# Asymptotic density

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

${\ displaystyle \ lim _ {n \ rightarrow \ infty} {\ frac {a (n)} {n}}}$

the asymptotic density of a subset . It is the counting function of . This indicates how many elements from are not larger than . It applies . ${\ displaystyle d (A)}$${\ displaystyle A \ subseteq \ mathbb {N}}$${\ displaystyle a (n)}$${\ displaystyle A}$${\ displaystyle A}$${\ displaystyle n}$${\ displaystyle 0 \ leq d (A) \ leq 1}$

## Upper and lower asymptotic density

For any one be and . ${\ displaystyle n \ in \ mathbb {N}}$${\ displaystyle A (n) = \ {1,2, \ dotsc, n \} \ cap A}$${\ displaystyle a (n) = | A (n) |}$

The upper asymptotic density of is then through ${\ displaystyle {\ overline {d}} (A)}$${\ displaystyle A}$

${\ displaystyle {\ overline {d}} (A) \ colon = \ limsup _ {n \ rightarrow \ infty} {\ frac {a (n)} {n}}}$

defined, where lim sup is the limes superior . Likewise, it 's through ${\ displaystyle {\ underline {d}} (A)}$

${\ displaystyle {\ underline {d}} (A) \ colon = \ liminf _ {n \ rightarrow \ infty} {\ frac {a (n)} {n}}}$

defined lower asymptotic density of . has an asymptotic density only if holds. In this case the limit exists ${\ displaystyle A}$${\ displaystyle A}$ ${\ displaystyle d (A)}$${\ displaystyle {\ underline {d}} (A) = {\ overline {d}} (A)}$

${\ displaystyle \ lim _ {n \ rightarrow \ infty} {\ frac {a (n)} {n}} = {\ underline {d}} (A) = {\ overline {d}} (A) = \ colon d (A)}$

and therefore can be defined by him . ${\ displaystyle d (A)}$

## Examples

• If exists for the set , then for the complementary set :${\ displaystyle d (A)}$${\ displaystyle A}$${\ displaystyle \ mathbb {N}}$ ${\ displaystyle {\ overline {A}}}$${\ displaystyle d ({\ overline {A}}) = 1-d (A)}$
• ${\ displaystyle d (\ mathbb {N}) = 1}$
• For any finite set of natural numbers we have:${\ displaystyle E}$${\ displaystyle d (E) = 0}$
• The following applies to the set of all square numbers :${\ displaystyle A = \ {n ^ {2}; n \ in \ mathbb {N} \}}$${\ displaystyle d (A) = 0}$
• The following applies to the set of all even numbers:${\ displaystyle A = \ {2n; n \ in \ mathbb {N} \}}$${\ displaystyle d (A) = 1/2}$
• More generally applies to every arithmetic sequence with a positive :${\ displaystyle A = \ {an + b; n \ in \ mathbb {N} \}}$${\ displaystyle a}$${\ displaystyle d (A) = 1 / a}$
• For the set of all prime numbers one obtains due to the prime number theorem :${\ displaystyle P}$${\ displaystyle d (P) = 0}$
• The set of all square-free natural numbers has the density with the Riemann zeta function .${\ displaystyle 6 / \ pi ^ {2} = 1 / \ zeta (2)}$ ${\ displaystyle \ zeta}$
• 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:${\ displaystyle A = \ bigcup \ limits _ {n = 0} ^ {\ infty} \ left \ {2 ^ {2n}, \ dotsc, 2 ^ {2n + 1} -1 \ right \}}$
${\ displaystyle {\ underline {d}} (A) = \ lim _ {m \ rightarrow \ infty} {\ frac {1 + 2 ^ {2} + \ dotsb + 2 ^ {2m}} {2 ^ {2m +2} -1}} = \ lim _ {m \ rightarrow \ infty} {\ frac {2 ^ {2m + 2} -1} {3 (2 ^ {2m + 2} -1)}} = {\ frac {1} {3}}}$
${\ displaystyle {\ overline {d}} (A) = \ lim _ {m \ rightarrow \ infty} {\ frac {1 + 2 ^ {2} + \ dotsb + 2 ^ {2m}} {2 ^ {2m +1} -1}} = \ lim _ {m \ rightarrow \ infty} {\ frac {2 ^ {2m + 2} -1} {3 (2 ^ {2m + 1} -1)}} = {\ frac {2} {3}}}$

## swell

• Melvyn B. Nathanson: Elementary methods in number theory (=  Graduate Texts in Mathematics . Volume 195 ). Springer, New York 2000, ISBN 0-387-98912-9 (English, zbmath.org ).
• 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 ( books.google.de - reading sample).
• Jörn Steuding: Probabilistic number theory. (PDF) In: psu.edu. citeseerx.ist.psu.edu, 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, zbmath.org ).