inequality of logarithmic and asymptotic density
For any we denote and .
Recall that the values
are called lower and upper asymptotic density of .
are called lower and upper logarithmic density of .
We have (we use the Landau notation). This follows from the fact that is Euler’s constant. Therefore we can use instead of in the definition of logarithmic density as well.
The sum in the definition of logarithmic density can be rewritten using Iverson’s convention as . (This means that we only add elements fulfilling the condition . This notation is introduced in [1, p.24].)
For any subset
We first observe that
There exists an such that for each it holds .
We denote . For we get
This inequality holds for any , thus .
For the proof of the inequality for lower densities we put . We get
and this implies . ∎
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.
- 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. http://www.math.uni-frankfurt.de/~steuding/steuding/prob.pdfProbabilistic number theory.
- 5 G. Tenenbaum. Introduction to analytic and probabilistic number theory. Cambridge Univ. Press, Cambridge, 1995.
|Title||inequality of logarithmic and asymptotic density|
|Date of creation||2014-03-24 9:16:11|
|Last modified on||2014-03-24 9:16:11|
|Last modified by||kompik (10588)|