙ, uₙ. In particular Sₙ/ n 0. Nevertheless, the question whether there exists a sequence aₙ=o (bₙ) such that Sₙ/aₙ converges to some limit remains a mystery. Note that the corresponding problem in the Steinhaus Setting was recently resolved by Gor1. In this paper make an attempt to shed some light on the convergence of Sₙ/aₙ. Additionally, we obtain explicit estimates on hight moments of Sₙ without restrictions on the size of the moment compared to n like in 1. 2Har19, which is of independent interest. This is achieved by a martingale argument together with the Burkholder inequality, and it has applications in a natural number theoretic combinatorial problem. Using martingale techniques we will also obtain exponential concentration inequalities for Sₙ (in the large deviations regime)