# square-free number

A square-free number is a natural number that contains no powers greater than 1 in its prime factorization. In other words, if $x$ is our number, and

 $x=\prod_{i=1}^{r}p_{i}^{a_{i}}$

is the prime factorization of $x$ into $r$ distinct primes, then $a_{i}\geq 2$ is always false for square-free $x$.

Note: we assume here that $x$ itself must be greater than 1; hence 1 is not considered square-free. However, one must be alert to the particular context in which “square-free” is used as to whether this is considered the case.

The name derives from the fact that if any $a_{i}$ were to be greater than or equal to two, we could be sure that at least one square divides $x$ (namely, $p_{i}^{2}$.)

## 1 Asymptotic Analysis

The asymptotic density of square-free numbers is $\frac{6}{\pi^{2}}$ which can be proved by application of a square-free variation of the sieve of Eratosthenes (http://planetmath.org/SieveOfEratosthenes2) as follows:

 $\displaystyle A(n)$ $\displaystyle=\sum_{k\leq n}[k\text{ is squarefree }]$ $\displaystyle=\sum_{k\leq n}\sum_{d^{2}|k}\mu(d)$ $\displaystyle=\sum_{d\leq\sqrt{n}}\mu(d)\sum_{\begin{subarray}{c}k\leq n\\ d^{2}|n\end{subarray}}1$ $\displaystyle=\sum_{d\leq\sqrt{n}}\mu(d)\left\lfloor{\frac{n}{d^{2}}}\right\rfloor$ $\displaystyle=n\sum_{d\leq\sqrt{n}}\frac{\mu(d)}{d^{2}}+O(\sqrt{n})$ $\displaystyle=n\sum_{d\geq 1}\frac{\mu(d)}{d^{2}}+O(\sqrt{n})$ $\displaystyle=n\frac{1}{\zeta(2)}+O(\sqrt{n})$ $\displaystyle=n\frac{6}{\pi^{2}}+O(\sqrt{n}).$

It was shown that the Riemann Hypothesis implies error term $O(n^{7/22+\epsilon})$ in the above [1].

## References

• 1 R. C. Baker and J. Pintz. The distribution of square-free numbers. Acta Arith., 46:73–79, 1985. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0535.10045Zbl 0535.10045.
 Title square-free number Canonical name SquarefreeNumber Date of creation 2013-03-22 11:55:33 Last modified on 2013-03-22 11:55:33 Owner akrowne (2) Last modified by akrowne (2) Numerical id 17 Author akrowne (2) Entry type Definition Classification msc 11A51 Synonym square free number Synonym square free Synonym square-free Synonym squarefree Related topic MoebiusFunction Related topic SquareRootsOfRationals