  In 2017 Tao proposed a variant Sarnak's M\"{o}bius disjointness conjecture
with logarithmic averaging: For any zero entropy dynamical system $(X,T)$,
$\frac{1}{\log N} \sum_{n=1} ^N \frac{f(T^n x) \mu (n)}{n}= o(1)$ for every
$f\in \mathcal{C}(X)$ and every $x\in X$. We construct examples showing that
this $o(1)$ can go to zero arbitrarily slowly. Nonetheless, all of our examples
satisfy the conjecture.
