**Notation:**
$$ \{x\}\ :=\ x-\lfloor x\rfloor $$

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

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**Remark:** &nbsp; Obviously, $\ \xi\ $ is more sensitive than $\ \tau$.

**Remark:** &nbsp; I've written a simple Perl program. It shows rather convincingly, that @GHfromMO was right, that my functions are not too sensitive to the arithmetical properties -- pretty soon, they make an impression of randomness.