Distribution
Let Q(x) denote the number of square-free (quadratfrei) integers between 1 and x. For large n, 3/4 of the positive integers less than n are not divisible by 4, 8/9 of these numbers are not divisible by 9, and so on. Because these events are independent, we obtain the approximation:
This argument can be made rigorous to yield:
(see pi and big O notation). Under the Riemann hypothesis, the error term can be reduced:
See the race between the number of square-free numbers up to n and round(n/ζ(2)) on the OEIS:
A158819 – (Number of square-free numbers ≤ n) minus round(n/ζ(2)).
The asymptotic/natural density of square-free numbers is therefore
where ζ is the Riemann zeta function and 1/ζ(2) is approximately 0.6079 (over 3/5 of the integers are square-free).
Likewise, if Q(x,n) denotes the number of n-free integers (e.g. 3-free integers being cube-free integers) between 1 and x, one can show
Read more about this topic: Square-free Integer
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