Natural Density

In number theory, asymptotic density (or natural density or arithmetic density) is one of the possibilities to measure how large a subset of the set of natural numbers is.

Intuitively, it is thought that there are more positive integers than perfect squares, since every perfect square is already positive, and many other positive integers exist besides. However, the set of positive integers is not in fact larger than the set of perfect squares: both sets are infinite and countable and can therefore be put in one-to-one correspondence. Nevertheless if one goes through the natural numbers, the squares become increasingly scarce. This notion will be described mathematically, and we will see that the squares have a 'density' which is lower than the density of the natural numbers.

If an integer is randomly selected from the set, then the probability that it belongs to A is the ratio of the number of elements of A in to the total number of elements in . If this probability tends to some limit as n tends to infinity, then this limit is referred to as the asymptotic density of A. This notion can be understood as a kind of probability of choosing a number from the set A. Indeed, the asymptotic density (as well as some other types of densities) is studied in probabilistic number theory.

Asymptotic density contrasts, for example, with the Schnirelmann density. A drawback of this approach is that the asymptotic density is not defined for all subsets of .