Natural density

The asymptotic density is a number-theoretic threshold that indicates the proportion of a subset of natural numbers to the set of natural numbers.

Simple definition

This is called the limit

The asymptotic density of a subset. It is the counting function of. This specifies how many elements are not larger than. It is true.

Is and for any.

The upper asymptotic density is then given by

Defined, where lim sup is the limit superior. Similarly, the by

Defined lower asymptotic density. only has an asymptotic density when applies. In this case, the limit exists

And therefore can be defined by it.

If exists for the amount, then for the respect complementary Quantity:
For an arbitrary finite set of natural numbers is valid:
For the set of all square numbers applies:
For the set of all even numbers is valid:
More generally, for each arithmetic sequence with positive a:
For the set of all prime numbers is obtained due to the prime number theorem:
The set of all square-free natural numbers has the density with the Riemann zeta function.
The density of abundant numbers is between 0.2474 and 0.2480.
The set of numbers whose binary representation has an odd number of digits, is an example of a set without asymptotic density. For the lower and upper asymptotic density applies in this case:

Melvyn B. Nathanson: Elementary Methods in Number Theory, 195 Springer -Verlag, 2000, ISBN 0,387,989,129th
Ostermann HH: Additive Number Theory I ( German ), 7 Springer -Verlag, Berlin -Göttingen -Heidelberg 1956.
Jörn Steuding: Probabilistic number theory. Retrieved on 6 October 2005.
Gérald Tenenbaum: Introduction to analytic and probabilistic number theory, 46 Cambridge University Press, Cambridge 1995.

Square-free integer

84798