Davenport–Erdős theorem


In number theory, the Davenport–Erdős theorem states that, for sets of multiples of integers, several different notions of density are equivalent.
Let be a sequence of positive integers. Then the multiples of are another set that can be defined as the set of numbers formed by multiplying members of by arbitrary positive integers.
According to the Davenport–Erdős theorem, for a set, the following notions of density are equivalent, in the sense that they all produce the same number as each other for the density of :
However, there exist sequences and their sets of multiples for which the upper natural density differs from the lower density, and for which the natural density itself does not exist.
The theorem is named after Harold Davenport and Paul Erdős, who published it in 1936. Their original proof used the Hardy–Littlewood tauberian theorem; later, they published another, elementary proof.