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Natural density of a set $S = (a_1,a_2,...) $ is defined (assuming it exists) as $$\lim_{n \to \infty}\frac{1}{n}\sum\limits_{k\in S, k\le n}1 $$ The logarithmic density of the same set is defined as $$\lim_{n \to \infty}\frac{1}{\log n}\sum\limits_{k\in S, k\le n}\frac{1}{k} $$

My first question is are there any examples of a set where the natural density does not exist but the logarithmic density does. My second question is, can anyone give rough properties (necessary or sufficient) a set must have in order for its natural density to exist.

P.S. I am aware that if you have a set with sufficiently large 'blocks' of elements and then sufficiently large 'gaps', its natural density cannot exist. However the task I am trying to complete is to prove that the natural density of a certain set does exist (without necessarily finding it) so any conditions this set must have is what I am really looking for.

Elie Bergman
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1 Answers1

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For your first question: consider the set $S=\{2,3, 8,9, ... , 15, 32, ..., 63, ..., 2^{2k-1}, ..., 2^{2k}-1, 2^{2k+1}, ... \}$.

For $2^{2k} \leq N < 2^{2s+1}$ we have

$1/\log N \sum_{n\leq N, n \in S} 1/n=1/\log N \cdot \sum_{j=0}^{k-1} \sum_{\tau=0}^{2^{2j+1} } 1/(2^{2j+1}+\tau)=1/2 + O(1/(\log N))$ and similarly for $2^{2s+1} \leq N < 2^{2s+2}$. So logarithmic density exists.

Natural density does not exist: the density on the interval $1,.., 2^{2k}$ converges to $2/3$ and on the interval $1, ..., 2^{2k+1}$ to $1/3$.

I have never seen the conditions for the existance of natural density. However, there are many theorems that answers the question whether the following limit exists $\lim_{n \to \infty} 1/n (f(1) + ... +f(n))$ (a natural mean) for some number-theoretic function $f$. Such questions are quite subtle. For example one can approach them using the analytic properties of the Mobius inverse of $f$. The following is true:

If $f(n)=\sum_{d|n}g(d)$ (such $g$ always exists by Mobius inversion) then if the series $\sum_{d=1}^{\infty}g(d)/d$ converges then $f$ has a natural mean.

You can try to apply these to the characteristic function of the set...

user68061
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  • Thank you for this, very helpful. You say that if g(1)/1+g(2)/2+... converges, f has a natural mean. Is the converse also true? – Elie Bergman Jan 15 '14 at 21:00
  • I don't think so. there is a notion of Besikovitch class $B^1$ in which it is still true, however the definition of this class is not very easy. And I believe that I have seen an example of the function from Besikovitch class that does not satisfy this condition. I can search for it, if you are interested. – user68061 Jan 15 '14 at 21:04
  • This sounds a bit (a slight understatement) beyond me. I also suspect that it is not true in the first place anyway. I am certainly interested though so if there is any other relevant information, I would be grateful to be pointed in the right direction. – Elie Bergman Jan 15 '14 at 21:20
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    Ah, I found a normal reference: Look at A.G. Postnikov ``Introduction to analytic number theory'' chapter 3. It contains information on the class $B^1$, the proof of the theorem above and much more. – user68061 Jan 15 '14 at 21:42
  • What would be the natural and logarithmic densities of a repeating sequence 4x+1, for example (7, 29, 117, 469, 1877, 7509, 30037...) ? I am pretty sure the natural density does not exist, or is 0 - but I, for the life of me, cannot figure out the logarithmic density :( – Predrag Stojadinović Jun 10 '23 at 09:34