asymptotic estimate


characteristic functionMathworldPlanetmathPlanetmathPlanetmath

An asymptotic estimate is an that involves the use of O, o, or . These are all defined in the entry Landau notationMathworldPlanetmathPlanetmath. Examples of asymptotic are:

nxμ2(n) =6π2x+O(x) (see convolution method for more details)
π(x) xlogx (see prime number theoremMathworldPlanetmath for more details)

Unless otherwise specified, asymptotic are typically valid for x. An example of an asymptotic that is different from those above in this aspect is

cosx=1-x22+O(x4) for |x|<1.

Note that the above would be undesirable for x, as the would be larger than the . Such is not the case for |x|<1, though.

Tools that are useful for obtaining asymptotic include:

If A, then an asymptotic for nxχA(x), where χA denotes the characteristic function ( of A, enables one to determine the asymptotic density of A using the


provided the limit exists. The upper asymptotic density of A and the lower asymptotic density of A can be computed in a manner using lim sup and lim inf, respectively. (See asymptotic density ( for more details.)

For example, μ2 is the characteristic function of the squarefreeMathworldPlanetmath natural numbersMathworldPlanetmath. Using the asymptotic above yields the asymptotic density of the squarefree natural numbers:


Title asymptotic estimate
Canonical name AsymptoticEstimate
Date of creation 2013-03-22 16:00:01
Last modified on 2013-03-22 16:00:01
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 13
Author Wkbj79 (1863)
Entry type Definition
Classification msc 11N37
Related topic AsymptoticEstimatesForRealValuedNonnegativeMultiplicativeFunctions
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