asymptotic estimate
An asymptotic estimate is an that involves the use of , , or . These are all defined in the entry Landau notation. Examples of asymptotic are:
(see convolution method for more details) | ||
(see prime number theorem for more details) |
Unless otherwise specified, asymptotic are typically valid for . An example of an asymptotic that is different from those above in this aspect is
Note that the above would be undesirable for , as the would be larger than the . Such is not the case for , though.
Tools that are useful for obtaining asymptotic include:
- •
-
•
Abel’s lemma
-
•
the convolution method (http://planetmath.org/ConvolutionMethod)
- •
If , then an asymptotic for , where denotes the characteristic function (http://planetmath.org/CharacteristicFunction) of , enables one to determine the asymptotic density of using the
provided the limit exists. The upper asymptotic density of and the lower asymptotic density of can be computed in a manner using and , respectively. (See asymptotic density (http://planetmath.org/AsymptoticDensity) for more details.)
For example, is the characteristic function of the squarefree natural numbers. 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 |
Related topic | DisplaystyleYOmeganOleftFracxlogXy12YRightFor1LeY2 |
Related topic | DisplaystyleXlog2xOleftsum_nLeX2OmeganRight |
Related topic | DisplaystyleSum_nLeXYomeganO_yxlogXy1ForYGe0 |