# logarithmic density

For any $A\subseteq\mathbb{N}$ we denote $S(n):=\sum\limits_{k=1}^{n}\frac{1}{k}$. The values

 $\overline{\delta}(A)=\liminf_{n\to\infty}\frac{\sum\limits_{k\in A;k\leq n}% \frac{1}{k}}{S(n)}\qquad\underline{\delta}(A)=\limsup_{n\to\infty}\frac{\sum% \limits_{k\in A;k\leq n}\frac{1}{k}}{S(n)}$

are called lower and upper logarithmic density of $A$. If $\overline{\delta}(A)=\underline{\delta}(A)$ we denote this value by $\delta(A)$ and call it the logarithmic density of $A$.

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

 $\delta(A)=\lim\limits_{n\to\infty}\frac{\sum\limits_{k\in A;k\leq n}\frac{1}{k% }}{S(n)},$

exists, then it is called logarithmic density of $A$.

By the well-known $\gamma=\lim\limits_{n\to\infty}S(n)-\ln n$ defining Euler’s constant, we can see that the denominator in the above definitions can be replaced by $\ln n$.

## References

• 1 M. Kolibiar, A. Legéň, T. Šalát, and Š. Znám. Algebra a príbuzné disciplíny. Alfa, Bratislava, 1992. (in Slovak)
• 2 H. H. Ostmann. Additive 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 LogarithmicDensity 2013-03-22 15:31:54 2013-03-22 15:31:54 kompik (10588) kompik (10588) 6 kompik (10588) Definition msc 11B05 InequalityOfLogarithmicAndAsymptoticDensity upper logarithmic density lower logarithmic density