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square-free number∗
akrowne†
2013-03-21 12:42:54
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=
r
Y
pai i
i=1
is the prime factorization of x into r distinct primes, then ai ≥ 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 ai were to be greater than or
equal to two, we could be sure that at least one square divides x (namely, p2i .)
∗ hSquarefreeNumberi created: h2013-03-21i by: hakrownei version: h30636i Privacy
setting: h1i hDefinitioni h11A51i
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Asymptotic Analysis
The asymptotic density of square-free numbers is π62 which can be proved by
application of a square-free variation of the sieve of Eratosthenes as follows:
X
A(n) =
[k is squarefree ]
k≤n
=
XX
µ(d)
k≤n d2 |k
=
X
µ(d)
√
d≤ n
=
X
1
k≤n
d2 |n
µ(d)
√
d≤ n
=n
X
jnk
d2
X µ(d)
√
+ O( n)
2
d
√
d≤ n
=n
X µ(d)
d≥1
d2
√
+ O( n)
√
1
+ O( n)
ζ(2)
√
6
= n 2 + O( n).
π
=n
It was shown that the Riemann Hypothesis implies error term O(n7/22+ ) in the
above [?].
References
[1] R. C. Baker and J. Pintz. The distribution of square-free numbers. Acta
Arith., 46:73–79, 1985. Zbl 0535.10045.
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