asymptotic estimate

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

$\displaystyle\sum_{n\leq x}\mu^{2}(n)$	$\displaystyle=\frac{6}{\pi^{2}}x+O(\sqrt{x})$	(see convolution method for more details)
$\displaystyle\pi(x)$	$\displaystyle\sim\frac{x}{\log x}$	(see prime number theorem for more details)

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

\cos x=1-\frac{x^{2}}{2}+O(x^{4})\text{ for }|x|<1.

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

Tools that are useful for obtaining asymptotic include:

•

the Euler-Maclaurin summation formula
•

Abel’s lemma
•

the convolution method (http://planetmath.org/ConvolutionMethod)
•

the Dirichlet hyperbola method

If $A\subseteq\mathbb{N}$ , then an asymptotic for $\displaystyle\sum_{n\leq x}\chi_{A}(x)$ , where $\chi_{A}$ denotes the characteristic function (http://planetmath.org/CharacteristicFunction) of $A$ , enables one to determine the asymptotic density of $A$ using the

\lim_{x\to\infty}\frac{1}{x}\sum_{n\leq x}\chi_{A}(x)

provided the limit exists. The upper asymptotic density of $A$ and the lower asymptotic density of $A$ can be computed in a manner using $\limsup$ and $\liminf$ , respectively. (See asymptotic density (http://planetmath.org/AsymptoticDensity) for more details.)

For example, $\mu^{2}$ is the characteristic function of the squarefree natural numbers. Using the asymptotic above yields the asymptotic density of the squarefree natural numbers:

$\begin{array}[]{ll}\displaystyle\lim_{x\to\infty}\frac{1}{x}\sum_{n\leq x}\mu^% {2}(n)&\displaystyle=\lim_{x\to\infty}\frac{1}{x}\left(\frac{6}{\pi^{2}}x+O(% \sqrt{x})\right)\\ \\ &\displaystyle=\lim_{x\to\infty}\frac{6}{\pi^{2}}+O\left(\frac{\sqrt{x}}{x}% \right)\\ \\ &\displaystyle=\frac{6}{\pi^{2}}\end{array}$

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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