The main question of this paper is the following: how much cancellation can the partial sums restricted to the k-free integers up to x of a \(\pm 1\) multiplicative function f be in terms of x? Building upon the recent paper by Liu (Acta Math Sin (Engl Ser) 39(12):2316–2328, 2023), we prove that under the Riemann Hypothesis for quadratic Dirichlet L-functions, we can get \(x^{1/(k+1)}\) cancellation when f is a modified quadratic Dirichlet character, i.e., f is completely multiplicative and for some quadratic Dirichlet character \(\chi \) , \(f(p)=\chi (p)\) for all but a finite subset of prime numbers. This improves the conditional results by Aymone et al. (Ramanujan J 59(3):713–728, 2022).
