View source for Asymptotic density

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A variant of the general concept of the [[density of a sequence]] of natural numbers; which measures how large a part of the sequence of all natural numbers belongs to the given sequence $A$ of natural numbers including zero. The ''(lower) asymptotic density'' of a sequence $A$ is expressed by the real number $\alpha$ defined by the formula

$$ \alpha=\liminf_{x\to\infty}\frac{A(x)}{x},$$

where

$$ A(x)=\sum_{\substack{a\in A\\0<a\leq x}}1,\quad x\geq 1.$$

The number

$$\beta=\limsup_{x\to\infty}\frac{A(x)}{x}$$

is known as the ''upper asymptotic density''. If the numbers $\alpha$ and $\beta$ coincide, their common value is called the ''natural density''. Thus, for instance, the sequence of numbers that are free from squares has the natural density $\delta=6/\pi^2$. The concept of an asymptotic density is employed in finding criteria for some sequence to be an [[asymptotic basis]].

====References====
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|valign="top"|{{Ref|HaRo}}||valign="top"|  H. Halberstam,   K.F. Roth,   "Sequences" , '''1''' , Clarendon Press  (1966) {{ZBL|0141.04405}} (repr. 1983) ISBN 0-387-90801-3 {{ZBL|0498.10001}} 
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