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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 $$\sup_{n,m,l \in \mathbb{N}} |f(n,l)-f(m,l)| = O(k).$$

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No. There are various results which give counterexamples.

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 $\ell \prod_{p\leq x}\left(1-\frac{1}{p}\right)\sim \frac{e^{-\gamma}x^2}{\log x}$.

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  • $\begingroup$ Thanks for your answer. What do you think about the correct order? Is it $O(k^2 \log k)$? $\endgroup$ – Martin Hicks Jun 12 at 19:35
  • $\begingroup$ 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. $\endgroup$ – Jan-Christoph Schlage-Puchta Jun 13 at 8:52

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