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=∏p|jXₚ 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√ln(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 Sₙ/bₙ→ 0 in L¹, where bₙ=(n1/2(ln(ln(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 [Theorem 1.2]{Har19}, 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)