inferior. One may say has asymptotic density if , in which case is equal to this common value. This definition can be restated in the following way: if this limit exists. These definitions may equivalently be expressed in the following way. Given a subset of , write it as an increasing sequence indexed by the natural numbers: Then and if the limit exists. A somewhat weaker notion of density is the upper Banach density of a set This is defined as Properties and examples For any finite set F of positive integers, d(F) = 0. If d(A) exists for some set A and Ac denotes its complement set with respect to , then d(Ac) = 1 − d(A). Corollary: If is finite (including the case ), If and exist, then If is the set of all squares, then d(A) = 0. If is the set of all even numbers, then d(A) = 0.5. Similarly, for any arithmetical progression we get For the set P of all primes we get from the prime number theorem that d(P) = 0. The set of all square-free integers has density More generally, the set of all nth-power-free numbers for any natural n has density where is the Riemann zeta function. The set of abundant numbers has non-zero density. Marc Deléglise showed in 1998 that the density of the set of abundant numbers is between 0.2474 and 0.2480. The set of numbers whose binary expansion contains an odd number of digits is an example of a set which does not have an asymptotic density, since the upper density of this set is whereas its lower density is The set of numbers whose decimal expansion begins with the digit 1 similarly has no natural density: the lower density is 1/9 and the upper density is 5/9. (See Benford's law.) Consider an equidistributed sequence in and define a monotone family of sets: Then, by definition, for all . If S is a set of positive upper density then Szemerédi's theorem states that S contains arbitrarily large finite arithmetic progressions, and the Furstenberg–Sárközy theorem states that some two members of S differ by a square number. Other density functions Other density functions on subsets of the natural numbers may be defined analogously. For example, the logarithmic density of a set A is defined as the limit (if it exists) Upper and lower logarithmic densities are defined analogously as