Proust s’est inspiré d’Henri Kowalski né en 1841, fils d’un officier polonais émigré en Bretagne. Il était à la fois compositeur de musique et concertiste.

or

Proust used as model Henri Kowalski, born in 1841, son of a Polish officer who emigrated to Brittany. He was a composer as well as a concert player.

to quote an authoritative list of Proust characters.

That Henri Kowalski is, it turns out, the son of Nepomus Adam Louis Kowalski, whose brother was Joachim Gabriel Kowalski, one of whose sons was Eugène Joseph Ange Kowalski, one of whose sons was Louis André Marie Joseph Kowalski, the fourth son of whom was my father. This puts me at genealogical distance (at most) six to a character from Proust. (Of course, rumors that Viradobetski was inspired by someone else can be safely discarded).

Wikipedia has a small page on Henri Kowalski, who was quite active and successful as a musician, and a rather impressive traveller. He spent thirteen years in Australia, leaving enough traces to be the subject of public lectures at the university of Melbourne. There are a few of his pieces on Youtube, for instance here. He also wrote a travel book which I now intend to read…

]]>In the first lecture, I mentioned a result of Fouvry as a motivation behind the study of other arithmetic functions in arithmetic progressions: roughly speaking, if one can prove that the exponent of distribution of the divisor functions ,…, is strictly larger than , then the same holds for the primes in arithmetic progressions.

This statement (which I will make more precise below, since there are issues of detail, including what type of distribution is implied) is very nice. But it turned out that quite a few people at the school were not aware of it before. The reason is probably to a large extent that, as of today (and as far as I know…), it has not been possible to use this mechanism to prove unconditional results about primes: the problem is that one does not know how to handle divisor functions beyond … One could in fact interpret this as saying that higher divisor functions are basically as hard as the von Mangoldt function when it comes to such questions.

The precise statement of Fouvry is Theorem 3 in his paper “Autour du théorème de Bombieri-Vinogradov” (Acta Mathematica, 1984). The notion of exponent of distribution of a function concerns a fixed residue class , and the average over moduli (with coprime to ) for some of the usual discrepancy

The actual assumptions concerning , , is a bit more than having this exponent of distribution : this must be true also for all convolutions

where is an arbitrary essentially bounded arithmetic function supported on a very short range for some .

This extra assumption is reasonable because since can be arbitrarily small, certainly all known methods to prove exponents of distribution larger than would accommodate this tweak.

As far as the proof is concerned, this Theorem 3 is actually rather “simple”: using the Heath-Brown identity, all the hard work is moved to the proof of an exponent of distribution beyond for the characteristic function of integers having no prime factors for and . This is much deeper, and involves all the machinery of dispersion and Kloostermania…

In addition, Fouvry mentioned to me the following facts, which I didn’t know, and which are very interesting from a historical point of view. First, this theorem of Fouvry is a strengthened version of the results of Chapter III of his *Thèse de Doctorat d’État* (Bordeaux, September 1981, supervised by J-M. Deshouillers and H. Iwaniec). At that time, Kloostermania was under construction and Fouvry had only Weil’s classical bound for Kloosterman sums at his disposal, and this original version required an exponent of distribution beyond for the functions . This illustrates the strength of Kloostermania!

Moreover, in this thesis, Fouvry used an iteration of Vaughan’s identity, instead of Heath-Brown’s identity, which only apparead in 1982. However, although this was less elegant, this iteration had the same property to transform a sum over primes into multilinear sums where all non smooth variables have small support near the origin.

Fouvry also suggests the following inverse challenge for *aficionados*: assuming an exponent of distribution for the sequence of primes, can one prove a similar exponent of distribution for all the divisor functions ?

This year, by a lucky coincidence, I can finally say something definite about these graphs, and indeed something quite interesting. What happened is that Will Sawin was visiting Switzerland in June to talk to Philippe Michel and me about some very nice results about trace functions, and he noticed a picture like the one above that I have on the whiteboard next to my office. He asked me what it was, and more or less immediately it became clear that we could try to determine, for instance, the distribution of the “middle point” (corresponding to the partial sum up to ). Then, in a few more hours, we realized that we should be able to go much further. After a few more months, I’ve just uploaded to arXiv the paper we wrote about this subject, which is also available on my web page. It is a combination of probability, algebraic geometry and analytic number theory which I personally find extremely appealing and pleasant.

In rough terms, what we show is that, if one normalizes the *Kloosterman paths* properly, as continuous maps , and consider those paths associated to all Kloosterman sums modulo a large prime , then these will, statistically, “look like” the graphs of a certain very specific random Fourier series.

Precisely, for a prime , an invertible residue class modulo and a real number , let’s denote by

the point at “time” on the graph joining the partial sums of

each segment being parameterized linearly by an interval of length . This corresponds to normalizing the pictures I used to draw by saying that they correspond to a parameterization by , and also by scaling by the factor , which has the effect that the endpoint is always a real number in , by the Weil bound for Kloosterman sums.

We interpret the data as a stochastic process; the “randomness” is in the residue class , which is supposed to be chosen uniformly at random among invertible classes modulo .

Our first result is convergence in the sense of finite distributions: we consider any fixed finite number of “times” , and prove that the random vectors

have a limit, in the sense of convergence in distribution, as tends to infinity.

This limit is explicit: it is

where is a very nice (a.s.) continuous stochastic process, defined in the following manner: take any sequence of independent random variables which are all Sato-Tate distributed, and put

a random Fourier series (here the term has to be interpreted as ).

**Remark.** It would be interesting to know if this series has already appeared in other contexts; our searches of the literature on random Fourier series have not yet found any previous occurence.

The proof is actually relatively short, but nevertheless this is a very deep property. Indeed, it depends in an essential way on Deligne's Riemann Hypothesis over finite fields, and on the computation of the monodromy groups of Kloosterman sheaves by Katz.

In fact, it is a very interesting instance where the application of the Riemann Hypothesis requires not only to understand *when* it gives cancellation in exponential sums, but also requires in an essential way the knowledge of the *main term* when there is no cancellation. This seems (to me at least) to require in an essential way the group-theoretic interpretation of the equidistribution of Kloosterman sums with respect to the Sato-Tate measure, because it is this interpretation that allows us to control arbitrary moments of partial sums of Kloosterman sums using the Goursat-Kolchin-Ribet criterion from group theory (roughly speaking, an algebraic subgroup of that surjects to each factor is the whole product group).

**Remark.** As a simple consequence, note that is Sato-Tate distributed, so this result contains the “vertical” Sato-Tate law of Katz.

Plotting some samples of this random Fourier series, by interpolating linearly between the values of samples of partial sums, we get pictures with a definite *air de famille*:

There is an obvious question that arises from this result, which probabilists will already have raised: since both our stochastic processes and the limit are continuous processes, we should really be trying to prove convergence in the space of continuous processes, i.e, when viewing the Kloosterman paths as random variables with values in the Banach space of continuous functions on .

We tried to do this. Using Kolmogorov’s criterion for tightness (and the previous result), one can see that the question is intimately related to non-trivial (average) estimates for short partial sums of Kloosterman sums *just around the Pólya-Vinogradov range*. We have not (yet) established the required estimate, which seem quite delicate, but it is very interesting to see a unification of two aspects of exponential sums which are extremely important in applications, namely equidistribution of Katz-Sato-Tate type, and estimate for short sums (of length about square root of the number of terms). Moreover, using this approach, we show easily that further average over the additive character defining Kloosterman sums does lead to convergence to as continuous processes. As a first application, we get a rather elementary proof of tail bounds for the supremum of the partial sums (with this extra average), using some nice facts about probability in Banach spaces (I may write a bit about this in a later post).

I talked about this result in Oxford a few weeks ago, on the occasion of a workshop on analytic number theory there (this might have been one of the first mathematical talks coming with its own app!).

We have dedicated this paper to the memory of Marc Yor.

]]>Is that a millenial-style coincidence worth cosmic pronouncements? Actually, not that much: since the dices are indistinguishable, the probability of a single throw of this type is

so about one and a half percent. And for two, assuming independence, we get a probability

or a bit more than one chance in five throusand. This is small, but not extraordinarily so.

(The dices are thrown from a cup, so the independence assumption is quite reliable here.)

]]>Here’s one ad for the most famous vegetarian restaurant in Zürich (probably the best in the world):

Here’s an ad for the Zürich public transportation system; the left-hand man is a right-wing politician, and the right-hand one a socialist (I had to look this up; I’m not really up to date on local politics), and the ad says that fortunately there is a tram stop on average every 300 meters…

Here’s a recent ad to warn against swimming in the wrong places in the river, despite the absence of sharks (*Hai* means shark in German):

of the building as it looked in 1870. The red square indicates where my office is located…

]]>(1) My lecture notes on representation theory (expanded) will appear in September, published in the Graduate Studies in Mathematics series; the preview material contains Chapter 1, and a fair bit of Chapter 2; the index is also available, and perusing it will give an idea of the range of topics mentioned.

(2) Henryk Iwaniec has also a new book coming, in October, containing his lectures notes on the Riemann zeta function. I haven’t seen it yet, but he told me that the highlight, in his opinion, is the second part which contains his personal treatment of the Levinson method for finding critical zeros of zeta. This should be quite interesting to read…

**Update** (August 29): the AMS web site confirms that my book is already available!

Did I like the book? This is probably far from the right question to ask in any case, but it’s therefore worth investigating in a post-modern spirit. Since I finished the whole of the seven volumes in about seven months, during which time I had to take care of many other activities, and also started (and often stopped) reading a fair number of other books, I was certainly finding something in Proust that kept me engaged in his work.

One technical aspect of Proust’s style that struck me was how he manages to capture the way that crucial events or characters will first appear informally and casually in life. Here’s for example the first time the narrator meets Gilberte while playing at the Champs-Élysées:

Retournerait-elle seulement aux Champs-Élysées? Le lendemain elle n’y était pas; mais je l’y vis, les jours suivants; je tournais tout le temps autour de l’endroit où elle jouait avec ses amies, si bien qu’une fois où elles ne se trouvèrent pas en nombre pour leur partie de barres, elle me fit demander si je voulais compléter leur camp, et je jouai désormais avec elle chaque fois qu’elle était là.

And here is the first appearance of the *jeunes filles*, among whome is Albertine:

J’aurais osé entrer dans la salle de balle, si Saint-Loup avait été avec moi. Seul je restai simplement devant le Grand-Hôtel à attendre le moment d’aller retrouver ma grand-mère, quand, presque encore à l’extrémité de la digue où elles faisaient mouvoir une tâche singulière, je vis s’avancer cinq ou six fillettes, aussi différentes, par l’aspect et par les façons, de toutes les personnes auxquelles on était accoutumé à Balbec, qu’aurait pu l’être, débarquée on ne sait d’où, une bande de mouettes qui exécute à pas comptés sur la plage — les retardataires rattrapant les autres en voletant — une promenade dont le but semble aussi obscur aux baigneurs qu’elles ne paraissent pas voir, que clairement déterminé dans leur esprit d’oiseaux.

This comes with no warning or no articifial build-up of *something is going to happen, drumroll, drumroll*.

I was also very touched by the last pages, which certainly affected my overall impression and reaction in a way that I’ve only felt before when finishing Faulkner’s *Absalom, Absalom* (after which I went into a rather intense faulknerian phase during my PhD years). It seems that to understand Proust (as much as I can…), I would have to re-read the whole text, in the light of these last pages. Is that the expected reaction? It could well be… Will I do it? Who knows…

On a different note, I was amused to see that while the first Pléiade edition is a rather straightforward edition of the novel with short notes and biographical information (rather like the Library of America editions of Faulkner, for instance), the second edition succombs

to editorial inflation on a magnificent scale: the variants, *esquisses*, notes and notes on the *esquisses*, take up more space than the actual text!

Here is the first volume of the old edition:

compared with the second of the new edition:

This can of course be helpful, as are certainly useful the 125 pages of *Liste des personnages cités* which allow you to quickly locate all the places where Rembrandt, or the Marquis de Norpois, or Saint Simon, or any other character, real or imagined, makes an appearance in the whole text.

(There is a similar list for names of places and names of works of arts, again real or imagined).

Another thing I noticed is that the first edition doesn’t use accented letters as capitals at the beginning of sentences, while the second does:

compared with

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