On a Tur\'an conjecture and random multiplicative functions

TL;DR

This paper investigates the probability that the sum of a random multiplicative function remains positive up to x, providing bounds and asymptotic estimates for the likelihood of negativity in the sum.

Contribution

It establishes a high probability bound for the positivity of the sum of a random multiplicative function and derives an asymptotic upper bound on the probability of negativity for large x.

Findings

Probability that the sum is positive for all x is at least 1 - 10^{-45}

Asymptotic upper bound on the probability of negativity is O(exp(-exp(log x / (C log log x))))

The probability of negativity is strictly less than 1 for large x.

Abstract

We show that if $f$ is the random completely multiplicative function, the probability that $\sum_{n\le x}\frac{f(n)}{n}$ is positive for every $x$ is at least $1-10^{-45}$, while also strictly smaller than $1$. For large $x$, we prove an asymptotic upper bound of $O(\exp(-\exp( \frac{\log x}{C\log \log x })))$ on the exceptional probability that a particular truncation is negative, where $C$ is some positive constant.