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

Abstract

We provide examples of multiplicative functions f supported on the k-free integers such that at primes $$f(p)=\pm 1$$ 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 $$o(x^{1/(k+\frac{1}{2})+\epsilon })$$ , for all $$\epsilon >0$$ . This generalizes to the k-free integers the results of Aymone (J. Number Theory, 212:113-121, 2020).