asymptotic estimate
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:
$\sum _{n\le x}}{\mu}^{2}(n)$  $={\displaystyle \frac{6}{{\pi}^{2}}}x+O(\sqrt{x})$  (see convolution method for more details) 
$\pi (x)$  $\sim {\displaystyle \frac{x}{\mathrm{log}x}}$  (see prime number theorem^{} for more details) 
Unless otherwise specified, asymptotic are typically valid for $x\to \mathrm{\infty}$. An example of an asymptotic that is different from those above in this aspect is
$$ 
Note that the above would be undesirable for $x\to \mathrm{\infty}$, as the would be larger than the . Such is not the case for $$, though.
Tools that are useful for obtaining asymptotic include:
 •

•
Abel’s lemma

•
the convolution method (http://planetmath.org/ConvolutionMethod)
 •
If $A\subseteq \mathbb{N}$, then an asymptotic for $\sum _{n\le x}}{\chi}_{A}(x)$, where ${\chi}_{A}$ denotes the characteristic function (http://planetmath.org/CharacteristicFunction) of $A$, enables one to determine the asymptotic density of $A$ using the
$$\underset{x\to \mathrm{\infty}}{lim}\frac{1}{x}\sum _{n\le x}{\chi}_{A}(x)$$ 
provided the limit exists. The upper asymptotic density of $A$ and the lower asymptotic density of $A$ can be computed in a manner using $lim\; sup$ and $lim\; inf$, 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:
$\begin{array}{cc}\underset{x\to \mathrm{\infty}}{lim}{\displaystyle \frac{1}{x}}{\displaystyle \sum _{n\le x}}{\mu}^{2}(n)\hfill & =\underset{x\to \mathrm{\infty}}{lim}{\displaystyle \frac{1}{x}}\left({\displaystyle \frac{6}{{\pi}^{2}}}x+O(\sqrt{x})\right)\hfill \\ & \\ & =\underset{x\to \mathrm{\infty}}{lim}{\displaystyle \frac{6}{{\pi}^{2}}}+O\left({\displaystyle \frac{\sqrt{x}}{x}}\right)\hfill \\ & \\ & ={\displaystyle \frac{6}{{\pi}^{2}}}\hfill \end{array}$
Title  asymptotic estimate 

Canonical name  AsymptoticEstimate 
Date of creation  20130322 16:00:01 
Last modified on  20130322 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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