  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.
