Fake Mu's

TL;DR

This paper investigates multiplicative functions with range {-1,0,1} called fake μ's, analyzing their bias behavior in summatory functions and characterizing conditions for persistent, apparent, or no bias.

Contribution

It introduces the concept of fake μ's, studies their bias properties, and characterizes functions with maximal, minimal, or no bias, including sign-changing behavior.

Findings

Identifies conditions for persistent and apparent bias in fake μ's.

Characterizes functions with maximal and minimal bias.

Shows sign changes in the normalized sum for functions with apparent bias.

Abstract

Let $f(n)$ denote a multiplicative function with range $\{-1,0,1\}$, and let $F(x) = \sum_{n\leq x} f(n)$. Then $F(x)/\sqrt{x} = a\sqrt{x} + b + E(x)$, where $a$ and $b$ are constants and $E(x)$ is an error term that either tends to $0$ in the limit, or is expected to oscillate about $0$ in a roughly balanced manner. We say $F(x)$ has persistent bias $b$ (at the scale of $\sqrt{x}$) in the first case, and apparent bias $b$ in the latter. For example, if $f(n)=\mu(n)$, the M\"{o}bius function, then $F(x) = \sum_{n\leq x} \mu(n)$ has $b=0$ so exhibits no persistent or apparent bias, while if $f(n)=\lambda(n)$, the Liouville function, then $F(x) = \sum_{n\leq x} \lambda(n)$ has apparent bias $b=1/\zeta(1/2)$. We study the bias when $f(p^k)$ is independent of the prime $p$, and call such functions fake $\mu's$. We investigate the conditions required for such a function to exhibit a persistent or apparent bias, determine the functions in this family with maximal and minimal bias of each type, and characterize the functions with no bias of either type. For such a function $F(x)$ with apparent bias $b$, we also show that $F(x)/\sqrt{x}-a\sqrt{x}-b$ changes sign infinitely often.