Consider the summatory function
$$f(x)=\sum\limits_{n=1}^x 1_{n\ne k^m}\tag{1}$$
where $1_{n\ne k^m}$ is the non-integer-power indicator function which returns $1$ when $n$ is a non-integer-power and $0$ otherwise. In order for $n$ to be considered a non-integer-power there must be a difference in at least one of the exponents in the prime factorization of $n=p_1^{e_1}\,p_2^{e_2}\,...\,p_r^{e_r}$ (i.e. $e_1\,...\,e_r$ must not all have the same value).
Here's a list of non-integer powers for $1\le n\le 100$.
$$\{12,18,20,24,28,40,44,45,48,50,52,54,56,60,63,68,72,75,76,80,84,88,90,92,96,98,99\}\tag{2}$$
The discrete plot of $f(x)$ illustrated in Figure (1) below illustrates the asymptotic for $f(x)$ seems to be nearly a linear function. In Figure (1) below $f(x)$ is evaluated at intervals of $100$ for a total of $100$ plot points.
Figure (1): Illustration of $f(x)$ defined in formula (1)
Question (1): Does the Riemann hypothesis predict a bound on the growth of $f(x)$?
Question (2): Can the Dirichlet series defined in formula (3) below be expressed in terms of the Riemann zeta function $\zeta(s)$?
$$F(s)=\sum\limits_{n=1}^\infty \frac{1_{n\ne k^m}}{n^s}\tag{3}$$