I apologize if this question is simple and the answer is obvious, but I was not able to find a reference that could be used immediately.

Let $k_1, k_2, k_3$ be three integers which are each at least $2$. Let $K \subset [-1,1]^3$ be a compact set with positive volume, and let $K(B) = \{\mathbf{x} \in \mathbb{R}^3 : \mathbf{x} = B \mathbf{y} \text{ for some } \mathbf{y} \in K\}$. Put $N(K(B)) = \#\{\mathbf{x} \in \mathbb{Z}^3 \cap K(B)\}$. Davenport showed that if $K$ is not too skew, then $N(K(B)) \sim V(K) B^3$, where $V(K)$ denotes the volume of $K$. Now put $N_{k_1, k_2, k_3}(K(B))$ to count the triples $(x_1, x_2, x_3) \in \mathbb{Z}^3$ in $K(B)$ for which $x_i$ is $k_i$-free, meaning if $p | x_i,$ then $p^{k_i} \nmid x_i$. Does it follow that

$$\displaystyle N_{k_1, k_2, k_3} (K(B)) \sim \frac{1}{\zeta(k_1) \zeta(k_2) \zeta(k_3)} V(K) B^3?$$

Here $\zeta(s)$ denotes the Riemann zeta function. The answer to this question is clear if $K = [-1,1]^3$.

More generally, I would like to know when the $k_i$-free conditions are not independent. For instance, a question I am interested in is to count those triples $(x_1, x_2, x_3)$ inside $K(B)$ for which there does not exist an integer $a$ greater than one such that $a^{k_i} | x_i, i = 1, 2,3$. Again, if $K = [-1,1]^3$, then it is easy to see that there are asymptotically

$$\displaystyle \frac{1}{\zeta(k_1 + k_2 + k_3)} V(K) B^3$$

such triples. I suspect the same is true for all `nice' sets $K$.

Any answer or reference is much appreciated.


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