Erratum to: On the pairs of multiplicative functions satisfying some relations

Abstract

This assertion is incorrect, as the following example shows: Let F (n) = (−1)n−1 (n = 1, 2, . . .), G(2n + 1) = χ4(2n + 1), where χ4 is the nonprincipal Dirichlet character (mod 4). Then F (n) and G(n) are multiplicative functions (G(n) can be arbitrary on the powers of 2), furthermore G(2n+ 1) = −F (n) (n ∈ N). (1.2) The failure comes from an incorrect assertion made in the proof of Lemma 8. Let H(n) = f(n) g(n) . Following the argument of the proof of Lemma 8, we obtain that H(3) = H(Q) if Q ≡ 3, 11 (mod 12), thus H(3) = H (3(4n+ 1)) = H(3)H(4n + 1) if (3, 4n + 1) = 1. For Q = 11 we have H(3) = H(11). Since ν, μ ≡ 3 (mod 4), (ν, μ) = 1 implies that H(νμ) = H(ν)H(μ) = 1, if (νμ, 3) = 1, it can occur only if H(n) = constant if n runs