The maximum difference between the number of elements in the two sets of equal length of consecutive numbers that divisible by some prime numbers

Suppose that $$p_1,...,p_k$$ are distinct prime numbers.

Let $$f(n,l)$$ be equal to the number of elements from set $$\{n+1,n+2,...,n+l\}$$ that are divisible by some $$p_1,...,p_k$$. Is it true that $$\sup_{n,m,l \in \mathbb{N}} |f(n,l)-f(m,l)| = O(k).$$

For example, Rankin's construction of large prime gaps boils down to the fact that if $$p_1, \ldots, p_k$$ denote all prime numbers below $$x$$, then there exists some $$n$$ with $$f\left(n, \frac{x\log x\log\log\log x}{(\log\log x)^2}\right)=0$$.
If $$n=x, \ell=x^2-x$$, then $$f(n, \ell)=\pi(x^2)+\mathcal{O}(x)\sim\frac{x^2}{2\log x}$$, but the average taken over all $$n$$ is $$\ell \prod_{p\leq x}\left(1-\frac{1}{p}\right)\sim \frac{e^{-\gamma}x^2}{\log x}$$.
• Thanks for your answer. What do you think about the correct order? Is it $O(k^2 \log k)$? – Martin Hicks Jun 12 at 19:35
• A bound involving $k^2$ might be achievable, but I don't think anybody has done it yet. For the upper bound Selberg's sieve might suffice. The lower bound is probably more difficult. A good starting point might be Iwaniec's work on Jacobsthal function. – Jan-Christoph Schlage-Puchta Jun 13 at 8:52