Condorcet, Dedekind, Minkowski

One of my great pleasures in life is to walk leisurely down from my office about 30 minutes before the train (to Paris, or Göttingen, or Basel, or what you will) starts, browse a few minutes in one of the second-hand bookstores on the way, and get on the train with some wonderfully surprising book, known or not.

A few months ago, I found “Condorcet journaliste, 1790-1794”,

Condorcet
Condorcet

which one cannot call a well-known book. It is the printed version of the 1929 thesis (at the École des Hautes Études Sociales) of Hélène Delsaux, and its main goal is to survey and discuss in detail all the journal articles that Condorcet, that particularly likable character of the French revolution (about the only one to be happily married, one of the very few in favor of a Republic from the outset, and — amid much ridicule — a supporter of vote for women), wrote during those years.

Condorcet was also known at the time as a mathematician; hence this remarkable quote from the book in question:

Il est généralement admis que rien ne dessèche le coeur comme l’étude approfondie des mathématiques…

or in a rough translation

It is a truth universally acknowledged that nothing shrivels the heart more than the deep study of mathematics… [Ed. Note: what about real estate?]

This book cost me seven Francs. More recently, my trip to the bookstore was crowned by the acquisition of a reprint of R. Dedekind’s Stetigkeit und irrationale Zahlen” and “Was sind und was sollen die Zahlen” (five Francs)

Dedekind
Dedekind

and of a first edition (Teubner Verlag, Leipzig, 1907) of Minkowski’s “Diophantische Approximationen”

Minkowski
Minkowski

for the princely sum of thirty-eight Francs.

The content of Minkowski’s book is not at all what the title might suggest. There are roughly two parts, one concerned with the geometry of numbers, and the second with algebraic number theory. In both cases, the emphasis is on dimensions 2 and (indeed, especially) 3, so cubic fields are at the forefront of the discussion in the second part. This leads to a much greater number of pictures (there are 82) than a typical textbook of algebraic number theory would have today. Here are two examples,

Minkowski
Minkowski
Minkowski
Minkowski

and here is Minkowski’s description of the Minkowski functional (or gauge) of a convex set:

Minkowski
Minkowski

AMS Open Math Notes

When I was attending the conference in honor of Alex Lubotzky’s 60th birthday, Karen Vogtmann, who was also there, told me of the Open Math Notes repository, a new project of the AMS that she was involved with. This is meant to be a collection of (mostly) lecture notes, such as many mathematicians write for a course, but which are not published (nor necessarily meant to be published). So they can be incomplete, they might contain mistakes, and may more generally be subject to all the slings and arrows that mathematical writing is heir to. (See the web site for more information, submission guidelines, etc…)

I think that this is a great idea, and am very happy that, as the web site is now public, two of my own lecture notes can be found among the inaugural set! The highlight of the current selection is however undoubtedly “A singular mathematical promenade”, by Étienne Ghys, his beautiful book on graphs of polynomials, Newton’s method, Puiseux expansions, divergent series, and much much else that I have yet to see (I’m only one-third through looking at it…)

Hopefully, the Open Math Notes collection will grow to contain many further texts. The example of the book of Ghys is already an illustration of how useful this may be — although it is also available on his home page, one doesn’t necessarily visit it frequently enough to notice it…

Two final whimsical remarks to conclude: (1) among the six authors currently represented [Update (four hours later): this has already changed!], three [Update: four] (at least) are French; (2) one of my set of notes promises a randonnée, and Ghys’s book is a promenade — clearly, one can think of mathematics as a journey…

Bagchi’s Theorem

Bagchi’s Theorem is a functional version of earlier results of Bohr and Jessen related to the statistical properties of the Riemann zeta function on a vertical line between the critical line and the region of absolute convergence. It seems that it is not as well-known as it could, partly because Bagchi proved it in his thesis, and did not publish a paper with this result (his only related paper explicitly states that he removed the probabilistic language that a referee did not like). It seems therefore useful to describe the result. I will then sketch the proof I gave last semester

Consider an open disc D contained in the region 1/2<\mathrm{Re}(s)< 1 (other compact regions may be considered, for instance an open rectangle). For any real number t, we can look at the function \zeta_t\colon s\mapsto \zeta(s+it) on D. This is a holomorphic function on D, continuous on the closed disc \bar{D}. What kind of functions arise this way? Bagchi proved the following (this is essentially Theorem 3.4.11 in his thesis):


Theorem. Let H denote the Banach space of holomorphic functions on D which are continuous on the closed disc. For T>0, define a probability measure \mu_T on H to be the law of the random variable t\mapsto \zeta_t, where t is uniformly distributed on [-T,T]. Then \mu_T converges in law, as T\to +\infty, to the random holomorphic function
Z(s)=\prod_{p}(1-X_pp^{-s})^{-1},
where (X_p) is a sequence of independent random variables indexed by primes, all uniformly distributed on the unit circle.

This is relatively easy to motivate: if we could use the Euler product
\zeta(s+it)=\prod_p (1-p^{-s-it})^{-1}
in D, then we would be led to an attempt to understand the probabilistic behavior of the sequence (p^{-it})_p, viewed as a random variable on [-T,T] with values in the infinite product \widehat{U} of copies of the unit circle indexed by primes. This is a compact topological group, and the easy answer (using the Weyl criterion) is simply that this sequence converges to the Haar measure on \widehat{U}. In other words, the random sequence (p^{-it}) converges in law to a sequence (X_p) of independent, uniform, random variables on the unit circle. Then it is natural to expect that Z_t should converge to the random function Z(s), which is obtained formally by replacing (p^{-it}) by its limit (X_p).

Bagchi’s proof is somewhat intricate, in comparison with this heuristic justification, especially if one notices that if D is replaced by a compact region in the domain of absolute convergence, then the same idea applies, and is a completely rigorous proof (one need only observe that the assignment of an Euler product
\prod_p (1-x_pp^{-s})^{-it}
to a sequence (x_p) of complex numbers of modulus one is a continuous operation in the region of absolute convergence.)

The proof I give in my script tries to remain closer to the basic intuition, and is indeed less involved (it avoids both a use of the pointwise ergodic theorem that Bagchi required and any use of tightness or weak-compactness). It makes it easy to see exactly what arithmetic ingredients are needed, beyond the convergence in law of (p^{-it})_p to the Haar measure on \widehat{U}. Roughly speaking, it goes as follows:

  1. One checks that the random Euler product Z(s) does exist (as an H-valued random variable), and that it has the Dirichlet series expansion
    Z(s)=\sum_{n\geq 1} X_nn^{-s}
    converging for \mathrm{Re}(s)> 1/2 almost surely, where (X_n)_{n\geq 1} is defined as the totally multiplicative extension of (X_p). This is done as Bagchi did using fairly standard probability theory and elementary facts about Dirichlet series.
  2. One shows that Z(s) has polynomial growth on vertical lines for \mathrm{Re}(s)> 1/2. This is again mostly elementary probability with a bit of Dirichlet series theory.
  3. Consider next smoothed partial sums of Z(s), of the type
    Z^{(N)}(s)=\sum_{n\geq 1}X_n\varphi(n/N)n^{-s},
    where \varphi is a compactly supported test function with \varphi(0)=1. Using again standard techniques (including Cauchy’s formula for holomorphic functions), one proves that
    \mathbf{E}(\sup_{s\in D}|Z(s)-Z^{(N)}(s)|)\ll N^{-\delta}
    for some \delta>0.
  4. One next shows that the smoothed partial sums of the zeta function
    \zeta^{(N)}(s)=\sum_{n\geq 1}\varphi(n/N)n^{-s}
    satisfy
    \mathbf{E}_T(\sup_{s\in D}|\zeta(s+it)-\zeta^{(N)}(s+it)|)\ll N^{-\delta}+NT^{-1}
    (the second term arises because of the pole), where \mathbf{E}_T(\cdot) denotes the expectation with respect to the uniform measure on [-T,T]. This step is also in Bagchi’s proof, and is essentially the only place where a specific property of the Riemann zeta function is needed: one requires the boundedness on average of \zeta(s) in vertical strips to the right of the critical line. The standard proof of this uses the Cauchy inequality and the mean-value property
    \frac{1}{2T}\int_{-T}^T|\zeta(\sigma+it)|^2dt\to \zeta(2\sigma)
    for any fixed \sigma with \sigma> 1/2. It is here that the bottleneck lies if one wishes to generalize Bagchi’s Theorem to any “reasonable” family of L-functions.
  5. Finally, we just use the definition of convergence in law: for any continuous bounded function f\colon H\to\mathbf{C}, we should prove that
    \mathbf{E}_T(f(\zeta_T))\to \mathbf{E}(f(Z)),
    where \zeta_T is the H-valued random variable giving the translates of \zeta(s), and Z is the random Dirichlet series. The minor tweak that is useful to notice (and that I wasn’t consciously aware of before) is that one may assume that f is Lipschitz: there exists a constant C such that
    |f(g_1)-f(g_2)|\leq C\sup_{s\in D}|g_1(s)-g_2(s)|
    (this is hidden in standard references — e.g., Billingsley’s — in the proof that one may assume that f is uniformly continuous; the functions used to prove this are in fact Lipshitz…).

    Now pick some parameter N>0, and write
    |\mathbf{E}_T(f(\zeta_T))-\mathbf{E}(f(Z))|\leq A_1+A_2+A_3,
    where
    A_1=|\mathbf{E}_T(f(\zeta_T))\to \mathbf{E}_T(f(\zeta_T^{(N)}))|\leq C\ \mathbf{E}_T(\sup_{s\in D}|\zeta(s+it)-\zeta^{(N)}(s+it)|),
    A_2=|\mathbf{E}_T(f(\zeta_T^{(N)}))\to \mathbf{E}(f(Z^{(N)}))|,
    A_3=|\mathbf{E}(f(Z^{(N)}))\to \mathbf{E}(f(Z))|\leq C\ \mathbf{E}(\sup_{s\in D}|Z(s)-Z^{(N)}(s)|).
    Fix \varepsilon>0. For some fixed N=N_0 big enough, A_3 is less than \varepsilon by Step 3, and A_1 is at most \varepsilon+N_0T^{-1}. For this fixed N_0, A_2 tends to 0 as T tends to infinity because of the convergence in law of (p^{-it}) to (X_p) — the sum defining the truncations are finite, so there is no convergence issue. So for all T large enough, we will get
    |\mathbf{E}_T(f(\zeta_T))\to \mathbf{E}(f(Z))|\leq 4\varepsilon.

Number Theory Days 2016

As usual, with Spring comes the annual Number Theory Days of EPFL and ETHZ, this time in Zürich during the week-end of April 15 and 16. The website is online, and the poster should be ready very soon (I will update the post when it is…)

The meeting is organized by the Forschungsinstitut für Mathematik, and (again as usual!) there is a certain amount of funding for local expenses made available by FIM for young researchers (graduate students and postdocs). Please register on the FIM web page before March 21 if you are interested!