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

TL;DR

This paper explores the properties of generalized Liouville functions defined on subsets of primes, demonstrating the ability to attain any asymptotic average value and providing estimates and exact results for specific cases.

Contribution

It introduces a generalized Liouville function for subsets of primes, showing the range of possible asymptotic averages and deriving asymptotic estimates and exact values for character-like functions.

Findings

For every alpha in [0,1], there exists a subset of primes with asymptotic average alpha.

Asymptotic estimates are provided for sums of these functions under certain restrictions.

Exact asymptotics and bounds are given for character-like Liouville functions.

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.