A note on P\'olya's observation concerning Liouville's function

TL;DR

This paper demonstrates that a specific weighted average of the Liouville function is negative, indicating that the function tends to be negative on average, which provides insight into its overall behavior.

Contribution

It introduces a new weighted mean of the Liouville function and shows it is negative, offering a novel perspective on the function's average properties.

Findings

Weighted mean of Liouville function is negative

Liouville function tends to be negative on average

Provides new insight into Liouville function's behavior

Abstract

We show that a certain weighted mean of the Liouville function lambda(n) is negative. In this sense, we can say that the Liouville function is negative "on average".