Does there exist a closed-form expression for the following function?

Question

I would like to find a closed-form expression for the function that is defined as follows:

$T_{s}(x) = x^{s}(1 - x^{s}), \text{for prime } x \\ T_{s}(x) = x^{s}, \text{otherwise}$

Answer 1

This is equivalent to asking for an indicator function for the primes: Given $T_s(x)$,

$$ \frac{T_s(x)-x^s}{x^s(1-x^s)-x^s}=x^{-s}-x^{-2s}T_s(x) $$

is an indicator function for the primes, and conversely given an indicator function $I(x)$ of the primes,

$$ T_s(x)=x^s(1-I(x)x^s) $$

is your function. Various indicator functions for the primes are known, e.g. here.