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

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

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

It was almost twenty years ago that I started drawing and looking at the graphs of Kloosterman sums (at least, that’s a likely date; I don’t remember when it began, but I put some drawings as whimsical illustrations — see, e.g., page 26 — in my PhD thesis to enliven it, and that was around 1997–1998).

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 $latex (p-1)/2$). 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 $latex [0,1]\rightarrow \mathbf{C}$, and consider those paths associated to all Kloosterman sums modulo a large prime $latex p$, then these will, statistically, “look like” the graphs $latex t\mapsto S(t)$ of a certain very specific random Fourier series.

Precisely, for a prime $latex p$, an invertible residue class $latex a$ modulo $latex p$ and a real number $latex t\in [0,1]$, let’s denote by
$latex K_p(a)(t)$
the point at “time” $latex t$ on the graph joining the partial sums of
$latex K\ell_p(a)=\frac{1}{\sqrt{p}}\sum_{1\leq n\leq p-1} e((ax+\bar{x})/p),$
each segment being parameterized linearly by an interval of length $latex 1/(p-1)$. This corresponds to normalizing the pictures I used to draw by saying that they correspond to a parameterization by $latex [0,1]$, and also by scaling by the factor $latex \sqrt{p}$, which has the effect that the endpoint is always a real number in $latex [-2,2]$, by the Weil bound for Kloosterman sums.

We interpret the data $latex (K_p(\cdot)(t))_{t\in [0,1]}$ as a stochastic process; the “randomness” is in the residue class $latex a$, which is supposed to be chosen uniformly at random among invertible classes modulo $latex p$.

Our first result is convergence in the sense of finite distributions: we consider any fixed finite number of “times” $latex 0\leq t_1<\cdots<t_k\leq 1$, and prove that the random vectors
$latex (K_p(t_1),\ldots, K_p(t_k)),$
have a limit, in the sense of convergence in distribution, as $latex p$ tends to infinity.

This limit is explicit: it is
$latex (S(t_1),\ldots, S(t_k)),$
where $latex (S(t))_{t\in [0,1]}$ is a very nice (a.s.) continuous stochastic process, defined in the following manner: take any sequence $latex (X_h)_h$ of independent random variables which are all Sato-Tate distributed, and put
$latex S(t)=\sum_{h\in\mathbf{Z}} \frac{e^{2i\pi ht}-1}{2i\pi h}X_h,$
a random Fourier series (here the term $latex h=0$ has to be interpreted as $latex tX_0$).

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 $latex \mathrm{SL}_2\times \cdots\times\mathrm{SL_2}$ that surjects to each factor is the whole product group).

Remark. As a simple consequence, note that $latex S(1)=X_0$ 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 $latex (K_p(\cdot)(t))_t$ and the limit $latex (S_t)_t$ 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 $latex [0,1]$.

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 $latex S(t)$ 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.

Despite everything, there is something to be said for the internet. Just a few days ago, I wanted to reference the work of Bagchi, who provided the probabilistic interpretation of Voronin’s Universality Theorem for the Riemann zeta function. However, the original was unpublished, and one of the few papers of Bagchi on this topic pointedly indicated that he had removed most probabilistic considerations (why? if it was at the request of a referee, I can only sigh). But fortunately, lo and behold, the original thesis (from 1981) can be found in a very decent scan from the Indian Statistical Institute!

Update (November 1, 2016): it seems that the web site of the library of the Indian Statistical Institute has been reorganized, and I can’t locate the PDF file of Bagchi’s Thesis anymore (sigh… maybe the internet has drawbacks after all.) Anyone looking for a copy can send me an email, and I’ll be happy to forward the PDF that I saved.

Second Update (July 20, 2017): the PDF is once more accessible here.