E. Kowalski's blog

Аналитическая теория чисел

Thanks to the recent Russian translation of my book with Henryk Iwaniec, I can now at least read my own last name in Cyrillic; I wonder what the two extra letters really mean…

Analytic Number Theory in Russian

February 9th, 2015 at 9:06 pm

Posted in Language,Mathematics

An ideal hypothetical list

A few months ago, for purposes that will remain clouded in mystery for the moment, I had the occasion to compose an ideal list of rare books of various kinds, which do not necessarily exist.

Here is what I came up with:

(i) “The Elements of the Most Noble game of Whist; elucidated and discussed in all details”, by A. Bandersnatch, Duke Dimitri, N. Fujisaki, A. Grothendieck, Y. Grünfiddler, J. Hardy, Jr., B. Kilpatrick and an Anonymous Person.

(ii) “Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom funften Grade”, by F. Klein; with barely legible annotations and initialed “HW” on the first page.

(iii) “Histoire Naturelle”, Volume XXXIII: Serpens, by George Louis Leclerc, Comte de Buffon, edition of 1798; initialed “A.K.” on the title page.

(iv) “Chansons populaires de Corse, Navarre et Outre-Quiévrain”, collected and commented by P. Lorenzini.

(v) “奥の細道” (Oku no Hosomichi), by Matsuo Bashò.

(vi) “On the care of the pig”, by R.T. Whiffle, KBE.

(vii) “La Chartreuse de Parme”, by Stendhal.

(viii) “Der tsoyberbarg”, by T. Mann; Yiddish translation by I.B. Singer of “Der Zauberberg”.

(ix) “Les problèmes d’un problème”, by P. Ménard; loose manuscript.

(x) Opera Omnia of L. Euler, volumes 1, 2, 7, 11, 13, 23, 24, 30, 56, 62, 64, 65 and 72.

(xi) “Discorsi sopra la seconda deca di Tito Livio”, by N. Macchiavelli.

(xii) “An account of the recent excavations of the Metropolitan Museum at Khróuton, in the vicinity of Uqbar”, by E.E. Bainville, OBE.

(xiii) Die Annalen der Physik, volumes 17, 18, 23 and 25.

(xiv) “Diccionaro y gramática de la lengua Tehuelve”, anonymous; attributed on the second page to “a Humble Jesuit of Rank”.

(xv) “Le roi cigale”, French translation by Jacques Mont–Hélène of an anonymous English romance.

(xvi) “Mémoires du Général Joseph Léopold Sigisbert Hugo”, by himself, with an Appendix containing the “Journal historique du blocus de Thionville en 1815, et des sièges de cette ville, Sierck et Rodemack en 1815”.

(xvii) “Le Comte Ory”, full orchestral score of the opera by G. Rossini with Libretto by E. Scribe and Charles-Gaspard Delestre-Poirson.

(xviii) “Absalom, Absalom”, by W. Faulkner; first edition, dedicated To R.C. on the second page.

(xix) “Ficciones”, by J-L. Borges, third edition with page 23 missing.

(xx) “Die Gottardbahn in kommerzieller Beziehung”, by G. Koller, W. Schmidlin, and G. Stoll.

(xxi) “The etchings of the Master Rembrandt van Rijn, faithfully reproduced in the original size”, anonymous.

(xxii) “The memoirs of General S.I. Kemidov”, by Himself.

(xxiii) “Catalogue raisonné des œuvres d’Anton Fiddler”, by W.B. Appel.

(xxiv) “Zazie dans le métro”, by R. Queneau.

(xxv) “The mystery of the green Penguin”, by E. Mount.

(xxvi) “Χοηφόροι” (The Libation Bearers), by Aeschylus; an edition printed in Amsterdam in 1648.

(xxvii) “The Saga of Harald the Unconsoled”, Anonymous, translated from the Old Norse by W.B. Appel.

(xxviii) “Uncle Fred in the Springtime”, by P.G. Wodehouse.

(xxix) “The Tempest”, by W. Shakespeare.

(xxx) “The 1926 Zürich International Checkers Tournament, containing all games transcribed and annotated according to a new system”, by S. Higgs.

(xxxi) “A day at the Oval”, by G.H. Hardy.

(xxxii) “Harmonices Mundi”, by J. Kepler, initialed I.N on the second page.

(xxxiii) “Stories of cats and gulls”, by G. Lagaffe.

(xxxiv) “Traité sur la possibilité d’une monarchie générale en Italie”, by N. Faria; loose handwritten manuscript on silk.

(xxxv) “Broke Down Engine”, 78 rpm LP record, interpreted by Blind Willie McTell.

(xxxvi) “Les plages de France, Belgique et Hollande”, by A. Unepierre.

(xxxvii) “La Légende du Cochon Voleur et de l’Oiseau Rageur”, traditional folktale, translated from the Arabic by P. Teilhard de Chardin.

(xxxviii) “Discours des Girondins”, collected and transcribed by a parliamentary committee under the auspices of the “Veuves de la révolution française”, published by Van-den-Broeck, Bruxelles in 1862.

January 29th, 2015 at 4:00 pm

More conferences

It seems that most of my posts these days are devoted to announcing conferences in which I am involved as organizer… Indeed, there are two coming up this year (actually three, if I count the MSRI summer school):

(1) May 14 and 15, we will have the Number Theory Days 2015 at EPF Lausanne; the speakers are Gaetan Chenevier, Henryk Iwaniec, Alena Pirutka, Chris Skinner and Zhiwei Yun; this is co-organized by Ph. Michel and myself.

(2) Immediately afterward, from May 18 to 22, comes a conference at FIM, co-organized by H. Iwaniec, Ph. Michel and myself, with the title of “Analytic Aspects of Number Theory”; the current list of speakers is to be found on the web page; here is the poster (which is based on a picture taken by Henryk around Zürich last Fall):

Analytic Aspects of Number Theory

Most importantly, there is a certain amount of funding available for local expenses of your researchers (doctoral and postdoctoral students). Applications can be made here (before Feburuary 6; the form states January 28, but this is an error that will be corrected).

January 29th, 2015 at 3:56 pm

Posted in ETH,Mathematics

Proust, my family and Australia

When I was reading Proust, I noted with some amusement the character named simply “Ski” in the first volume of his appearance, a sculptor and amateur musician who is later revealed to be properly called “Viradobetski”, an actual name which was too complicated for the dear Madame Verdurin to try to remember. I just learnt from my better educated brother that

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.

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…

November 25th, 2014 at 7:10 pm

A (not so well-known) theorem of Fouvry, and a challenge

A few weeks ago, as already mentioned, I was in Oxford for the LMS-CMI summer school on bounded gaps between primes. My mini-course on this occasion was devoted to the ideas and results underlying Zhang’s original approach, based on expanding the exponent of distribution of primes in arithmetic progressions to large moduli.

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 $d_1$,…, $d_6$ is strictly larger than $1/2$, 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 $d_3$… 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 $f(n)$ concerns a fixed residue class $a$, and the average over moduli $q\leq x^{\theta}$ (with $q$ coprime to $a$) for some $\theta>1/2$ of the usual discrepancy
$\sum_{q\leq x^{\theta}} \Bigl|\sum_{n\equiv a\text{mod } q}f(n)-\frac{1}{\varphi(q)}\sum_{n}f(n)\Bigr|.$

The actual assumptions concerning $d_i$, $1\leq i\leq 6$, is a bit more than having this exponent of distribution $>1/2$: this must be true also for all convolutions
$d_i\star \lambda$
where $\lambda(n)$ is an arbitrary essentially bounded arithmetic function supported on a very short range $1\leq n\leq x^{\delta_i}$ for some $\delta_i>0$.

This extra assumption is reasonable because since $\delta_i$ can be arbitrarily small, certainly all known methods to prove exponents of distribution larger than $1/2$ 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 $1/2$ for the characteristic function of integers $n$ having no prime factors $\leq z$ for $n\leq x$ and $z\leq x^{1/6-\varepsilon}$. 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 $1/2$ for the functions $d_1, d_2, \ldots, d_{12}$. 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 $\theta>1/2$ for the sequence of primes, can one prove a similar exponent of distribution for all the divisor functions $d_k$?

November 4th, 2014 at 9:17 pm

Posted in Mathematics