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'square-free number'
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| Title of object: |
square-free number |
| Canonical Name: |
SquareFreeNumber |
| Type: |
Definition |
| Created on: |
2001-10-30 15:58:35 |
| Modified on: |
2002-12-27 07:35:50 |
| Classification: |
msc:11A51 |
| Synonyms: |
square-free number=square free number
square-free number=square free
square-free number=square-free |
Revision comment (for changes between this and next version):
| Fixed typo ``a sieve of Eratosthenes'' -> ``the sieve of Eratosthenes''. |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic} |
Content:
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 $x$.
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 the $x$ (namely, $p_i^2$.)
The asymptotic density of square-free numbers is $\frac{6}{\pi^2}$ which can be proved by application of a square-free variation of a sieve of Eratosthenes as follows:
\begin{align*}
A(n)&=\sum_{k\leq n} [n \text{ is a squarefree}]\\
&=\sum_{k\leq n} \sum_{d^2 | k} \mu(d)\\
&=\sum_{d \leq \sqrt{n}} \mu(d) \sum_{\substack{k \leq n\\d^2 | n}} 1 \\
&=\sum_{d \leq \sqrt{n}} \mu(d) \left\lfloor{\frac{n}{d^2}}\right\rfloor\\
&=n \sum_{d \leq \sqrt{n}} \frac{\mu(d)}{d^2}+O(\sqrt{n})\\
&=n \sum_{d} \frac{\mu(d)}{d^2}+O(\sqrt{n})\\
&=n \frac{1}{\zeta(2)} + O(\sqrt{n}) = n \frac{6}{\pi^2}+O(\sqrt{n}).
\end{align*}
It was shown that Riemann Hypothesis implies error term $O(n^{7/22+\epsilon})$ in the above. |
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