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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 $$ \cos x=1-\frac{x^2}{2}+O(x^4) \text{ for } |x|<1. $$ 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 $|x|<1$ , 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 $$ \lim_{x \to \infty} \frac{1}{x} \sum_{n \le x} \chi_A(x) $$ 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:
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