Littlewood on real functions

Speaking of MSRI, I was there for two weeks last month for a summer school on analytic number theory and gaps between primes (co-organized with D. Koukoulopoulos, J. Maynard and K. Soundararajan; the videos are already available online, as well as the exercise sheets prepared by the two TAs, Z. Brady and B. Löffel.) And as usual when visiting an English-speaking town, I spent a fair amount of time in second-hand bookstores (I have developed from my graduate student days the theory that certain categories of books should never be bought new; for instance, can one improve on this old-fashioned cover


of a whodunit by D.L. Sayers, which at least makes it clear to the dimmest reader who indeed did it?)

From Half-Price Books and Black Oak Books, I was well rewarded. In particular, the latter boutique has a rather remarkable selection of mathematical books. I skipped the three copies of Gauss sums, Kloosterman sums and monodromy groups (I got mine for five dollars from the Rutgers University special-deals cart a long time ago), but acquired Freiman’s Foundations of a structural theory of set addition for 7 or 8 dollars (the review by B. Gordon that I link to is quite fun to read), and found a little gem in Littlewood’s rather obscure book The elements of the theory of real functions:


Actually, the obscurity of this book is maybe understandable. It’s rather depressing to read

These lectures are intended to introduce third year and the more advanced second year men to the modern theory of functions

in the preface. But more importantly maybe, despite the promising title, the content of the book has very little to do with functions, real or otherwise. It’s really a semi-rigorous treatment of set theory up to very basic facts about subsets of \mathbf{R}^n, that does little to excite or attract attention. (Maybe the citation above reflects the fact that women students in Littlewood’s time were simply too clever to find this of any interest, and prefered to spend their time reading Gödel or Bourbaki?)

Despite the claim further in the preface that Littlewood aimed at excluding as far as possible anything that could be called philosophy, the fluffiness of the statements reminds me strongly of that discipline. Indeed, to see the bland statement

In Prop. 19 take the class to be O itself. We obtain a blank contradiction.

shows that we are not in the most fastidious of company from the purely mathematical point of view.

Part of the charm of this book is the really weird terminology and the dizzying array of apparently pointless notational abbreviations (example at random: Prop. 5, p. 56 states Given any series S of \Xi_0 terms, and an \eta, E, there is a sub-series of E similar to S). Of course, the first edition is apparently from 1925, when some poetic license might have been permitted as far as set theory and topology are concerned, but one doesn’t need to be overly formalist to raise one’s eyebrows when understanding (in a “completely revised” third edition of 1954) that a “class” denotes what everyone else calls a set, and that a “set” is what everybody calls a subset of \mathbf{R}^n. At least this would go a long way towards explaining the poor track-record of Cambridge Students from that period at having the faintest idea what every mathematician outside England was saying, as far as set theory and topology was concerned. Maybe this was the best way to ensure that they would think about more interesting things?

MSRI Analytic Number Theory Semester in 2017

As already indicated by T. Tao on his blog, there will be a semester-length program on analytic number theory from january to june 2017 at MSRI, co-organized with C. David, A. Granville, Ph. Michel, K. Soundararajan. The mathjobs page for applying to the semester is now open (until December 1; because, apparently, of NSF data-gathering requirements, all applications are done through this website…) We encourage all interested mathematicians to apply!

Latest adventures

The last two weeks were quite eventful…

First I spent four days in China for the conference in honor of N. Katz’s 71st birthday. I was lucky with jetlag and was able to really enjoy this trip, despite its short length. The talks themselves were quite interesting, even if most of them were rather far from my areas of expertise. I talked about my work with W. Sawin on Kloosterman paths; the slides are now online.

I only had time to participate in one of the excursions, to the Forbidden City,

Forbidden City
Forbidden City

were I took many pictures of Chinese Dragons…

Chinese Dragon
Chinese Dragon

That same evening, with F. Rodriguez Villegas and C. Hall, I explored a small part of the Beijing subway,

Subway map
Subway map

trying to interpret and recognize various Chinese characters, before spending a fair amount of time in a huge bookstore


(where I got some comic books in Chinese for fun).

Upon coming back on Thursday, I first found in my office the two volumes of the letters between Serre and Tate that the SMF has just published, and which I had ordered a few days before taking the plane. Reading the beginning of the first volume was very enjoyable in the train on Friday morning from Zürich to Lausanne, where the traditional Number Theory Days were organized this year. All talks were excellent again — we’re now looking forward to next year’s edition, which will be back in Zürich! And I’ll write later some more comments about the Serre-Tate letters…

And then, from last Monday to Friday, we had in Zürich the conference “Analytic Aspects of Number Theory”, organized by H. Iwaniec, Ph. Michel and myself with the help of FIM. It was great fun, and there were really superb and impressive talks. One interesting experience was the talk by J. Bellaïche : for health reasons, he couldn’t travel to Zürich, but we organized his talk by video (using a software called Scopia), watching it from a teleconference room at ETH. This went rather well.

New versions, new bugs

Ph. Michel found the first bug in the new version of Kloostermania: when entering a modulus for a sum from the menu, there was no check that it is prime (or adjustment to make it so). This is now corrected in version 1.01, to be found in the usual place.

As another bonus feature, a double tap on the screen will cycle between the types of sums presented: Kloosterman sum to Birch sum to both to Kloosterman…

Kloostermania turns 1.0

Since the Pocket Kloostermania has remained unchanged for almost five years, an update is probably welcome! This is now available, in time to honor N. Katz on the occasion of this 71st birthday (which I will help celebrate in China next week).

Important changes in the new version:

(1) It was recompiled — hence a new modern look instead of a style reminiscent of the dark ages –, and the resulting binary should work on any Android system with version 4.0.3 or later;

(2) To keep up with recent progress, the program now displays Kloosterman sums and/or Birch sums, instead of Kloosterman and/or Salié sums. On the other hand, the moduli used are now restricted to primes, to simplify things a bit, and only one parameter is used: the sums displayed are

(3) In addition to being able to change the modulus by swiping horizontally, a vertical swipe on the screen scrolls among the values of the parameter a. As was the case in the previous version (and even more because phones are faster now), the scrolling is usually too fast for a single step, so tapping once close to one of the edges of the window displaying the sum will perform just one step of the corresponding move (e.g., tapping close to the right of the screen goes to the next prime modulus). The value of the sum and its parameters and then displayed quickly.

(4) The plots are presented in orthonormal configurations, to give a more faithful representation of the paths in the plane;


(5) In the “About” dialog, a single click will display (if a PDF viewer is installed) the paper of W. Sawin and myself that explains the limiting distribution of Kloosterman (and Birch) paths…

(6) It is possible to save (or rather to “share” in the usual Android way) a picture of the sums currently displayed, as a PNG file.

(6) And the launcher icon is better.


The installation file is available right now on the updated Kloostermania page!