How negative can ∑n≤ xf(n)/n be?

Turán observed that logarithmic partial sums ∑n≤ xf(n)/n of completely multiplicative functions (in the particular case of the Liouville function f(n)=λ(n) ) tend to be positive. We develop a general approach to prove two results aiming to explain this phenomena. Firstly, we show that there exist constants C, x₀≥ 1, such that for any completely multiplicative function f satisfying -1≤ f(n)≤ 1 , we have {equation*}∑n≤ xf(n)/n≥ -{C(logloglog{x})^2}{(loglog{x})}, x≥ x_0.{equation*} This improves a previous bound due to Granville and Soundararajan. Secondly, we show that if f is a typical (random) completely multiplicative function f:N→ \-1,1\ , the probability that ∑n≤ xf(n)/n is negative for a given large x , is O(exp(-exp(log x· logloglog x/Clog log x))). This improves on recent work of Angelo and Xu.