Fake mu’s

Abstract:

Let $\digamma (n)$ denote a multiplicative function with range $\{-1,0,1\}$, and let $F(x) = \sum _{n=1}^{\left \lfloor x\right \rfloor } \digamma (n)$. 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 $\digamma (n)=\mu (n)$, the Möbius function, then $F(x) = \sum _{n=1}^{\left \lfloor x\right \rfloor } \mu (n)$ has apparent bias $0$, while if $\digamma (n)=\lambda (n)$, the Liouville function, then $F(x) = \sum _{n=1}^{\left \lfloor x\right \rfloor } \lambda (n)$ has apparent bias $1/\zeta (1/2)$. We study the bias when $\digamma (p^k)$ 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. 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.