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Homesquare-free number

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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 $a_{i}\geq 2$ 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 as follows:

$\displaystyle A(n)$ | $\displaystyle=\sum_{{k\leq n}}[k\text{ is squarefree }]$ | ||

$\displaystyle=\sum_{{k\leq n}}\sum_{{d^{2}|k}}\mu(d)$ | |||

$\displaystyle=\sum_{{d\leq\sqrt{n}}}\mu(d)\sum_{{\substack{k\leq n\\ d^{2}|n}}}1$ | |||

$\displaystyle=\sum_{{d\leq\sqrt{n}}}\mu(d)\left\lfloor{\frac{n}{d^{2}}}\right\rfloor$ | |||

$\displaystyle=n\sum_{{d\leq\sqrt{n}}}\frac{\mu(d)}{d^{2}}+O(\sqrt{n})$ | |||

$\displaystyle=n\sum_{{d\geq 1}}\frac{\mu(d)}{d^{2}}+O(\sqrt{n})$ | |||

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

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

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.

## Mathematics Subject Classification

11A51*no label found*

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## Corrections

form by Wkbj79 ✓

emphasis on defined terms; suppress link by Mathprof ✓

clarify by Mathprof ✓

what about n=1? by alozano ✓

## Comments

## some questions about squarefree numbers

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

## Re: some questions about squarefree numbers

Sorry. I should have looked up asymptotic density before posting. That answered most of my questions.

Wkbj79

## form

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.

## Re: form

Oops! This was supposed to be a correction. I've filed it properly. :-)

## Question about Square-free integers

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?)