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asymptotic estimate∗ Wkbj79† 2013-03-21 20:42:30 characteristic function An asymptotic estimate is an estimate that involves the use of O, o, or ∼. These are all defined in the entry Landau notation. Examples of asymptotic estimates are: X µ2 (n) n≤x π(x) √ 6 x + O( x) 2 π x ∼ log x = (see convolution method for more details) (see prime number theorem for more details) Unless otherwise specified, asymptotic estimates are typically valid for x → ∞. An example of an asymptotic estimate that is different from those above in this aspect is x2 cos x = 1 − + O(x4 ) for |x| < 1. 2 Note that the above estimate would be undesirable for x → ∞, as the error would be larger than the estimate. Such is not the case for |x| < 1, though. Tools that are useful for obtaining asymptotic estimates include: • the Euler-Maclaurin summation formula • Abel’s lemma • the convolution method • the Dirichlet hyperbola method If A ⊆ N, then an asymptotic estimate for X χA (x), where χA denotes the n≤x characteristic function of A, enables one to determine the asymptotic density of A using the formula 1X lim χA (x) x→∞ x n≤x ∗ hAsymptoticEstimatei created: h2013-03-21i by: hWkbj79i version: h38027i Privacy setting: h1i hDefinitioni h11N37i † 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 provided the limit exists. The upper asymptotic density of A and the lower asymptotic density of A can be computed in a similar manner using lim sup and lim inf, respectively. (See asymptotic density for more details.) For example, µ2 is the characteristic function of the squarefree natural numbers. Using the asymptotic estimate above yields the asymptotic density of the squarefree natural numbers: √ 1 6 1X 2 x + O( x) µ (n) = lim lim x→∞ x x→∞ x π2 n≤x 6 +O x→∞ π 2 = lim = 6 π2 2 √ x x