On the positivity of some weighted partial sums of a random multiplicative function

Inspired in the papers by Angelo and Xu, Q.J Math., 74, pp. 767-777, and improvements by Kerr and Klurman, arXiv:2211.05540, we study the probability that the weighted sums of a Rademacher random multiplicative function, ∑n≤ xf(n)n-σ, are positive for all x≥ x_σ≥ 1 in the regime σ→1/2⁺. In a previous paper by the author, when σ≤ 1/2 this probability is zero. Here we give a positive lower bound for this probability depending on x_σ that becomes large as σ→1/2⁺. The main inputs in our proofs are a maximal inequality based in relatively high moments for these partial sums combined with a Hal\'asz-Bonami's moment inequality, and also explicit estimates for the partial sums of non-negative multiplicative functions.