  We provide examples of multiplicative functions $f$ supported on the
squarefree integers, such that on primes $f(p)=\pm1$ and such that
$M_f(x):=\sum_{n\leq x} f(n)=o(\sqrt{x})$. Further, by assuming the Riemann
hypothesis (RH) we can go beyond $\sqrt{x}$-cancellation.
