Trace functions, a survey

At last count, my series of works with Étienne Fouvry and Philippe Michel on trace functions and their applications consists of seven research papers or preprints, amounting to a bit more than 200 pages. To these are added a number of works-in-progress or partial notes (some with results we did not need or use and so took out of earlier drafts of our papers, some with worked-out examples or remarks, etc). We have a relatively firm project of writing a monograph account of the whole theory and applications, which we view in part as a way of making accessible some of the deep consequences of the Riemann Hypothesis over finite fields of Deligne, but this is clearly a long-term project. In the meantime, we have written a first short survey, the first draft of which can be found on my web page. This is in fact the written (and slightly expanded) account of a Colloquio de Giorgi that I gave at the

Scuola Normale Superiore in Pisa earlier this year, and we have included an unusual representation of the fundamental domain of the modular group acting on the upper half-plane as an homage and acknowledgement of this occasion.

Fouvry 60

We are currently enjoying in Marseille the warmth and delights of a French Mediterranean Bouillabaisse while celebrating analytic number theory and the achievements of É. Fouvry, on the occasion of his 60th birthday.

I think everyone who has been in contact with any of his papers has immense respect for his scientific work. All those of us who have been fortunate enough to talk with him beyond purely scientific matters will also attest to his exemplary intellectual honesty, rectitude, generosity and — also important to my mind — to his sense of humor.

In analytic number theory, we play day to day in a wild down-to-earth jungle. We also all know that somewhere there is a Garden of Eden, where the Riemann Hypothesis roams free, and we hope to go there one day. Fewer know that there is a place even beyond, a Nirvana where even the Riemann Hypothesis is but a shadow of a deeper truth. And fewer still are those who have set foot in this special place. É. Fouvry did, and he was among the very first ones, if not the very first; and more people have walked on the moon than been there.

A few years ago, I wrote a nominating letter for Étienne’s application to the Institut Universitaire de France. There is one sentence that I wrote which still seems to me to summarize best my feelings about this part of his work: Rarely in history was so much owed by so many arithmeticians to so few. This is even truer today than it was then. Reader, if you care at all about prime numbers, recall that without É. Fouvry and very few others (two of whom are with us in Marseille), you might well never have known that the gaps between successive primes do not grow to infinity.

Who needled Buffon?

I already mentioned Buffon’s needle a long time ago, with a picture of the page of my grandfather’s own copy of the multi-volume Histoire naturelle. Recently, I had a closer look at what Buffon actually wrote, and realized that, far from dropping unit-lengths needles to compute π, he was suggesting dropping well-calibrated needles to compute 1/2, or more precisely to create a fair game between two players, one of which bets on whether a needle dropped by the other will cross the boundaries of the strips of wood on the floor?

In fact, since he uses the “value” π=22/7 just a few pages before,

one can see that computing π must have been very far from his mind. The question thus arises: who did first interpret Buffon’s question as one that may lead to an estimate for π?

Equally recently, I noticed the following variant of the problem, which may already be known: suppose a player is allowed to throw a dart randomly anywhere on the complex upper half-plane (this may seem too “infinite”, but the disc model of the hyperbolic plane will work just as well if you want to throw darts to a circular target…), but that the dart is instantaneously moved, as soon as it touches the target, to the corresponding point of the fundamental domain of $SL_2(\mathbf{Z})$

(in principle, this is an operation that can be performed very efficiently by some variant of the euclidean algorithm, though in practice there are numerical stability problems for points very close to the boundary of the fundamental domain). Now the bet with the other player is whether the point in the fundamental domain will have imaginary part larger than some fixed a>0.

As Buffon, we ask: which value of a will ensure that this is a fair game? If we assume that the point in the fundamental domain is uniformly distributed for its probability Poincaré measure, which is $3\pi^{-1} y^{-2}dxdy$, it is easy to see that the answer is $a=6/\pi$, so we have some kind of hyperbolic Buffon needle. To make this into a “computing π procedure”, one may for instance “throw darts” successively at the points

$x_j=\frac{j+i}{N}$

on the segment of horocycle at height $1/N$, for some large $N\geq 1$, and look at the corresponding points $y_j$ in the fundamental domain. Then it is is a well-known result (which follows here most easily from non-trivial bounds on Hecke eigenvalues of Maass cusp forms) that as $N$ grows, the points $y_j$ become equidistributed in the fundamental domain for the Poincaré measure, and hence that the proportion among them which have imaginary part larger than $a$ converges to $1/a$. Hence if we look at the median of this sequence of imaginary parts, we will find that it converges to $6/\pi$ as $N$ tends to infinity…