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

Abstract

Define {\em the Liouville function for $A$}, a subset of the primes $P$, by $\lambda_{A}(n) =(-1)^{\Omega_A(n)}$ where $\Omega_A(n)$ 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 $$L_A(n):=\sum_{k\leq n}\lambda_A(n)\quad{and}\quad R_A:=\lim_{n\to\infty}\frac{L_A(n)}{n}.$$ We show that for every $\alpha\in[0,1]$ there is an $A\subset P$ such that $R_A=\alpha$. Given certain restrictions on $A$, asymptotic estimates for $\sum_{k\leq n}\lambda_A(k)$ are also given. With further restrictions, more can be said. For {\em character--like functions} $\lambda_p$ ($\lambda_p$ agrees with a Dirichlet character $\chi$ when $\chi(n)\neq 0$) exact values and asymptotics are given; in particular $$\quad\sum_{k\leq n}\lambda_p(k)\ll \log n.$$ Within the course of discussion, the ratio $\phi(n)/\sigma(n)$ is considered.