On a Turán conjecture and random multiplicative functions

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.

V3: accepted version, typos and minor mistakes corrected