Positivity of partial sums of a random multiplicative function and corresponding problems for the Legendre symbol

TL;DR

This paper investigates the positivity of partial sums of a random multiplicative function and related problems for Legendre symbols, providing asymptotic probabilities and bounds that improve upon previous results.

Contribution

It establishes new asymptotic probabilities for partial sums of random multiplicative functions and relates these to Legendre symbol sums, improving existing bounds.

Findings

Probability that all partial sums are nonnegative approaches 1 under certain conditions

New upper bounds for the probability that sum of Legendre symbol terms is negative

Improved bounds assuming a conjecture related to Halász's theorem

Abstract

Let $f(n)$ be a random completely multiplicative function such that $f(p) = \pm 1$ with probabilities $1/2$ independently at each prime. We study the conditional probability, given that $f(p) = 1$ for all $p < y$, that all partial sums of $f(n)$ up to $x$ are nonnegative. We prove that for $y \ge C \frac{(\log x)^2 \log_2 x}{\log_3 x}$ this probability equals $1 - o(1)$. We also study the probability $P_x'$ that $\sum_{n \le x} \frac{f(n)}{n}$ is negative. We prove that $P_x' \ll \exp \left( - \exp \left( \frac{\log x \log_4 x}{(1 + o(1)) \log_3 x} \right) \right)$, which improves a bound given by Kerr and Klurman. Under a conjecture closely related to Hal\'asz's theorem, we prove that $P_x' \ll \exp(-x^{\alpha})$ for some $\alpha > 0$. Let $\chi_p(n) = \left( \frac{n}{p} \right)$ be the Legendre symbol modulo $p$. For a prime $p$ chosen uniformly at random from $(x, 2x]$, we express the probability that all partial sums of $\frac{\chi_p(n)}{n}$ are nonnegative in terms of the probability that partial sums of $\frac{f(n)}{n}$ are nonnegative.