Count of multiples Is is possible to estimate the total number of distinct multiples $N_{\text{mult}}(p, x)$ of primes $p_k \le p$ in the general case of arbitrary $p$ and range [1, x]? Or, alternatively, the number of integers in [1, x] that aren't divisible by any $p_k \le p$?
An estimate is straightforward for the first several primes via PIE but appears to be more challenging in the general case as the number of factors increases.
 A: Let $\Phi(x,y)$ denote the number of integers up to $x$ which are not divisible by any prime up to $y$. Let $u:=\log x/\log y$.
Then Theorem 3 in Section of III.6.2 of Tenenbaum’s Introduction to analytic and probabilistic number theory (Cambridge University Press, 1995) gives the asymptotic formula
$$ \Phi(x,y)=\frac{x\omega(u)-y}{\log y}+O\left(\frac{x}{(\log y)^2}\right),\qquad x\geq y\geq 2,$$
where $\omega:[1,\infty)\to[1/2,1]$ is the Buchstab function satisfying (cf. Corollary 3.1 after the theorem)
$$ \omega(u)=e^{-\gamma}+O(u^{-u/2}),\qquad u\geq 1.$$
As you can see this is a delicate question, $y$ is your $p$ and the relationship between $x$ and $y$ is crucial. The Buchstab function $w(u)$ controls this, it decreases from $w(0)=1$ to $w(2)=1/2,$ then increases again, eventually being asymptotic to
$e^{-\gamma},$ as $u$ gets large, which is related to Mertens third theorem
$$\prod_{p \leq n} \left(1-\frac{1}{p}\right) =\frac{ e^{-\gamma}+o(1)}{ \log n},$$
yielding that roughly
$$\frac{ e^{-\gamma}X}{ \log n},$$
integers $\leq X$ are not divisible by any prime $\leq n,$ if $X$ is much larger than $n.$
