In number theory, the Davenport–Erdős theorem states that, for sets of multiples of integers, several different notions of density are equivalent.[1][2][3]
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.[1][2][3]
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
:[1][2][3]
- The lower natural density, the inferior limit as
goes to infinity of the proportion of members of
in the interval
.
- The logarithmic density or multiplicative density, the weighted proportion of members of
in the interval
, again in the limit, where the weight of an element
is
.
- The sequential density, defined as the limit (as
goes to infinity) of the densities of the sets
of multiples of the first
elements of
. As these sets can be decomposed into finitely many disjoint arithmetic progressions, their densities are well defined without resort to limits.
However, there exist sequences
and their sets of multiples
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.[4]
The theorem is named after Harold Davenport and Paul Erdős, who published it in 1936.[5] Their original proof used the Hardy–Littlewood tauberian theorem; later, they published another, elementary proof.[6]
- Behrend sequence, a sequence
for which the density
described by this theorem is one
Ahlswede, Rudolf; Khachatrian, Levon H. (1997), "Classical results on primitive and recent results on cross-primitive sequences", The Mathematics of Paul Erdős, I, Algorithms and Combinatorics, vol. 13, Berlin: Springer, Theorem 1.11, p. 107, doi:10.1007/978-3-642-60408-9_9, MR 1425179 Hall, Richard R. (1996), Sets of multiples, Cambridge Tracts in Mathematics, vol. 118, Cambridge University Press, Cambridge, Theorem 0.2, p. 5, doi:10.1017/CBO9780511566011, ISBN 0-521-40424-X, MR 1414678 Tenenbaum, Gérald (2015), Introduction to Analytic and Probabilistic Number Theory, Graduate Studies in Mathematics, vol. 163 (3rd ed.), Providence, Rhode Island: American Mathematical Society, Theorem 249, p. 422, ISBN 978-0-8218-9854-3, MR 3363366