asymptotic estimate

characteristic function

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:

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:

• 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 ${\displaystyle \sum _{n\le x}}{\chi }_A(x)$, 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: