inequality of logarithmic and asymptotic density

For any A we denote A(n):=|A{1,2,,n}| and S(n):=k=1n1k.

Recall that the values

d¯(A)=lim infnA(n)n  d¯(A)=lim supnA(n)n

are called lower and upper asymptotic density of A.

The values

δ¯(A)=lim infnkA;kn1kS(n)  δ¯(A)=lim supnkA;kn1kS(n)

are called lower and upper logarithmic density of A.

We have S(n)lnn (we use the Landau notationMathworldPlanetmathPlanetmath). This follows from the fact that limnS(n)-lnn=γ is Euler’s constant. Therefore we can use lnn instead of S(n) in the definition of logarithmic density as well.

The sum in the definition of logarithmic density can be rewritten using Iverson’s convention as k=1n1k[kA]. (This means that we only add elements fulfilling the condition kA. This notation is introduced in [1, p.24].)

Theorem 1.

For any subset AN




We first observe that


There exists an n0 such that for each nn0 it holds d¯(A)-εA(n)nd¯(A)+ε.

We denote C:=1+S(n0). For nn0 we get

δ¯(A)=lim supnD(n)lnnd¯(A)+ε.

This inequalityMathworldPlanetmath holds for any ε>0, thus δ¯(A)d¯(A).

For the proof of the inequality for lower densities we put C:=k=1n0-1A(k)k(k+1)-(d¯(A)-ε)S(n0). We get


and this implies δ¯(A)d¯(A). ∎

For the proof using Abel’s partial summation see [4] or [5].

Corollary 1.

If a set has asymptotic density, then it has logarithmic density, too.

A well-known example of a set having logarithmic density but not having asymptotic density is the set of all numbers with the first digit equal to 1.

It can be moreover proved, that for any real numbers 0α¯β¯β¯α¯1 there exists a set A such that d¯(A)=α¯, δ¯(A)=β¯, δ¯(A)=β¯ and d¯(A)=α¯ (see [2]).


  • 1 R. L. Graham, D. E. Knuth, and O. Patashnik. Concrete mathematics. A foundation for computer science. Addison-Wesley, 1989.
  • 2 L. Mišík. Sets of positive integers with prescribed values of densities. Mathematica Slovaca, 52(3):289–296, 2002.
  • 3 H. H. Ostmann. Additive Zahlentheorie I. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1956.
  • 4 J. Steuding. number theory.
  • 5 G. Tenenbaum. Introduction to analytic and probabilistic number theory. Cambridge Univ. Press, Cambridge, 1995.
Title inequality of logarithmic and asymptotic density
Canonical name InequalityOfLogarithmicAndAsymptoticDensity
Date of creation 2014-03-24 9:16:11
Last modified on 2014-03-24 9:16:11
Owner kompik (10588)
Last modified by kompik (10588)
Numerical id 8
Author kompik (10588)
Entry type Theorem
Classification msc 11B05
Related topic AsymptoticDensity
Related topic LogarithmicDensity