# 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 ${a}_{i}\ge 2$ 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:

$A(n)$ | $={\displaystyle \sum _{k\le n}}[k\text{is squarefree}]$ | ||

$={\displaystyle \sum _{k\le n}}{\displaystyle \sum _{{d}^{2}|k}}\mu (d)$ | |||

$={\displaystyle \sum _{d\le \sqrt{n}}}\mu (d){\displaystyle \sum _{\begin{array}{c}k\le n\\ {d}^{2}|n\end{array}}}1$ | |||

$={\displaystyle \sum _{d\le \sqrt{n}}}\mu (d)\lfloor {\displaystyle \frac{n}{{d}^{2}}}\rfloor $ | |||

$=n{\displaystyle \sum _{d\le \sqrt{n}}}{\displaystyle \frac{\mu (d)}{{d}^{2}}}+O(\sqrt{n})$ | |||

$=n{\displaystyle \sum _{d\ge 1}}{\displaystyle \frac{\mu (d)}{{d}^{2}}}+O(\sqrt{n})$ | |||

$=n{\displaystyle \frac{1}{\zeta (2)}}+O(\sqrt{n})$ | |||

$=n{\displaystyle \frac{6}{{\pi}^{2}}}+O(\sqrt{n}).$ |

It was shown that the Riemann Hypothesis^{} implies error term $O({n}^{7/22+\u03f5})$ 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 |