Equivalence of notions of density for sets of multiples of integers
In number theory, the Davenport–Erdős theorem states that, for sets of multiples of integers, several different notions of density are equivalent.
Let formula_1 be a sequence of positive integers. Then the multiples of formula_2 are another set formula_3 that can be defined as the set formula_4 of numbers formed by multiplying members of formula_2 by arbitrary positive integers.
According to the Davenport–Erdős theorem, for a set formula_3, the following notions of density are equivalent, in the sense that they all produce the same number as each other for the density of formula_3:
However, there exist sequences formula_2 and their sets of multiples formula_3 for which the upper natural density (taken using the superior limit in place of the inferior limit) differs from the lower density, and for which the natural density itself (the limit of the same sequence of values) 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.
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