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

Abstract

Define the Liouville function for A, a subset of the primes P,by λA(n) = (-1)ωA(n),where ω 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 LA(x):=∑λ a (n)and R A:= lim/ν→∞ La(ν)/ν. It is known that for each α ∈ [0, 1] there is an A ∪ P such that RA = a. Given certain restrictions on the sifting density of A, asymptotic estimates for ∑n≤x λa (n) can be given. With further restrictions, more can be said. For an odd prime p, define the character-like function λp as λp(pk + i) = (i/p) for i =1,...,p- land k ≥ 0, and λp (p) = 1, where(i/p) is the Legendre symbol (for example, λ3 is defined by λ3(3k +1) = 1, λ3(3k + 2) =- 1(k ≥ 0) and λ3(3) = 1). For the partial sums of character-like functions we give exact values and asymptotics; in particular, we prove the following theorem. Theorem. If p is an odd prime, then max [Eqation Present] This result is related to a question of Erdos concerning the existence of bounds for number-theoretic functions. Within the course of discussion, the ratio φ (n)/ σ(n) is considered. © 2010 by the authors.