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)+ϵ) , for all ϵ >0 . This generalizes to the k -free integers the results of Aymone (J. Number Theory, 212:113-121, 2020).