Multiplicative functions supported on the k-free integers with small partial sums

We provide examples of multiplicative functions f supported on the k-free integers such that at primes f (p) = +/- 1 and such that the partial sums of f up to x are o(x(1/k)). Further, if we assume the Generalized Riemann Hypothesis, then we can improve the exponent 1/k: There are examples such that the partial sums up to x are o(x(1/(k+ 1/2)+epsilon)), for all epsilon > 0. This generalizes to the k-free integers the results of Aymone.