  Tur\'an observed that logarithmic partial sums $\sum_{n\le x}\frac{f(n)}{n}$
of completely multiplicative functions (in the particular case of the Liouville
function $f(n)=\lambda(n)$) tend to be positive. We develop a general approach
to prove two results aiming to explain this phenomena. Firstly, we show that
for every $\varepsilon>0$ there exists some $x_0\ge 1,$ such that for any
completely multiplicative function $f$ satisfying $-1\le f(n)\le 1$, we have
$$\sum_{n\le x}\frac{f(n)}{n}\ge -\frac{1}{(\log\log{x})^{1-\varepsilon}},
\quad x\ge x_0.$$ This improves a previous bound due to Granville and
Soundararajan. Secondly, we show that if $f$ is a typical (random) completely
multiplicative function $f:\mathbb{N}\to \{-1,1\}$, the probability that
$\sum_{n\le x}\frac{f(n)}{n}$ is negative for a given large $x,$ is
$O(\exp(-\exp(\frac{\log x\cdot \log\log\log x}{C\log \log x}))).$ This
improves on recent work of Angelo and Xu.
