I have two very related questions:
If $f(N)$ is the number of square-free integers in the interval $[1, N]$, it is well known that $$f(N) \sim \frac{6}{\pi^{2}} N.$$
My first question is, if we impose the additional condition that the integer is not divisible by any prime smaller than $ N^{1/k}$, for some fixed integer $k \geq 2$, then what is the precise asymptotic? Clearly it is at least $\frac{N}{\log{N}}$ by the prime number theorem. Furthermore, by an asymptotic for squarefree integers divisible by precisely $c$ prime factors here we see that it is at most $\frac{k_{1}N(\log \log{N})}{\log{N}}$ with $k_{1}$ a constant depending on $k$.
Secondly, there's a connection with a product over all primes. That is, analogously to finding the density of square-free integers, we have that the density of squarefree integers not divisible by any prime less than $N^{1/k}$ is
$$ \prod_{p < N^{1/k}} \left( 1 – \frac{1}{p} \right) \prod_{p > N^{1/k}} \left( 1 – \frac{1}{p^2} \right). $$ Now the second term converges to some constant, and the first one decreases, by Mertens' theorem, like $\frac{c_{2}}{\log{N}}$. This looks similar to what I mentioned before, with the $\log{N}$ appearing in the denominator, but this doesn't a priori tell us anything about what happens inside the interval $[1, N]$ – only an interval $[1, M]$ where $M$ is sufficiently large (and this might be much larger than $N$). So, is there a way to formulate this so that one can find the asymptotic up to $N$ by looking at this product (or perhaps, see how large relative to $N$ that $M$ must be)?
Best Answer
Forgetting the squarefree condition for a moment, the number of integers up to $N$ that are not divisible by any primes less than $N^{1/k}$ is asymptotic to $$ \omega(k) \frac N{\log N} \sim e^\gamma \omega(k) N \prod_{p\le N^{1/k}} \bigg( 1-\frac1p \bigg), $$ where $\omega$ is the Buchstab function. (In particular, the first half of the density you derive in your answer is not correct: there is a correction factor of the form $e^\gamma \omega(k)$.)
Now the number of integers up to $N$ that are divisible by the square of a particular prime $p$ is at most $N/p^2$. So the number of integers up to $N$ that are divisible by the square of a prime greater than $N^{1/k}$ is at most $$ \sum_{p>N^{1/k}} \frac N{p^2} < N \sum_{n>N^{1/k}} \frac1{n^2} < N \cdot \frac1{N^{1/k}}. $$ Therefore the above asymptotic also holds for squarefree numbers not divisible by small primes.