Asymptotic density

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The asymptotic density is a number-theoretic limit that indicates the proportion of a subset of natural numbers in the set of natural numbers.

Simple definition

One calls the limit value

the asymptotic density of a subset . It is the counting function of . This indicates how many elements from are not larger than . It applies .

Upper and lower asymptotic density

For any one be and .

The upper asymptotic density of is then through

defined, where lim sup is the limes superior . Likewise, it 's through

defined lower asymptotic density of . has an asymptotic density only if holds. In this case the limit exists

and therefore can be defined by him .


  • If exists for the set , then for the complementary set :
  • For any finite set of natural numbers we have:
  • The following applies to the set of all square numbers :
  • The following applies to the set of all even numbers:
  • More generally applies to every arithmetic sequence with a positive :
  • For the set of all prime numbers one obtains due to the prime number theorem :
  • The set of all square-free natural numbers has the density with the Riemann zeta function .
  • The density of abundant numbers is between 0.2474 and 0.2480.
  • The set of all numbers whose binary representation has an odd number of digits is an example of a set with no asymptotic density. In this case, the following applies to the lower and upper asymptotic density:


  • Melvyn B. Nathanson: Elementary methods in number theory (=  Graduate Texts in Mathematics . Volume 195 ). Springer, New York 2000, ISBN 0-387-98912-9 (English, ).
  • Hans-Heinrich Ostmann: Additive number theory (=  results of mathematics and its border areas . Volume 7 ). Part One: General Investigations . Springer-Verlag, Berlin / Göttingen / Heidelberg 1956, ISBN 978-3-662-11030-0 ( - reading sample).
  • Jörn Steuding: Probabilistic number theory. (PDF) In:, accessed February 7, 2016 .
  • Gérald Tenenbaum: Introduction to analytic and probabilistic number theory (=  Cambridge studies in advanced mathematics . Volume 46 ). Cambridge university press, Cambridge 1995, ISBN 0-521-41261-7 (French, ).