Inspired by the papers by Angelo and Xu (Q J Math 74:767–777), and improvements by Kerr and Klurman (arXiv:2211.05540), we study the probability that the weighted sums of a Rademacher random multiplicative function, \(\sum _{n\le x}f(n)n^{-\sigma }\) , are positive for all \(x\ge x_\sigma \ge 1\) in the regime \(\sigma \rightarrow 1/2^+\) . In a previous paper by Heap, Zhao and the author, and by the author, when \(0\le \sigma \le 1/2\) this probability is zero. Here we give a positive lower bound for this probability depending on \(x_\sigma \) that becomes large as \(\sigma \rightarrow 1/2^+\) . The main inputs in our proofs are a maximal inequality based in relatively high moments for these partial sums combined with a Bonami–Halász’s moment inequality, and also explicit estimates for the partial sums of non-negative multiplicative functions.
