A Hal\'{a}sz-type asymptotic formula for logarithmic means and its consequences

TL;DR

This paper proves a new asymptotic formula for logarithmic means of multiplicative functions, leading to significant progress on longstanding problems and applications in number theory.

Contribution

It establishes a sharp asymptotic formula for logarithmic means of multiplicative functions and derives new bounds and probabilistic results, improving previous work and solving open problems.

Findings

Improved lower bound for the logarithmic mean of completely multiplicative functions.

Proved that the probability of negativity in Rademacher multiplicative functions decays exponentially.

Constructed examples showing the optimality of the bounds for small absolute values of logarithmic means.

Abstract

We establish an asymptotic formula for the logarithmic mean value of a 1-bounded multiplicative function that is sharp in many cases of interest. We derive from it a variety of applications, making progress on several old problems. As a first application, we show that if $f$ is a completely multiplicative function taking values in $[-1,1]$ then there is a constant $c > 0$ such that for every $x \geq 3$, $$ L_f(x) := \sum_{n \leq x} \frac{f(n)}{n} > -\frac{c}{(\log x)^{1-2/\pi}}, $$ thus significantly improving on a 20-year-old result of Granville and Soundararajan. We also show that the exponent of $\log x$ in this result can be improved to $-1+o(1)$, as long as $f$ does not ``behave like'' the Liouville function $\lambda$ in a precise sense. As a second application, we show that for a Rademacher random completely multiplicative function $\mathbf{f}$, the probability that $L_{\mathbf{f}}(x)$ is negative is $O(\exp(-x^c))$ for some $c \in (0,1)$, thus establishing a previously conjectured bound. Finally, we obtain a converse theorem for small absolute values $|L_f(x)|$, and construct examples $f$ that show that it is (essentially) best possible.