A classic advice about writing papers and books is to write the introduction last. I must admit that it makes excellent sense, and in fact, I’m sure I’ve told as much to students. However, I find that I’m usually sorely tempted to write the Introduction first, and that I end up doing this quite often (especially when the project involved is not a joint paper).

There is an advantage in this approach: if I write the introduction early, most often I do not know the precise technical statements that will come out of the arguments, so I am forced to try to explain the motivation, the main points and the qualitative interest of the paper, instead of focusing on the minutia of the actual theorem, which may well be of less importance. Of course, this is partly a consequence of working in a field (analytic number theory) where it is very frequent that the final theorem involves (for instance) some parameters whose value is not particularly important, but where it is instead crucial that it is positive, etc. Some other fields afford much cleaner statements: something like “for every elliptic curve E/Q, the group E(Q) is an abelian group of finite type”, or like “two compact hyperbolic manifolds M and N of dimension at least 3 are isometric if and only if they have isomorphic fundamental groups” can not really be made clearer by trying to focus on any larger picture…

The disadvantages, on the other hand, are in fact quite real: one may write and polish with enthusiasm an introduction (so it becomes suitable for a O’Henry award) only to realize when coming to the point of writing the proofs that a fatal mistake lurked somewhere in the arguments only sketched previously. Or one may find new ideas or points of view when writing the proofs in question that lead to a complete change of emphasis of the paper (e.g., going from proving a special case of a statement to a more general one), and require a complete overhaul of the finely chiseled prose of the already completed introduction…

Indeed, both have happened to me, except of course that the literary quality of my drafts are far from deserving any award. The elephant cemetery section of my LaTeX directories contains at least three sad and melancholy beginnings of papers that will most likely never be revived, and I don’t know how many times I ended up re-working the introduction to my book on the large sieve (the final version of which states, quite accurately, that this project started as a planned short paper on extending previous results about the large sieve for Frobenius over finite fields to work in small sieve contexts…)

## More mathematical terminology: friable

Today’s terminological post will be a contribution to the French-led insurgency that tries to replace the denomination “smooth number” (or “smooth integer“) with the much better “friable number” (in French, “nombre friable” instead of “nombre lisse“).

Of course, many readers may wonder “what is this anyway?”. And part of the point is that the better choice may lead such a reader to guess fairly accurately what is meant (possibly with a hint that this has to do with multiplicative properties of integers), whereas playing a game of “Define a smooth number” with a wide group of mathematicians may probably lead to wildly different interpretations.

So here is the definition: a positive integer n is called y-friable (or smooth, if you still insist) if all the prime divisors of n are at most y. The idea is that y should be much smaller than n, so that this means (intuitively) that n only has “small” prime factors. But the definition makes sense for all y, and for instance, any integer n is n-friable, a 2-friable integer is a power of 2, etc.

I do not wish to discuss the properties of those integers (only their name), so let me just refer to this survey by Granville for a discussion of their basic properties and of their applications to computational number theory.

The adjective “friable” (Capable of being easily crumbled or reduced to powder, OED) seems perfect to describe this type of integers: it is evocative and conveys not only something of the technical definition, but also a lot of the intuitive meaning and applicability. The other contender, “smooth”, has several problems (in fairness, it has at least one positive aspect: whatever we call them, the integers without large prime factors are extremely useful in many parts of analytic and algorithmic number theory, and the underlying current that smoothness is something desirable is not usurped): (1) it is much too overloaded (search for “smooth” without more precision in Math Reviews: 68635 hits as of tonight; for “friable”, only 19); (2) whichever meaning of smooth you want to carry from another field, it does not really mean anything here; (3) not to mention that, chronologically speaking, the terminology was already preempted by the smooth integers of Moerdijk and Reyes, which are the solutions of the equation sin π x=0 in the real line of suitable topoi (such as the smooth Zariski topos, apparently).

The chronology of the use of these words, as it appears from Math Reviews at least, seems quite interesting: the first mention it finds of “smooth numbers” in the number-theoretic meaning is in the title of a paper of Balog and Pomerance, published in 1992. However, the notion is of course quite a bit older: the standard paraphrase was “integers without large prime factors”, with many variants (as can be seen from the bibliography of Granville’s survey, e.g, A. A. Buchstab, “On those numbers in an arithmetic progression all prime factors of which are small in order of magnitude”, 1949; Balog and Sarkozy, “On sums of integers having small prime factors, I”, 1984; Harman, “Short intervals containing numbers without large prime factors”, 1991; etc : clearly, something needed to be done…).

As for “friable”, the first number-theoretic use (interestingly, the six oldest among the 19 occurences of “friable” in Math Reviews also refer to other contexts than number theory, namely some studies of models of friable materials, from 1956 to 1987) is in a review (by G. Martin in 2005) of a paper of Pomerance and Shparlinski from 2002, though “smooth” is used instead in the paper. The first occurence in a paper (and so, possibly, in print) is in one by G. Tenenbaum and J. Wu, published in 2003. It must be said that, for the moment, only French writers seem to use the right word (Tenenbaum, de la Bretèche, Wu, and their students)… G. Martin consistently uses it in his reviews, despite having to recall that this is the same as smooth numbers; however, he uses “smooth” in the title and body of his paper on friable values of polynomials (published in 2002, admittedly, and the abstract on his web page uses mostly “friable” instead…).

## English comparative and the sieve

One of my favorite constructions in the English language is that bizarre form of comparative that makes it possible to speak of the “Shorter Oxford English Dictionary”, without any mention of what this estimable dictionary (two long and heavy volumes…) is actually compared to. Does this grammatical construction have a name? Does it exist in other languages? Certainly it is completely inexistent in French, and makes for rather thorny translation puzzles: how should a number theorist translate, in French, the name of Gallagher’s remarkably clever larger sieve? [The construction is actually particularly twisted here, since the implicit comparison point of Gallagher is, of course, already known as the large sieve…]

For those readers who have never heard of the larger sieve, here is the idea and the explanation for the name (which is very clearly explained in Gallagher’s paper): recall that a basic sieve problem (for integers) is to estimate the number of integers remaining from (say) an interval

$1,2,\ldots, N$

after removing all those n which, reduced modulo some prime p in some set (for instance, all those up to z=Nδ for some δ>0) always stay away from a given subset Ωp of primes: in other words, one wishes to know the cardinality of the sifted set

$S=\{n\leq N\,\mid\, n\text{ mod p}\notin \Omega_p\text{ for all }p\leq z\}.$

Classically (and also not so classically), the first examples were those were one tries to get S to be essentially made of primes, or twin primes, etc. In that case, the size of Ωp is bounded as p grows. There situations are called small sieves.

Then Linnik introduced the large sieve which is efficient for situations where the size of Ωp is not bounded, and typically grows to infinity with p: basic examples are the set of quadratic residues (or non-residues), or the set of primitive roots modulo p.

And then came the larger sieve: Gallagher’s method works better than the large sieve when Ωp is extremely large, so that the integers in S have few possible reductions modulo primes (roughly speaking, the larger sieve is better when the number of excluded classes is larger than half of the residue classes modulo p; so quadratic non-residues are borderline, and indeed both the large and the larger sieve give the correct upper bound — up to a constant — for the number of squares up to N). More precisely, Gallagher shows that

$|S|\leq N/D$

where

$N=\sum_{p\leq z}{\log p}-\log N$

and

$D=\sum_{p\leq z}{\frac{\log p}{p-|\Omega_p|}}-\log N,$

provided the denominator D is positive.

As the number of classes excluded increases, the efficiency of this inequality becomes extremely impressive: if

$|\Omega_p|>p-p^{\theta}$

with θ>0, the number of elements of S becomes at most a power of log(N), whereas the large sieve gives a power of N. For an arithmetico-geometric application of a new variant of the larger sieve in number fields in a situation where the numerology is of this type, you can read a recent paper of J. Ellenberg, C. Elsholtz, C. Hall and myself.

[I should mention that it was C. Elsholtz who first mentioned the larger sieve to me a few years ago: the method is not as well known as it should, since it is extremely simple — Gallagher deals with it in nine lines, and our version is not much more complicated, though it is a bit more involved since it works with heights in the number field to sieve elements which are not necessarily integers. The basic argument and its applications can provide excellent exercises and problems for any introductory number-theory course.]

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:

## 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…]