Abstract Turán observed that logarithmic partial sums ₍ ₗf (n) /n of completely multiplicative functions (in the particular case of the Liouville function f (n) = (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*₍ ₗf (n) n -C (x) ² (), x x₀. 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 ₍ ₗf (n) /n is negative for a given large x, is O ( (- (x x/C x) ) ). This improves on recent work of Angelo and Xu.
Kerr et al. (Mon,) studied this question.