S ep 2 00 8 COMPLETELY MULTIPLICATIVE FUNCTIONS TAKING VALUES IN { − 1 , 1 }

λ(n) := (−1). So λ(1) = λ(4) = λ(6) = λ(9) = λ(10) = 1 and λ(2) = λ(5) = λ(7) = λ(8) = −1. In particular, λ(p) = −1 for any prime p. It is well-known (e.g. See §22.10 of [7]) that Ω is completely additive, i.e, Ω(mn) = Ω(m) + Ω(n) for any m and n and hence λ is completely multiplicative, i.e., λ(mn) = λ(m)λ(n) for all m,n ∈ N. It is interesting to note that on the set of square-free positive integers λ(n) = μ(n), where μ is the Möbius function. In this respect, the Liouville λ–function can be thought of as an extension of the Möbius function. Similar to the Möbius function, many investigations surrounding the λ–function concern the summatory function of initial values of λ; that is, the sum