Identifying a Number Theoretic Function

Question

Does the arithmetic function $\xi_{s}:\mathbb{N}\to\mathbb{C}$ $$\xi_s(n)=\sum_{p\mid n}p^{-s}$$ exist in literature? If it makes any difference I only care about square-free inputs $n$ so it may be that it simplifies to an existing function in that case, if it isn't already one. In this special case of $n$ square-free I managed to derive $$\xi_s(n)=\sigma_{-s}(n)(\omega(n)-1)^{-1}\left(\left[\sum_{p\mid n}\sum_{q\mid n}\frac{q^s}{p^s(q^s+1)}\right]-1\right)$$ but I wouldn't call this an improvement.

Answer 1

The approximation $\xi_1(n)\sim\log\log\log n$ is quite famous but $\xi_s$ is not a multiplicative function.
We have $$ \sum_{n\geq 1}\frac{\xi_s(n)}{n^t}=\sum_p \frac{1}{p^s}\sum_{n:p\mid n}\frac{1}{n^t}=\sum_p\frac{\zeta(t)}{p^{s+t}}\tag{1}$$ and in terms of the prime zeta function $$ \sum_{n\geq 1}\frac{\xi_s(n)}{n^t} = \zeta(t)\,P(s+t)=\zeta(t)\sum_{m\geq 1}\frac{\mu(m)}{m}\log\zeta(ms+mt).\tag{2} $$