Kowalski – Page 69 – E. Kowalski's blog

The harmonic series and surrounding folklore

Speaking of divergent series, I’d like to mention one of my favorites bits of informal folklore: this starts with a slogan

The only “truly” divergent series is the harmonic series
$latex H=1+\frac{1}{2}+\cdots +\frac{1}{n}+\cdots,$

which, of course, deserves plenty of quotes. But the idea is quite sound: it is that, given about any other series with real or complex terms (which does not involve this one in a hidden manner, of course, e.g., as an “irreducible summand”), such as

$latex \sum_{m\geq 1}{(-1)^{m+1}\frac{m^2}{4m^2-1}},$

it is quite often possible, by some kind of trick or feint, to assign a “value” (i.e., a real or complex number) to the series in a way which is reasonable and useful for certain purposes. [A famous example is the (apparently) ultra-divergent series

$latex \sum_{m\geq 0}{(-1)^m m!}=1-1+2-6+24-120+720-\cdots,$

which was considered by Wallis and Euler: its “value” is

$latex e\int_0^1{e^{-1/x}x^{-1}dx}=e\int_1^{+\infty}{y^{-1}e^{-y}dy}=0.5963473623231\ldots$

(see this article for an explanation of some of Euler’s ideas to do this “computation”).]

One setting in which the philosophy above has been refined to a precise (conjectural) statement is the theory of L-functions (over number fields). Indeed, observe that (formally) we have

$latex H=\zeta(1),$

where ζ(s) is the Riemann zeta function, which is only defined properly for the real part of s larger than one by the series

$latex \zeta(s)=\sum_{n\geq 1}{\frac{1}{n^s}}.$

Then, after translating the basic insight about the harmonic series from Dirichlet series to L-functions, one gets the following folklore conjecture:

If a reasonable (say, automorphic, or “motivic”) L-function L(s) over a number field k has a pole at s=1 (when normalized so that the functional equation relates values at s and 1-s), then L(s) is divisible by the Riemann zeta function, in the sense that
$latex L_1(s)=\frac{L(s)}{\zeta(s)}$
does not acquire poles at any of the complex zeros of the Riemann zeta function.

This is quite a rich problem: it contains a famous conjecture of Artin (the Dedekind zeta function of a number field should be divisible by the Riemann zeta function), and applied to the Rankin-Selberg convolution, it suggests the existence of the symmetric-square L-functions of modular forms — indeed, the first crucial result towards the “automorphic” existence of the latter (due to Gelbart and Jacquet), was the proof of this divisibility property by Shimura).

Now for the most amazing thing concerning this folklore property (at least to me): it seems that it also works modulo primes! Let me explain this (an explanation which I heard from Jean-Pierre Serre): another renowned formula of Euler for the values of the zeta function at even integers can be rephrased, after using the functional equation

$latex \pi^{-s/2}\Gamma(s/2)\zeta(s)=\pi^{-(1-s)/2}\Gamma((1-s)/2)\zeta(1-s),$

as the formula

$latex \zeta(1-2n)=1+2^{2n-1}+3^{2n-1}+\cdots + k^{2n-1}+\cdots =-\frac{1}{2n}B_{2n},$

where the Bernoulli numbers B_2n are rational numbers defined by the power series expansion

$latex 1-\frac{1}{2}x+\sum_{n\geq 1}{B_{2n}\frac{x^n}{n!}}=\frac{x}{e^x-1}$

(and, of course, the middle expression for ζ(1-2n) is purely formal: it is one more example of a divergent series that can be given a convincing value).

The idea now is that the last expression (-1/2n B_2n) can be reduced modulo a prime p, provided p does not divide 2n and does not divide the denominator of the Bernoulli number. Now, lo and behold, the primes dividing the denominators of Bernoulli numbers are known: they are exactly the primes such that

$latex p-1\mid 2n$

or equivalently such that

$latex k^{2n-1}\equiv k^{-1}\text{ mod } p$

for all integers k coprime with p (those for which the inverse modulo p makes sense…). So the Bernoulli number can not be reduced modulo primes exactly when (either p divides 2n or) formally we have

$latex -\frac{1}{2n}B_{2n}\equiv 1+2^{2n-1}+3^{2n-1}+\cdots + k^{2n-1}+\cdots\equiv 1+2^{-1}+\cdots +k^{-1}+\cdots,$

the divergent harmonic series again! (One must omit the terms with k divisible by p of course, but since this is purely formal, why not?).

I have no idea if there is a good explanation for this coincidence, but it is remarkably beautiful, and it certainly gives a convincing argument for the fact that the numerators of Bernoulli numbers are much more mysterious: so are the zeros of the Riemann zeta function, in comparison with its pole…)

Euler for a day, or “my” formula for pi

Here it is:

$latex \pi=16\sum_{m\geq 1}{(-1)^{m+1}\frac{m^2}{4m^2-1}}.$

Yes, it’s a divergent series, but I’m sure Euler would like it even more. (Actually, the probability that this formula is not somewhere in his works, or in Ramanujan’s, is close to zero, though I came upon it fairly accidentally today — maybe I’ll explain how it came about naturally at some later time).

Amusingly, both Pari/GP (numerically, using sumalt) and Maple (symbolically, after setting _EnvFormal:=true;) can confirm the “formula” as-is… (I didn’t try with Mathematica).

La Pléiade

In the spirit of fairness and balance, after my ode to an American magazine, I would like now to mention my admiration for one of the great achievements of the French publishing world: La Bibliothèque de la Pléiade. This is one of the collections edited by Gallimard, maybe the greatest French publishing house, which is dedicated to producing definitive editions of the best of the world’s literature. There is a strong emphasis on French-language writers, of course (including thirteen volumes of Voltaire’s correspondance), but by no means an exclusivity (as can be seen in the catalogue: note Spanish-language writers, such as Borgès, Russian masters like Dostoievski, Boulgakov or Tolstoy, Italian writers like Machiavelli, and of course many English-speaking ones, such as Faulkner, Melville or the Brontë family). This is one of the great differences with the natural reference point in the American world, the Library of America. The other main difference is that, besides the text itself, the Pléiade aims to provide extensive (sometimes exhaustive) editorial information on the author and the work, with notes, introductions and discussions, bibliographies, sometimes early versions or other relevant sources, etc. The books themselves (like those of the Library of America) are beautifully produced, on the thinnest paper (papier bible), so each volume is routinely longer than 1000 pages without being much bigger or heavier than a (fairly fat) paperback. The font is the elegant Garamond, with its intricate ligatures.

Being in Paris earlier this week, I visited one of the many bookstores, and noticed that the second part of the new complete Pléiade edition of Shakespeare’s works, the Histories, had just appeared; I therefore snatched the two volumes without more ado, to add to the Tragedies which were published a few years ago.

Now, it might seem slightly ridiculous to spend a lot of money on a French edition of Shakespeare (however beautiful the italic font in the scenic indications), and this was a valid criticism of the earlier edition (dating to the 1950’s), but the new one is in fact bilingual. And I will venture the opinion that reading Shakespeare in a bilingual version makes very good sense: one can try to read the “original” version as much as possible, but in case the syntax or grammar becomes decidedly perplexing on the page, the translation gives a backup. If the translation is written from the point of view of actual theatrical experience, then the solutions which are offered to the many ambiguities in the texts (which can most often not be fully translated) are likely to make more sense and to flow more smoothly than isolated glosses or paraphrases in footnotes, even if they can not convey all the possible meanings. In the new Pléiade edition, the main translator is Jean-Michel Déprats, and most of the translations were indeed used for actual representations in France before they appeared; so even if one can not always be sure of reading Shakespeare’s intended meaning, at least one gets something which may be the next best thing: some well-defined meaning, coming from a writer with enormous theatrical experience. And I’m sure that anyone who has seen a few plays of Shakespeare on the stage knows how different the experience may be from reading them. (My personal favorite memory is a magical version of The Tempest in the Théâtre des Bouffes du Nord, in Paris, directed by Peter Brook in 1990, in a translation of J-C. Carrière).

Now, lest any scholar of the Elizabethan theatre jump on my word “original” in the previous paragraphs, I emphasize the quotation marks: just as in any modern English edition, there has, very often, been a real choice of which text to use (Good Quarto, Bad Quarto, First Folio, and what you will). The whole history behind those various versions can be quite fascinating, and the very detailed notes explain which was used, what principles were applied in terms of localized corrections, etc: again, very solid scholarship comparable to those detailed editions one can find in English. There is also a separate genealogical tree of the relevant Kings, Queens, Princes, Princesses, Dukes, and other divers Noblemen and Noblewomen, included in the first volume of the Histories, which is certainly quite useful…

Here’s a picture of the two-volume Histories:

and here’s one of the text of Richard III:

and the genealogical tree:

Teaching weak topologies

I can’t say that I really have anything like a “teaching philosophy” — maybe because I was lucky and never had to write a teaching statement. However, experimentally, I realize that there are two prominent features in my teaching that I end up feeling fairly strongly about, partly from an impression that they were not emphasized enough when I was myself a student: one is that the development of the course and the underlying theory should be emphasized and motivated enough that it does not feel like the teaching of some disembodied Voice Of God or Deus Ex Machina (even if the motivation is not historical, and indeed the motivations I give are often based on a fair amoung of hindsight); and two, that whenever a theorem of some importance is proved, it should be clear where and how its assumptions are used.

I’ve been thinking a bit about this second point because I’m currently teaching weak topologies in my Functional Analysis class, and in particular I just finished proving the Banach-Alaoglu Theorem that states that the closed unit ball in the dual V’ of a Banach space is compact for the weak-star topology. Here one basic point I wanted to make in class was that the analogue statement is not true (in general) for the closed unit ball of V itself, with the weak topology. I was then surprised to see, in browsing through various textbooks, that none of those I saw explained explicitly why the argument doesn’t work in this situation (it is implicit, of course, in all those that stated, e.g., that this second statement is equivalent with the reflexivity of V, but I think that for a student beginning in Functional Analysis, this may not be sufficient, because the proof of this equivalence is not really obvious). Sometimes, the “broken” step was just glossed over, but sometimes it was simply left as an exercise. In the worst case, the written proof did not even spell out which of the steps of the proof requires that the weak-star topology be used, leaving this also as an exercise (the beginning of the standard proof works just as well with the norm topology)!

[Note: The standard proof of Banach-Alaoglu proceeds by embedding the closed unit ball B in the compact product space

$latex C=\prod_{||v||\leq 1}{\{z\,\mid\, |z|\leq 1\}$

$latex i(\lambda)=(\lambda(v))_v,\text{ for }\lambda\in B;$

then one shows that i is injective, continuous (this works even for the norm topology on B), that the inverse of i, defined on i(B), is also continuous (this requires that the weak-star topology be used, and fails for the norm topology if V is not finite-dimensional); then, finally, one shows that i(B) is closed in C, and this works because we deal with B in V’ instead of the closed ball of V itself — the basic issue is that knowing that (for some sequence in V) we have

$latex \lambda(v_n)\rightarrow w_{\lambda}$

for all continuous linear functionals λ on V does not always imply that we can find some w such that

$latex w_{\lambda}=\lambda(w).$

A simple example where this fails (i.e., w does not exist), is provided by taking

$latex V=c_0\text{ with sup norm}$

(the space of sequences converging to 0) and

$latex v_n=(1,\ldots, 1,0,\ldots )$

where there are n ones at the beginning: from the fact that the dual of V can then be identified with the space l₁ of absolutely convergent series, one checks that

$latex \lambda(v_n)\rightarrow \sum_{n\geq 1}{u_n}$

for all linear functionals λ represented by the sequence (u_n) in l₁. But

$latex \sum_{n\geq 1}{u_n}$

is not of the form

$latex \lambda(w)$

for any sequence w in c₀.]

The New Yorker

When the conversation turns to anti-américanisme primaire, as it will every once in a while in France, the first argument I use if I intend to display a contrary argument is the New Yorker. Indeed, this magazine is so much above the level of the available French weeklies that (to use a cliché), it’s not even funny. Not only is the content much better — more art, more poetry, more humor, more fiction, less French politics –, but the difference is even stronger from the purely visual point of view (typography, art, design: no need to be able to read French or English to see which editors/writers/readers have better taste). This is especially shameful, considering that the French penchant for style over substance would seem to guarantee that we would (at least) do much better in this respect. However, it is not for any of the French magazines that J.J. Sempé draws covers, but for The New Yorker. (Sempé is known in France in particular for inventing the second most important fictional Nicolas in history).

I think I first read about The New Yorker in the introduction to a French translation of Woody Allen’s short humor pieces for the magazine (if you’ve never read any of them, I suggest googling for “Gossage Vardebedian”), in which (the introduction) it was identified as “the most snobbish magazine in the world”, which immediately piqued my curiosity. However, I think I read an issue for the first time when I went for a month to the US to work with Henryk Iwaniec in 1992. Then in late 1993, I decided to start subscribing from France. At that time, issues arrived there about one month after publication, so that reading the jazz programme at the Café Carlyle, for instance, was a somewhat quixotic thing to do, but most of the articles were of lasting enough interest that this delay was not much a problem.

My interest for this magazine has been considered somewhat obsessive at times. It is not true that I brought my fledgling (three years old) collection to the US when I went there for Graduate School, but I must admit that I did ship back to France all issues accumulated during that period (and the resulting post-doc), and then had them also sent to Switzerland, together with the issues of the last eight years or so (they are now in storage somewhere in Zürich).

Frankly, my justification for this accumulation was not quite convincing: it is not really useful to have physical issues of The New Yorker somewhere in the basement in a random order, since (until recently) it did not really help to remember vaguely that, say, there was a hilarious story about a mathematics class by some Irish author sometime during the first (or was it second?) Clinton administration — the time to locate it would still be discouraging to consider. Moreover, I couldn’t help feeling terribly jealous of older subscribers who could reach (if they knew where they were located) for issues containing stories by I.B. Singer, for instance, and read them whenever they wanted.

In principle, this two problems were solved a few years ago when The New Yorker published a set of eight DVD’s containing all issues of the magazine (until that date, of course; it has been updated regularly). I bought it immediately, but the fact that the DVD’s were encrypted, and the reader program did not work under Linux was something of a problem. Because we had a Mac in the house, it was still theoretically possible to take advantage of the archive, but in practice it was very inconvenient (except for the fact that the search database was a standard SQLite database, and could thus very well be queried from my Linux computers; so I could say very quickly when the Gossage-Vardebedian papers were published — January 22, 1966 –, but actually reading it involved complicated manipulations and printing to PDF from a very slow Mac whose DVD reader was broken, etc.)

But, at last, this is old history: just recently, The New Yorker started making available both a digital edition (which is convenient, but not so important for me), and the complete archive online, available more or less as in the DVD set, as exact reproductions of the actual magazine (so even the ads, etc, are exactly as in the printed edition, which is quite wonderful actually). Better yet: both services are available free to subscribers.

[Note: I am aware that many older subscribers believe the magazine went downhill starting about 1990; but I can’t really be held responsible for not reading it before, and (1) now I can; (2) it is still much better than the French weeklies…]