A multiplicative function $f$ is said to be resembling the M\"{ o}bius function if $f$ is supported on the square-free integers, and $f(p)=\pm 1$ for each prime $p$. We prove $O$- and $\Omega$-results for the summatory function $\sum_{ n\leq x} f(n)$ for a class of these $f$ studied by Aymone, and the point is that these $O$-results demonstrate cancellations better than the square-root saving. It is proved in particular that the summatory function is $O(x^{ 1/3+\varepsilon})$ under the Riemann Hypothesis. On the other hand it is proved to be $\Omega(x^{ 1/4})$ unconditionally. It is interesting to compare these with the corresponding results for the M\"{ o}bius function.
Categories: math.NT