For any $A\subseteq\N$ we denote $A(n):=\abs{A\cap\{1,2,\ldots,n\}}$ and $S(n):=\sum\limits_{k=1}^n \frac 1k$ .

Recall that the values $$\ol d(A)=\liminf_{n\to\infty} \frac{A(n)}n \qquad \ul d(A) = \limsup_{n\to\infty} \frac{A(n)}n$$ are called lower and upper asymptotic density of $A$ .

The values $$\ol \delta(A)=\liminf_{n\to\infty} \frac{\sum\limits_{k\in A; k\leq n} \frac 1k}{S(n)} \qquad \ul \delta(A) = \limsup_{n\to\infty} \frac{\sum\limits_{k\in A; k\leq n} \frac 1k}{S(n)}$$ are called lower and upper logarithmic density of $A$ .

We have $S(n)\sim \ln n$ (we use the Landau notation). This follows from the fact that $\limti n S(n)-\ln n=\gamma$ is Euler's constant. Therefore we can use $\ln n$ 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 $\sum_{k=1}^n \frac 1k [k\in A]$ . (This means that we only add elements fulfilling the condition $k\in A$ . This notation is introduced in [1, p.24].)

Proof. We first observe that

There exists an $n_0\in\N$ such that for each $n\geq n_0$ it holds $\ul d(A)-\ve \leq \frac{A(n)}n \leq \ol d(A) + \ve$ .

We denote $C:=1+S(n_0)$ . For $n\geq n_0$ we get

This inequality holds for any $\ve>0$ , thus $\ol\delta(A)\leq \ol d(A)$ .

For the proof of the inequality for lower densities we put $C':=\sum_{k=1}^{n_0-1} \frac{A(k)}{k(k+1)}-(\ul d(A)-\ve)S(n_0)$ . We get

and this implies $\ul\delta(A)\geq\ul d(A)$ .

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

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 $0\leq\ul \alpha\leq\ul \beta\leq \ol \beta \leq \ol \alpha \leq 1$ there exists a set $A\subseteq\N$ such that $\ul d(A)=\ul\alpha$ , $\ul\delta(A)=\ul\beta$ , $\ol\delta(A)=\ol\beta$ and $\ol d(A)=\ol\alpha$ (see [2]).

Bibliography

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Concrete mathematics. A foundation for computer science.
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L. Mišík.
Sets of positive integers with prescribed values of densities.
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H. H. Ostmann.
Additive Zahlentheorie I.
Springer-Verlag, Berlin-Göttingen-Heidelberg, 1956.

4

J. Steuding.
Probabilistic number theory.

5

G. Tenenbaum.
Introduction to analytic and probabilistic number theory.
Cambridge Univ. Press, Cambridge, 1995.