How negative can $\sum_{n\le x}\frac{f(n)}{n}$ be?

TL;DR

This paper investigates the potential negativity of logarithmic partial sums of completely multiplicative functions, providing bounds and probabilistic estimates that extend previous results and deepen understanding of their behavior.

Contribution

It introduces new bounds on the negativity of these sums and probabilistic estimates for random functions, improving prior bounds and advancing theoretical understanding.

Findings

Sum of completely multiplicative functions is bounded below by a slowly decreasing function.

Probability of negativity for random functions decreases super-exponentially with x.

Improves previous bounds by Granville, Soundararajan, Angelo, and Xu.

Abstract

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.