Peano paragraphing

Every mathematician has heard of the Peano axioms of arithmetic. Here is a lesser known contribution of Giuseppe Peano: the “Peano paragraphing method”. This is a numbering system for sections/subsections/etc in books where the different items are identified by a decimal number (e.g., 9.132), where the integral part is the chapter number, and the decimal part is arranged in increasing order within each chapter. So for instance 9.301 is a subsubsection lying between 9.3 and 9.31.

I had noticed this system in Titchmarsh’s book “The theory of functions”, from 1932, without understanding it (it is not explained, nor attributed to Peano). Then I saw it again just recently as I was looking up a reference in Whittaker and Watson’s “A course of modern analysis” from 1927, where the explanation and attribution are given in a remark at the beginning. This greatly clarified my previous perplexity in navigating the book of Titchmarsh, which I had found extremely confusing; for instance in Chapter 9, we have

9.1, 9.11 up to 9.15, 9.2, 9.3, 9.31, 9.32, 9.4, 9.41 to 9.45, 9.5, 9.51 to 9.55, 9.6, 9.61, 9.62, 9.621 to 9.623, 9.7…

Looking into other classical books, I can see this system in Watson’s treatise on Bessel functions, but it is not used in either Hardy and Wright’s “Introduction to the theory of numbers”, nor in Titchmarsh’s “The theory of the Riemann zeta function”. It is also absent from Zygmund’s “Trigonometric series” (which, on the other hand, uses a continuous numbering scheme X.Y (Chapter.Item) both for equations, theorems, etc), and from Hardy and Rogosinsky’s “Fourier series”.

Note finally that it seems rather euphemistic to say that this is “lesser known”: neither Google nor Wikipedia seem to be able to give a reference or explanation!


A quip of S. Lang states that “analysis is number theory at the place infinity”.  (Which implies, correctly, that analytic number theory is some particularly exalted form of number theory).

The equally quipful E. Witten goes rather further in reducing mathematics to its essentials: during the conference organized by the Mathematics Department of Princeton University in honor of the 250th anniversary of Princeton University, he said something like: “Most of 20th century mathematics is the study of the harmonic oscillator”. (This can be seen, in a slightly different and weakened form, on page 120 of the Google Book preview linked above; my memory is that he did state, during his lecture, something closer to what I wrote; but that was a while ago, so I may be over-reacting in hindsight…)

P.S. For the obligatory etymological epilogue: the word “one-upmanship” is quite recent (1952), but “quip” goes back to the early 16th century. I didn’t know about the charming derivative “quipful” before looking in the OED.

An exercise

Here is a striking example of the unfortunate (but probably unavoidable) dispersion of mathematics: at the end of the abstract of V. Arnold’s talk at the recent conference in honor of A. Douady, one can read:

The Cesaro mean values K̂ of the numbers K(n) tend, as n tends to ∞, to a finite limit K̂(∞)=lim 1/n ∑m=1n K(m) = 15/π2. This theorem, deduced from the empirical observation of the coincidence of 20 first digits, is now proved, using the formula K̂(∞)=ζ(2)/ζ(4)

Here, K(n) is defined before in the abstract as the expression

K(n)=\prod_{p\mid n}{(1+1/p)}

Arnold’s achievements as mathematician are about as impressive as it can get. But the statement here is a completely elementary exercise in analytic number theory, and has been for at least one century (i.e., Dirichlet, or Chebychev, could do it in a few minutes, if not Euler).  Here’s the proof in Chebychev style:

K(n)=\sum_{d\mid n}{\mu(d)^2/d}

hence, exchaning the sum over n and the sum over d, we get

\sum_{n<X}{K(n)}=\sum_{d<X}{\mu(d)^2d^{-1} [X/d]}

and replacing the integral part by X/d+O(1), this is clearly asymptotic to CX with

C=\sum_{d\geq 1}{\mu(d)^2d^{-2}}

which is an absolutely convergent series. As an Euler product it is


as desired.

A combinatorial dichotomy

I have just read about the following very nice dichotomy: suppose we have an infinite set X, and a collection of subsets C in X; suppose further we look at all subsets F of X of finite size n, and at the subsets of F which can be defined by intersection of F with an element of C. How rich can this collection of subsets of F be, as F varies over all subsets of fixed (but growing) size?

For example, X could be the real line, and C the collection of half-lines ]-∞,a]. Then if we enumerate the finite subset F in increasing order

x_1\lt x_2 \lt \ldots \lt x_n

the subsets we obtain by intersecting with C are rather special: apart from the emptyset, we have only the n subsets of elements in increasing order up to some a:

x_1\lt x_2 \lt \ldots \lt x_r \le a

with r≤n. In particular, there are only n+1 subsets of F which can be obtained in this manner, much less than the 2n subsets of F.

As a second example, if C is the collection of all open subsets of the real line, we can clearly obtain all subsets of F as intersection of an element of C and F.

The dichotomy in question is that, in fact, those two examples are typical in the following sense: either one can, for all n, find an n element set F and recover all its subsets as intersection with C (as in the second example); or for any n and any F in X of size n, the number of subsets obtained from F by intersecting with C is bounded by a polynomial function of n. So it is not possible to have intermediate behavior (subexponential growth which is not polynomial), and this is certainly surprising at first (at least for me).

This very nice fact is due to Vapnik-Chernovenkis and Shelah, independently (published in 1969 and 1971, respectively). What is quite remarkable is that the first authors were interested in basic probability theory (they found conditions for the “empirical” probability of an event in the standard Bernoulli model to converge uniformly to the mathematical probability over a collection of events, generalizing the weak law of large numbers), while Shelah was dealing with model-theoretic properties of various first-order theories (in particular, stability).

In fact, these references are given by L. van den Dries (Notes to Chapter 5 of “Tame topology and O-minimal structures”, which is where I’ve read about this, and which is available in the Google preview), but whereas it’s easy to find the result in the paper of Vapnik-Chernovenkis, I would be hard put to give a precise location in Shelah’s paper where he actually states this dichotomy! This is a striking illustratation both of the unity and divergence of mathematics…

The proof of the dichotomy, as one can maybe expect (given the information that it is true), is clever but fairly simple, and gives rather more precise information than what I stated. Let’s say that C is a rich uncle of a finite set F if any subset of F is the intersection of F with a subset in C. We must show that either C is a rich uncle for at least one finite subset of every order, or else C only represents relatively few subsets of any finite subset.

First, a lemma states that:

Given a finite set of size n and a collection D of subsets of F, which contains (strictly) more subsets than there are subsets of F of size up to (but strictly less than) some d, one can always find in F a subset E of size d such that D is a rich uncle of E.

Note that this is best possible, because if we take D to be those substes F of size up to (and excluding) d, it certainly can not be a rich uncle of a set of size d.

If we grant this lemma, the proof of the dichotomy proceeds as follows: assume we are not in the first case, so for some d, C is a rich uncle for no subset of order d. Let n>d be given (to get polynomial behavior, we can restrict to this case), and let F be a subset of order n. The lemma (applied with D the intersections of C with F), and the definition of d, imply by contraposition that the number of subsets of F which are obtained by intersection from C is less than the number of subsets of a set of order n which are of order d at most. But this is a polynomial function of n, of degre d.

As for the lemma, I leave the proof as an exercise (see page 80 in the book of van den Dries, which is also in the preview), with a hint: proceed by induction on n. (One is tempted, in view of the statement, to use the pigeon-hole principle to say that D must contain one subset at least of order d, but the proof by induction doesn’t use that).

Now I am going to try and think if I can find some other application of this fact…

An exotic preprint server

In number theory, there is (as far as I know) no currently active preprint server, besides arXiv, and there hasn’t been one since quite a few years (there was an algebraic number theory preprint archive which was incorporated into arXiv around 1998, if I remember right). So I was intrigued to hear a colleague mention a very active preprint server for Linear Algebraic Groups and Related Structures, which contains a fair number of works of great interest which are not available on arXiv (some may be on individual web pages, but not all; for instance, among the more recent ones, neither preprint 292 nor 291 is on arXiv, but they are both quite interesting).

As befits the subject, there are of course plenty of papers there on exceptional groups and other exotic structures.