Chebyshev like inequalities for the number of prime powers up to $ x $

Question

Denoting by $ \Pi_{k}(x) $ the number of integers of the form $ p^{k} $ up to $ x $ where $ p $ is a prime, are there known inequalities similar to Chebyshev's ones for $ \Pi_{k}(x) $, namely of the form $ A_{k}\frac{x^{c_{k}}}{\log^{d_{k}}x}<\Pi_{k}(x)<B_{k}\frac{x^{c_{k}}}{\log^{d_{k}}x} $ for all $ x>x_{k} $, with explicit values of the considered constants depending only on $ k $?

Answer 1

Clearly if $p^k < x$ then $p < x^{1/k}$ hence the number of prime powers $p^k$ less than a given number $x$ is equal to the number of primes $\le x^{1/k}$. This gives

$$ \Pi_k(x) = \pi(x^{1/k}) $$

With this formulation, you can now use all the known upper and lower bounds on the prime counting function (such as the ones given by Dusart) to estimate $\Pi_k(x)$.