Abstract. Define the Liouville function for A, a subset of the primes P , by λ A (n) = (−1) Ω A (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. Denoten .
It is known that for eachGiven certain restrictions on the sifting density of A, asymptotic estimates for n≤x λ A (n) can be given. With further restrictions, more can be said. For an odd prime p, define the character-like function λ p as λ p (pk + i) = (i/p) for i = 1, . . . , p − 1 and k ≥ 0, and λ p (p) = 1, where (i/p) is the Legendre symbol (for example, λ 3 is defined by λ 3 (3k + 1) = 1, λ 3 (3k + 2) = −1 (k ≥ 0) and λ 3 (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, thenThis result is related to a question of Erdős concerning the existence of bounds for number-theoretic functions. Within the course of discussion, the ratio φ(n)/σ(n) is considered.