Fake mu's

Let (F)(n) denote a multiplicative function with range {- 1, 0, 1}, and let F(x) = Sigma(left perpendicularxright perpendicular)(n=1) (F)(n). Then F(x)/root x = a root x + b + E(x), where a and b are constants and E(x) is an error term that either tends to 0 in the limit or is expected to oscillate about 0 in a roughly balanced manner. We say F(x) has persistent bias b (at the scale of root x) in the first case, and apparent bias b in the latter. For example, if (F)(n) = mu(n), the Mobius function, then F(x) = Sigma(left perpendicularxright perpendicular)(n=1) mu(n) has apparent bias 0, while if (F)(n) = lambda.(n), the Liouville function, then F(x) = Sigma(left perpendicularxright perpendicular)(n=1) lambda(n) has apparent bias 1/zeta(1/2). We study the bias when F(p(k)) is independent of the prime p, and call such functions fake mu's. We investigate the conditions required for such a function to exhibit a persistent or apparent bias, determine the functions in this family with maximal and minimal bias of each type, and characterize the functions with no bias. For such a function F(x) with apparent bias b, we also show that F(x)/root x-root vx-b changes sign infinitely often.