Probabilistic number theory

During the fall semester, besides the Linear Algebra course for incoming Mathematics and Physics students, I was teaching a small course on Probabilistic Number Theory, or more precisely on a few aspects of probabilistic number theory that I find especially enjoyable.

Roughly speaking, I concentrated on a small number of examples of statements of convergence in law of sequences of arithmetically-defined probability measures: (1) the Erdös-Kac Theorem; (2) the Bohr-Jessen-Bagchi distribution theorems for the Riemann zeta function on the right of the critical line ; (3) Selberg’s Theorem on the normal behavior of the Riemann zeta function on the critical line ; (4) my own work with W. Sawin on Kloosterman paths. My goal in each case was to do with as little arithmetic, and as much probability, as possible. The reason for this is that I wanted to get and give as clear a picture as possible of which arithmetic facts are involved in each case.

This led to some strange and maybe pedantic-looking discussions at the beginning. But overall I think this was a useful point of view. I think it went especially well in the discussion of Bagchi’s Theorem, which concerns the “functional” limit theorem for vertical translates of the Riemann zeta function in the critical strip, but to the right of the critical line. The proof I gave is, I think, simpler and better motivated than those I have seen (all sources I know follow Bagchi very closely). In the section on Selberg’s Theorem, I used the recent proof found by Radziwill and Soundararajan, which is very elegant (although I didn’t have much time to cover full details during the class).

I have started writing notes for the course, which can be found here. Note that, as usual, the notes are potentially full of mistakes! But those parts which are written are fairly complete, both from the arithmetic and probabilistic point of view.

Bilinear forms with Kloosterman sums

Philippe Michel, Will Sawin and I have just finished the first draft of a paper where we prove estimates for general bilinear forms of the type
\sum_{m\leq M}\sum_{n\leq N}\alpha_m\beta_n K(mn)
where (1) K(x)=\mathrm{Kl}_k(mn;p) is a (normalized) hyper-Kloosterman sum (for k=2, this is a classical Kloosterman sum) modulo a prime p; and (2) the ranges M and N are such that we have non-trivial bounds even if M=N is a bit smaller than \sqrt{p} in logarithmic scale. In other words, we obtain non-trivial results below the “Pólya-Vinogradov range”.

The basic strategy to get this result is not new: it was devised by Fouvry and Michel a number of years ago (inspired at least in part by earlier work of Friedlander-Iwaniec and by the Vinogradov-Karatsuba-style “shift” method to estimate certain short exponential sums). What was missing (despite the strong motivation provided by applications that were known to follow from such a result, one of which is described in a recent preprint of Blomer, Fouvry, Milicevic, Michel and myself) was a way to prove certain estimates for (complete) sums over finite fields, of the type

\sum_{r\in \mathbf{F}_p} R(r,\lambda_1,b_1,b_2,b_3,b_4)\overline{R(r,\lambda_2,c_1,c_2,c_3,c_4)}\ll p^{3/2}


R(r,\lambda,b_1,b_2,b_3,b_4)= \sum_{s\in\mathbf{F}_p}e\Bigl(\frac{\lambda s}{p}\Bigr)K(s(r+b_1))K(s(r+b_2))\overline{K(s(r+b_3))K(s(r+b_4))}

unless the parameters (\lambda_1,\lambda_2, b_i, c_i) are in some “diagonal” positions. And we cannot afford too many diagonal cases…

The main contribution of our paper (much of which comes from the ideas of Will!) is to find a relatively robust approach to such estimates.

This relies, as one can expect, from extensive algebraic-geometric arguments to apply the Riemann Hypothesis over finite fields. In fact, from this point of view, this paper is by far the most complicated I’ve ever been involved in. We use, among other things:

  • The Riemann Hypothesis over finite fields, in its most general version of Deligne — indeed, we use it multiple times;
  • The interpretation of the sum over s (in the sum R above) as itself a trace function of sbome sheaf on the space of parameters (\lambda,r,b_1,b_2,b_3,b_4); this follows from the formalism of étale cohomology, which is also used in many other ways (e.g., to detect irreducibility of sheaves by properties of the top-degree cohomology);
  • A very general version of the Euler-Poincaré characteristic formula in étale cohomology – this comes from SGA5;
  • The formalism and properties of vanishing and nearby cycles in étale cohomology, and in particular their relations with local monodromy representations of sheaves on curves;
  • The global \ell-adic Fourier transform of Deligne as well as the local Fourier transform of Laumon;
  • A special case of the homogeneous Fourier transform of Laumon (which we might be able to avoid, although with an argument involving perverse sheaves);
  • Katz’s theory of Kloosterman and hypergeometric sheaves, in particular with respect to the computation of their geometric monodromy groups (and its implication through the Goursat-Kolchin-Ribet Criterion), but also (and equally importantly) with respect to their local monodromy properties;
  • The diophantine criterion for geometric irreducibility (which is again a case of the Riemann Hypothesis)…

Many of these are results and ideas that I was aware of but had never actually used before, and I learnt a lot by seeing how Will exploited and combined them.  I will try to write a few more posts later to (attempt to) explain and motivate them (and how we use them) from an analytic nunber theorist’s viewpoint.  The theory of vanishing cycles, in particular, should have many more applications in extending the range of applicability of Deligne’s Riemann Hypothesis to problems in analytic number theory.

The paper is dedicated to Henryk Iwaniec, who has been over the years the most eloquent and powerful advocate for a deeper use of the work of Deligne (and Katz and others) in applications to analytic number theory.


I just came back from a week on the Italian island of Ventotene, where I participated in a very nice conference on “Manifolds and groups”. This is of course not my usual topic, and besides giving a minicourse on expanders and coverings (with a focus on geometric applications of expanders having to do with coverings…), I learnt a lot of interesting things. I particularly enjoyed the other two minicourses, by T. Gelander on invariant random subgroups and by R. Sauer on Lücks’s Approximation Theorem. (For both of these, as well as for a number of other talks, the slides are available on the conference web page; my own — handwritten — notes on my course should appear there soon, after I scan them).

Being an island, Ventotene is reached by boat. One of the interesting things about the trip to Ventotene was to observe how birds


would follow us until a very definite point, and then suddenly disappear.

Also, the instructions to launch a lifeboat are rather daunting…


The organizers had scheduled a lot of free time during the conference besides the scientific programme. I was therefore able to take a few pictures, such as some of the local lizards,


and some of the wonderful local cats.


The end of the week also coincided with the beginning of the ten-day long celebration of the island’s patron saint, Santa Candida. Among the festivities were the evening launches


(over a week) of huge hot-air balloons (“Mongolfieri”), with some fireworks


(note the amusing effects of my camera’s fireworks scene setting without tripod…). Some Galois-theoretic persiflage was also notable…


Update The scans of my lectures can be found here.

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?