# If the natural density (relative to the primes) exists, then the Dirichlet density also exists, and the two are equal

On p. 76 of the 1996 edition of Serre's A Course in Arithmetic, one reads the following (inline) remark:

One can prove that, if $A$ has natural density $k$, the analytic density of $A$ exists and is equal to $k$.

Here, $A$ is a subset of $\bf P$ (the set of all positive rational primes), and the natural density of $A$ is actually the natural density of $A$ relative to $\bf P$, viz. the limit $$\lim_{n \to \infty} \frac{|A \cap [1,n]|}{|\mathbf P \cap [1,n]|}$$ (if it exists), while the analytic density of $A$ is actually the analytic density relative to $\bf P$, viz. the limit $$\lim_{s \to 1^+} \frac{\sum_{p \in A} p^{-s}}{\sum_{p \in \mathbf P} p^{-s}}$$ (again, if it exists). Here are then my questions:

Q1. Was it Serre the first who made this observation explicit? Q2. Do you know of a paper or book where a proof is provided? Serre doesn't even give a hint about it.

Notes (added later).

On Q1: In the light of Lucia's comment below, let me make it clear that I myself find it very unreasonable that the result hadn't been known before Serre's remark in the 1970 French edition of his book (p. 126). I'd just like to find out if it was Serre the first who made it explicit.

On Q2: I have my own proof, but would appreciate a reference. The reason is that something sensibly stronger is true, and I'm hoping to understand from the inspection of the proof he may have had in mind if this is intentional (e.g., it is evident from the proof he may have had in mind that something sensibly stronger is true, but he just didn't care), or not.

Edit (Feb 09, 2016). For future reference, I think it can be useful to make order and summarize, here in the OP, what has emerged from the answers and comments of those who have so far contributed to this discussion:

1) As expected, it wasn't Serre the one who first made explicit the relation between the analytic and natural densities relative to the primes. The result is already stated on p. 118 of:

E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Erster Band, Teubner: Leipzig, 1909,

where a detailed proof is also presented. This answers both Q1 and Q2.

2) Franz Lemmermeyer, in a comment to the OP, had suggested since the outset that the result should have appeared almost surely in some of Landau's books. This was confirmed by so-called friend Don in his answer (here), where it's also reported that the result was mentioned on p. 225 of the 1st edition of:

H. Hasse, Vorlesungen über Zahlentheorie, Die Grundlehren der mathematischen Wissenschaften 59, Springer-Verlag: Berlin, 1950.

Interestingly enough, Hasse made a mistake here, by stating that not only the existence of the natural density (relative to the primes) implies that the analytic density (always relative to the primes) also exists, and the two are then equal: He went on asserting that also the converse is true! As still noted by so-called friend Don, the mistake was fixed in the 2nd (1964) edition of the book (p. 236), and it was mentioned in a comment to his answer that we know by now that Hasse was really wrong, for an example attributed by Serre to a private communication from E. Bombieri (p. 126 in the 1970 French edition of A Course in Arithmetic, or p. 76 in the 1996 English edition) proves the existence of a set of primes that has analytic (relative) density, but not natural (relative) density.

3) Comparison results in the same spirit of those considered in this question, but involving densities on $\mathbf N^+$, are not so rare in the literature. Most notably, it is known (and easy to prove by Abel's summation formula) that the upper analytic density (on $\mathbf N^+$) is not greater than the upper logarithmic density, which is in turn not greater than the upper asymptotic density, see, e.g., Theorem 2 in Section III.1.3 of:

G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge Stud. Adv. Math. 46, Cambridge Univ. Press: Cambridge, 1995.

It follows at once that the existence of the natural density (on $\mathbf N^+$) implies the existence of the logarithmic density, and the existence of the logarithmic density implies the existence of the analytic density.

4) On the other hand, it is known that upper and lower asymptotic and natural densities are pretty much independent from each other, in a sense that was first made precise by L. Mišík in:

L. Mišík, Sets of positive integers with prescribed values of densities, Math. Slovaca 52 (2002), No. 3, pp. 289-296. see here for further reading on the subject.

You may want to read the comments to Question 103111: Prescribed values for the uniform density for a more accurate account of Mišík's results and generalizations theoreof.

5) Furthermore, it is known that the existence of the analytic density (on $\mathbf N^+$) implies that also the logarithmic density exists, and the two are then equal. This is a non-trivial result, which goes back at least to H. Davenport and P. Erdős, who make an implicit reference to it in the proof of Theorem 1 from:

H. Davenport and P. Erdős, On sequences of positive integers, Acta Arith. 2 (1936), No. 1, 147-151.

The proof is based on the Hardy-Littlewood tauberian theorem. All of this was pointed out by so-called friend Don in a comment to GH from MO's answer (here). An alternative proof, that rather uses Karamata's tauberian theorem, is given by Tenenbaum in his book (Theorem 3 in Section III.1.3). The same Tenenbaum mentioned in a private communication that the special case of Karamata's theorem needed here goes back to:

O. Szász, Münchner Sitzungsberichte (1929), 325-340.

6) Last but not least, Christian Elsholtz added some further elements to the story (here).

• This follows just by partial summation, and would have been known to everyone. For example, how did Chebyshev know that the limit in the prime number theorem, if it exists, must be one? – Lucia Feb 7 '16 at 0:00
• @Lucia. Agree. Yet, I can't find any explicit occurrence of the same remark in any other place (before Serre's book), which sounds strange to me, all the more that I happen to know people attributing it to him. Also, the proof you refer to (by partial summation only), while being so simple, results into an inequality that is sensibly coarser than what can be actually proved. That's why I'd like to see how the proof has been written down, if it ever was. – Salvo Tringali Feb 7 '16 at 0:25
• Try Hasse's lectures on class field theory from the 1930s. I'd be very surprised if it wasn't contained in Landau's books. – Franz Lemmermeyer Feb 7 '16 at 17:25
• @FranzLemmermeyer. I was hoping for a comment of yours. In the English translation of the 1923-1958 correspondence between Artin and Hasse, you note (p. 118): "When Artin writes that Chebotarev’s density theorem is "less precise" then he probably refers to the fact that Artin had used the natural density, whereas Chebotarev stated his result using the Dirichlet density. Any set of primes with a natural density also has a Dirichlet density (and both of them are equal), but the converse need not hold." So, if your interpretation of Artin's comment is correct, and it seems very plausible, (...) – Salvo Tringali Feb 7 '16 at 22:18
• No problem. Your edit gave me an excuse for posting a question on Meta: meta.mathoverflow.net/questions/2729. As I try to explain over there, the terms "Dirichlet density" and "natural density" mean different things in different contexts, so the redundancy in the old title was intentional. – Salvo Tringali Feb 8 '16 at 21:17

## 3 Answers

UPDATED: As Franz suspected, `es steht schon bei Landau.'

On p. 118 of the first volume of Landau's Handbuch, one finds the following theorem: Let $f(s)=\sum_{n\ge 1} a_n/n^s$ be a Dirichlet series (with real coefficients $a_n$) that converges for $s>1$. Let $S(x)=\sum_{n \le x} a_n$. Then $$\limsup_{s\downarrow 1} \frac{f(s)}{\log\frac{1}{s-1}} \le \limsup_{x\to\infty} \frac{S(x)}{x/\log{x}},$$ and $$\liminf_{s\downarrow 1} \frac{f(s)}{\log\frac{1}{s-1}} \ge \liminf_{x\to\infty} \frac{S(x)}{x/\log{x}}.$$ This implies the assertion in question. The copy of the Handbuch I am looking at carries a date of 1909.

OLD VERSION:

Hasse mentions in the first (1950) edition of his Vorlesungen uber Zahlentheorie that if the natural density of a set of primes exists, so does the Dirichlet density, and they are equal. He calls the proof "ver­hält­nis­mä­ßig einfach" (relatively easy), saying it is a generalization of Abel's continuity theorem. So he clearly has in mind the partial summation proof Lucia alluded to.

More interestingly, Hasse claims the converse is true (Dirichlet density exists ==> natural density exists); this is corrected in the 2nd (1964) edition. The error is discussed in Paul Bateman's MathSciNet review.

• Very interesting, thank you! I won't have access to Bateman's review until tomorrow, but there is a counterexample to Hasse's claim that is by now relatively well-known: Serre (on p. 76 of the 1996 edition of A Course in Arithmetic) credits it to a private communication by Bombieri, and more details can be found in: D. I. A. Cohen and T. M. Katz, Prime Numbers and the First Digit Phenomenon, J. Number Theory 18 (1984), 261-268. – Salvo Tringali Feb 7 '16 at 22:14
• Incidentally, could you please provide a page for Hasse's remark? As I can't read German, this would help me much. – Salvo Tringali Feb 7 '16 at 22:17
• You want p. 225 in the 1st edition, and p. 236 in the 2nd. – so-called friend Don Feb 7 '16 at 23:16
• OK, I went ahead and updated the answer. – so-called friend Don Feb 8 '16 at 23:53

This statement is proved in detail in Tenenbaum's book "Introduction to analytic and probabilistic number theory". See Theorem 2 in Section III.1.2. See also Theorem 3 in Section III.1.3, where it is proved that the analytic density is the same as the logarithmic density.

Personally, I find the mentioned Theorem 2 rather simple, while Theorem 3 somewhat subtle. Also, I don't know who made these observations first.

Added. Actually, Theorem 2 compares the logarithmic density with the natural density, while Theorem 3 compares the logarithmic density with the analytic density. Also, these are not relative densities but true densities, but I am sure a generalization would be straightforward.

• Now I feel sorry for not having mentioned it in the OP, but I did already know all of this (my question is motivated by ideas matured in the shadow of Tenenbaum's section on densities). And I agree with you about the substantial difference in the difficulty levels of Theorems 2 and 3. However, I'm not convinced that a generalization of Theorem 3 to relative densities (if true!) would be that straightforward, while this is certainly the case with Theorem 2. – Salvo Tringali Feb 7 '16 at 18:54
• Done! ;) Will keep you posted and update this comment if and when Tenenbaum replies. – Salvo Tringali Feb 8 '16 at 23:14
• Here is an excerpt of Tenenbaum's reply: "[...] This result is essentially of Tauberian nature, since the easy part is just Abel summation. I think the first proof of the special case of Karamata's theorem needed here is due to Otto Szász, Münchener Sitzungberichte (1929), 325-340. (This is just two years before Karamata's theorem was published.) [...] As far as a formulation in terms of densities of integer sequences is concerned, it might be in my book that it appears explicitly for the first time, but one cannot be sure: this had been part of the 'oral tradition' for ages." – Salvo Tringali Feb 9 '16 at 13:39
• The observation that analytic density ==> logarithmic density appears (but is not emphasized) in the proof of Theorem 1 of this 1936 paper of Erdos and Davenport: renyi.hu/~p_erdos/1936-04.pdf. They cite a 1914 theorem of Hardy and Littlewood. – so-called friend Don Feb 9 '16 at 15:20
• @SalvoTringali: Since nothing about their particular set is used, it's obvious that Erdos and Davenport knew the general case. (See also the exposition in Hall's monograph Sets of Multiples.) But I agree that they did not spell out the general case. – so-called friend Don Feb 9 '16 at 18:33

Scholz (in Jahresbericht der Deutschen Mathematiker-Vereinigung 45, 1935, p. 110, (Aufgabe 207) https://www.digizeitschriften.de/dms/toc/?PID=PPN37721857X_0045

https://www.digizeitschriften.de/download/PPN37721857X_0045/PPN37721857X_0045___log39.pdf

poses the problem below.

Given a set $M$ of positive integers $m_1<m_2< m_3 <...$ with natural density $\lim_{n \rightarrow \infty} \frac{n}{m_n}=h$.

Take the set $P=\{ p,...\}$ of those integers (e.g. in base 10) with $m_1, m_2, ...$ digits, which obviously have no natural density. Prove that it has a Dirichlet density $\lim_{s \rightarrow 1} (s-1) \sum_P p^{-s}=h$. ...

This shows that Scholz was certainly aware of the distinction between natural and Dirichlet density, by oscillating examples, similar to those alluded to (by Salvo) in the more modern literature.

For completeness, the Solution by Stöhr is in the 1938 volume https://www.digizeitschriften.de/dms/toc/?PID=PPN37721857X_0048

https://www.digizeitschriften.de/download/PPN37721857X_0048/PPN37721857X_0048___log26.pdf

• Hi Christian, and thanks for the pointers. Just a note: The densities in the OP are relative to the primes, and there is no obvious way, at least in general, to turn conclusions concerning densities defined on $\mathbf N^+$ into (sensible) statements concerning densities relative to a subset of $\mathbf N^+$ (see also the answer by GH from MO in this same thread). Thus, knowing that there is a distinction between the analytic and natural densities on $\mathbf N^+$ does not prove much about the analytic and natural densities on the primes (again, not in principle). – Salvo Tringali Feb 8 '16 at 19:04