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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 † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 1 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. 2