# 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:

 $\displaystyle\sum_{n\leq x}\mu^{2}(n)$ $\displaystyle=\frac{6}{\pi^{2}}x+O(\sqrt{x})$ (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:

• Abel’s lemma

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

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 AsymptoticEstimate 2013-03-22 16:00:01 2013-03-22 16:00:01 Wkbj79 (1863) Wkbj79 (1863) 13 Wkbj79 (1863) Definition msc 11N37 AsymptoticEstimatesForRealValuedNonnegativeMultiplicativeFunctions DisplaystyleYOmeganOleftFracxlogXy12YRightFor1LeY2 DisplaystyleXlog2xOleftsum_nLeX2OmeganRight DisplaystyleSum_nLeXYomeganO_yxlogXy1ForYGe0