How negative can $\boldsymbol{\sum}_{\textbf{n}\le \textbf{x}}\frac{\textbf{f}(\textbf{n})}{\textbf{n}}$ be?

<jats:p> Turán observed that logarithmic partial sums <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0305004125101837_inline3.png"/> <jats:tex-math>$\sum_{n\le x}{f(n)}/{n}$</jats:tex-math> </jats:alternatives> </jats:inline-formula> of completely multiplicative functions (in the particular case of the Liouville function <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0305004125101837_inline4.png"/> <jats:tex-math>$f(n)=\lambda(n)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> ) tend to be positive. We develop a general approach to prove two results aiming to explain this phenomena. </jats:p> <jats:p> Firstly, we show that there exist constants <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0305004125101837_inline5.png"/> <jats:tex-math>$C, x_0\ge 1,$</jats:tex-math> </jats:alternatives> </jats:inline-formula> such that for any completely multiplicative function <jats:italic>f</jats:italic> satisfying <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0305004125101837_inline6.png"/> <jats:tex-math>$-1\le f(n)\le 1$</jats:tex-math> </jats:alternatives> </jats:inline-formula> , we have <jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" position="float" xlink:href="S0305004125101837_eqnU1.png"/> <jats:tex-math>\begin{equation*}\sum_{n\le x}\frac{f(n)}{n}\ge -\frac{C(\log\log\log{x})^2}{(\log\log{x})}, \quad x\ge x_0.\end{equation*}</jats:tex-math> </jats:alternatives> </jats:disp-formula> This improves a previous bound due to Granville and Soundararajan. Secondly, we show that if <jats:italic>f</jats:italic> is a typical (random) completely multiplicative function <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0305004125101837_inline7.png"/> <jats:tex-math>$f:\mathbb{N}\to \{-1,1\}$</jats:tex-math> </jats:alternatives> </jats:inline-formula> , the probability that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0305004125101837_inline8.png"/> <jats:tex-math>$\sum_{n\le x}{f(n)}/{n}$</jats:tex-math> </jats:alternatives> </jats:inline-formula> is negative for a given large <jats:italic>x</jats:italic> , is <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0305004125101837_inline9.png"/> <jats:tex-math>$O(\exp(-\exp({\log x\cdot \log\log\log x}/{C\log \log x}))).$</jats:tex-math> </jats:alternatives> </jats:inline-formula> This improves on recent work of Angelo and Xu. </jats:p>