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square-free number

(Definition)

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

$\displaystyle x = \prod_{i=1}^r p_i^{a_i}$

is the prime factorization of into distinct primes, then $a_i \ge 2$ is always false for square-free .

Note: we assume here that 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 were to be greater than or equal to two, we could be sure that at least one square divides (namely, .)

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 as follows:

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

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.
Zbl 0535.10045.

"square-free number" is owned by akrowne. [ full author list (2) ]

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