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asymptotic estimate (Definition)

An asymptotic estimate is an estimate that involves the use of $ O$, $ o$, 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 $ A$, enables one to determine the asymptotic density of $ A$ 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 $ A$ and the lower asymptotic density of $ A$ 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}



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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
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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 MSC11N37 (Number theory :: Multiplicative number theory :: Asymptotic results on arithmetic functions)

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