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 Ω-results for the summatory function ∑_(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+ε))under the Riemann Hypothesis.On the other hand it is proved to be Ω(x^(1/4))unconditionally.It is interesting to compare these with the corresponding results for the Mobius function.