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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, $p_i^2$ .)

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 Aaron Krowne, Boris Bukh.

How to Cite This Entry

Aaron Krowne, Boris Bukh. "square-free number" (version 12). PlanetMath.org. Freely available at http://planetmath.org/SquareFreeNumber.html.

Classification

AMS MSC:

11A51 (Number theory :: Elementary number theory :: Factorization; primality)

Pending Errata and Addenda

None. [ View all 6 ]

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Discussion

Question about Square-free integers by evangelion1987 on 2008-10-05 09:41:13

From this post, I realized that
the number of square-free integers smaller than x approaches
6x/pi^2, but what about the number of square-free integers
smaller than x that is "DIVISIBLE" by some prime number pi
smaller than x?
Is there a way of approximating it?
(For instance, is there a way of approximating the number of
square-free integers smaller than 10^30 that are divisible by
2 or 3 or 11?)

[ reply | up ]

form by Warren Buck on 2006-06-26 05:49:25

In the very last line of the string of equalities, there are two equals signs. I think it would look better if the last statement (the result you're trying to prove) were on a line of its own.

[ reply | up ]

Re: form by Warren Buck on 2006-06-26 05:51:11

some questions about squarefree numbers by Warren Buck on 2003-03-13 02:35:57

Let s(n) denote the function that takes positive integers as its input and yields the number of (positive) squarefree numbers less than or equal to its input as its output. Is there a function like s(n) that already exists? Is there a general formula for s(n)? What is the limit as n approaches infinity of s(n)/n?

Wkbj79

[ reply | up ]

Re: some questions about squarefree numbers by Warren Buck on 2003-03-13 02:39:06

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square-free number

Asymptotic Analysis

References

How to Cite This Entry

Classification

Tags

Pending Errata and Addenda

Interact

Discussion