logarithmic density

For any A we denote S(n):=k=1n1k. The values

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

are called lower and upper logarithmic density of A. If δ¯(A)=δ¯(A) we denote this value by δ(A) and call it the logarithmic density of A.

Logarithmic density can be equivalently defined as follows: If the limit


exists, then it is called logarithmic density of A.

By the well-known γ=limnS(n)-lnn defining Euler’s constant, we can see that the denominator in the above definitions can be replaced by lnn.


  • 1 M. Kolibiar, A. Legéň, T. Šalát, and Š. Znám. Algebra a príbuzné disciplíny. Alfa, Bratislava, 1992. (in Slovak)
  • 2 H. H. Ostmann. AdditivePlanetmathPlanetmath Zahlentheorie I. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1956.
  • 3 J. Steuding. http://www.math.uni-frankfurt.de/ steuding/steuding/prob.pdfProbabilistic number theory.
  • 4 G. Tenenbaum. Introduction to analytic and probabilistic number theory. Cambridge Univ. Press, Cambridge, 1995.
Title logarithmic density
Canonical name LogarithmicDensity
Date of creation 2013-03-22 15:31:54
Last modified on 2013-03-22 15:31:54
Owner kompik (10588)
Last modified by kompik (10588)
Numerical id 6
Author kompik (10588)
Entry type Definition
Classification msc 11B05
Related topic InequalityOfLogarithmicAndAsymptoticDensity
Defines upper logarithmic density
Defines lower logarithmic density