# 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 $d_k$?