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asymptotic estimate

(Definition)

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

$\displaystyle \sum_{n \le x} \mu^2(n)$	$\displaystyle = \frac{6}{\pi^2}x+O(\sqrt{x})$	(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 estimates are typically valid for $x \to \infty$ . An example of an asymptotic estimate that is different from those above in this aspect is

$\displaystyle \cos x=1-\frac{x^2}{2}+O(x^4)$ for $\displaystyle \vert x\vert<1.$

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

Tools that are useful for obtaining asymptotic estimates include:

If $A \subseteq \mathbb{N}$ , then an asymptotic estimate for $\displaystyle \sum_{n \le x} \chi_A(x)$ , where $\chi_A$ denotes the characteristic function of , enables one to determine the asymptotic density of using the formula

$\displaystyle \lim_{x \to \infty} \frac{1}{x} \sum_{n \le x} \chi_A(x)$

provided the limit exists. The upper asymptotic density of

and the lower asymptotic density of

can be computed in a similar manner using $\limsup$ and $\liminf$ , respectively. (See asymptotic density for more details.)

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

$\begin{displaymath}\begin{array}{ll} \displaystyle \lim_{x \to \infty} \frac{1}{... ...x} \right) \ \ & \displaystyle =\frac{6}{\pi^2} \end{array}\end{displaymath}$

"asymptotic estimate" is owned by Wkbj79.

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See Also: asymptotic estimates for real-valued nonnegative multiplicative functions, $\displaystyle \sum_{n \le x} y^{\Omega(n)}=O\left( \frac{x(\log x)^{y-1}}{2-y} \right)$ for $1 \le y<2$ , $\displaystyle x\log^2x=O\left(\sum_{n \le x} 2^{\Omega(n)} \right)$ , $\displaystyle \sum_{n \le x} y^{\omega(n)}=O_y(x(\log x)^{y-1})$ for $y \ge 0$

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Cross-references: natural numbers, squarefree, lower asymptotic density, upper asymptotic density, limit, asymptotic density, Dirichlet hyperbola method, Abel's lemma, Euler-Maclaurin summation formula, prime number theorem, convolution method, Landau notation
There are 3 references to this entry.

This is version 10 of asymptotic estimate, born on 2006-06-13, modified 2008-03-14.
Object id is 8027, canonical name is AsymptoticEstimate.
Accessed 1582 times total.

Classification:

AMS MSC:

11N37 (Number theory :: Multiplicative number theory :: Asymptotic results on arithmetic functions)

Pending Errata and Addenda

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