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Question: Is there a function $f(n) \rightarrow \infty$, such that infinitely often the interval $[n,n+\frac{f(n) \log(n)}{\log{\log(n)}}]$ does not contain a squarefree integer?

Additional information: If we consider the simultaneous congruences:

$$ x \equiv 0 \ (\text{mod} \ 2^2)\\ x \equiv 1 \ (\text{mod} \ 3^2) \\ \cdots \\ x \equiv k-1 \ (\text{mod} \ p_{k}^2) $$

Which has a solution not bigger than $ \Pi_{i=1}^{k} p_i^2 \leq e^{2(1+\varepsilon)k\log k} $.

This way we can see that there is a constant such that there are no squarefree numbers in the interval $[n,n+c\frac{\log(n)}{\log{\log(n)}}]$ infinitely often. We can optimize the above equations, for example on the right hand side we don't have to consider numbers divisible by $4$ after the first equation. Or numbers congruent to $1$ modulo 9 after the second equation etc. but that just gives a better constant!

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Interesting question. The literature seems to have quite a lot about upper bounds. The best upper bounds I could find were $[n,n+cn^{1/5}\log n]$ (due to Filaseta) - clearly much bigger than your lower bound. Granville showed that the abc conjecture implies an upper bound of $n^\epsilon$ for any $\epsilon>0$. –  Anthony Quas Nov 18 '13 at 2:13

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up vote 10 down vote accepted

Erdos has mentioned this lower bound in several places, adding always that he's never been able to improve it. For example see http://renyi.hu/~p_erdos/1951-13.pdf (page 107; in fact he gives an explicit constant here that he says he cannot improve), and page 8 of http://hsb.org.hu/~p_erdos/1981-21.pdf .

A standard Borel-Cantelli type heuristic suggests that the gaps should be bounded by some constant times $\log n$, analogous to the Cramer conjectures for gaps between primes. I don't know if anyone has written down such a conjecture in this context. But I did find a paper by Kevin McCurley where he considers the least square-free number in an arithmetic progressions, and formulates a Borel-Cantelli type conjecture in this context. In analytic number theory, the situations of arithmetic progressions and short intervals are usually very similar, and one could adapt McCurley's argument to write down conjectures in the short interval case. McCurley's paper is here: http://www.ams.org/journals/tran/1986-293-02/S0002-9947-1986-0816304-1/S0002-9947-1986-0816304-1.pdf .

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