Notation: $$ [x]\ :=\ x-\lfloor x\rfloor $$

APF-functions $\ \tau(n)\ $ for $\ 2<n\in\mathbb N,\ $ and $\ \xi(n)\ $ for $\ 3<n\in\mathbb N,\ $ are defined as follows:

$$ \tau(n)\ :=\ \sum_{k=2}^{n-1}\,\left[\frac nk\right]\qquad\qquad\text{and}\qquad \qquad\xi(n)\ :=\ \sum_{k=2}^{\lfloor\sqrt n\rfloor}\ \left[\frac nk\right] $$

These functions are sensitive to their arguments being or not a prime. Locally, primes seem (how true is it?) dominate over their neighborhoods. In this spirit,

Question:   Do you already know or can you prove non-obvious results about the APF-functions $\ \tau\ $ and $\ \xi\,?$