Completely multiplicative functions taking values in $\{-1,1\}$

Abstract

Define {\em the Liouville function for $A$}, a subset of the primes $P$, by $\hat{I}{»}_A(n)=(-1{)}^{\hat{I}{\text{c◯}}_A(n)}$ where $\hat{I}{\text{c◯}}_A(n)$ is the number of prime factors of $n$ coming from $A$ counting multiplicity. For the traditional Liouville function, $A$ is the set of all primes. Denote $$L_A(n):=\sum _{k\le n}\hat{I}{»}_A(n)and{R}_A:=\underset{n\to \infty }{\lim }\frac{L_A(n)}{n}.$$ We show that for every $\hat{I}\pm \in [0,1]$ there is an $A\subset P$ such that $R_A=\hat{I}\pm$. Given certain restrictions on $A$, asymptotic estimates for ${\sum }_{k\le n}\hat{I}{»}_A(k)$ are also given. With further restrictions, more can be said. For {\em character--like functions} $\hat{I}{»}_p$ ($\hat{I}{»}_p$ agrees with a Dirichlet character $\ddot{I}‡$ when $\ddot{I}‡(n)\ne 0$) exact values and asymptotics are given; in particular $$\sum _{k\le n}\hat{I}{»}_p(k)\ll \log n.$$ Within the course of discussion, the ratio $\ddot{I}•(n)/\ddot{I}ƒ(n)$ is considered.