On a Turán Conjecture and random multiplicative functions

Abstract

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.