SCHWEITZER 2008 Problem 5

Let $A$ be an infinite subset of the set of natural numbers, and denote by $\tau_A(n)$ the number of divisors of $n$ in $A$ . Construct a set $A$ for which $$\sum_{n\le x}\tau_A(n)=x+O(\log\log x)$$ and show that there is no set for which the error term is $o(\log\log x)$ in the above formula.

(translated by Miklós Maróti)