Multiplicative Functions Resembling the Mobius Function

A multiplicative function f is said to be resembling the Mobius function if f is supported on the square-free integers, and f(p) = +/- 1 for each prime p. We prove O-and Omega-results for the summatory function Sigma(n <= x) f(n) for a class of these f, and the point is that these O-results demonstrate cancellations better than the square-root saving. It is proved in particular that the summatory function is O(x(1/3+epsilon)) under the Riemann Hypothesis. On the other hand it is proved to be Omega (x(1/ 4)) unconditionally. It is interesting to compare these with the corresponding results for the Mobius function.