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

TL;DR

This paper investigates the probability that weighted partial sums of a random multiplicative function remain positive beyond a certain point as the parameter approaches 1/2 from above, providing new lower bounds.

Contribution

It establishes a positive lower bound for the probability of positivity of weighted sums of a Rademacher multiplicative function near the critical line, improving understanding of their behavior.

Findings

Probability of positivity increases as σ approaches 1/2 from above.

Provides explicit lower bounds depending on the cutoff point x_σ.

Uses maximal inequalities and moment estimates to derive results.

Abstract

Inspired by 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, $\sum_{n\leq x}f(n)n^{-\sigma}$, are positive for all $x\geq x_\sigma\geq 1$ in the regime $\sigma\to1/2^+$. In a previous paper by Heap, Zhao and the author, and by the author, when $0\leq \sigma\leq 1/2$ this probability is zero. Here we give a positive lower bound for this probability depending on $x_\sigma$ that becomes large as $\sigma\to1/2^+$. The main inputs in our proofs are a maximal inequality based in relatively high moments for these partial sums combined with a Bonami--Hal\'asz's moment inequality, and also explicit estimates for the partial sums of non-negative multiplicative functions.