Thanks to the recent conference celebrating the 25th anniversary of the Zahlentheorie Seminar, and even more as this week’s conference for Fouvry’s 60th Birthday in Marseille, I have been able to talk with a number of people about Zhang’s result on bounded gaps between primes, especially Philippe Michel, Paul Nelson and Étienne Fouvry. All generic “our”s and “we”s below refer to these discussions.

We have concentrated exclusively on the critical “ternary-divisor” part of the argument, and attempted many variants and reconfigurations. Our goal is not really to do better than Zhang (in terms of exponents), but to see by direct experience what works and what doesn’t.

Quite a few of these attacks failed, which of course helps in building some understanding. But two or three directions are promising, and one definitely does seem to lead to a different version of the argument. We have not yet written full details, but the approach succeeds (i.e., it definitely breaks the barrier $latex 1/2$ of the Riemann Hypothesis, which is the whole game) in getting an exponent of distribution $latex 4/7$ for the ternary divisor function with nicely factorable moduli, at a level of informality which inspires great confidence. (When applied in the context of my previous papers with Fouvry and Michel, this type of sketches give fully consistent exponents and uniformity in comparison with the final papers.)

Here are just some remarks. We view the basic problem as understanding

$latex \sum_r\sum_s \sum_{mn_1n_2n_3\equiv a \bmod{rs}} 1$

where $latex q=rs$ is the squarefree modulus, which should therefore (to beat RH) be of size a bit larger than $latex X^{1/2}$ (by a small positive power of $latex X$), and $latex m$ is fixed (though one can try to exploit average over it, and it will be important in the end that it be relatively small). In fact, quite a few attempts purposely express this as a special case of

$latex \sum_r\sum_s \frac{1}{\sqrt{rs}}\sum_{n} d_3(n) K(n)$

for some general “trace function” (these are more general than usually discussed, both because the modulus is not prime, and because the $latex L^2$-normalized characteristic function of a residue class modulo a prime is only the trace function of a perverse sheaf, and not a standard lisse sheaf, but that last point is not particularly problematic). The idea is that we want to exploit the general context and insights that we have developped about these objects.

In contrast to what I mentioned in my last post, it does seem that the factorization of moduli in the ternary case is crucial, but the cancellation from Ramanujan sums might not be (although of course extra cancellation can not hurt…)

On the other hand, in trying to work with a generalish $latex K$, we understand now this extra cancellation at a higher level: it is not a super-specific fact about Ramanujan sums, but the latter is a concrete illustration of a fundamental phenomenon concerning trace functions of the so-called (pointwise pure) “middle-extension” sheaves, due to Deligne: at a singularity of such a sheaf, the Frobenius eigenvalues have lower weights than at “generic” points (see Lemma 1.8.1 in Weil II), and often strictly lower. (On the automorphic side, it corresponds to the well-known fact that the Satake parameters at a ramified prime are smaller in absolute value than the Ramanujan-Petersson bound at the unramified primes.)
In this picture, the Ramanujan sums correspond to the singularity at $latex 0$ of the basic Kloosterman sheaf. (Of course one doesn’t need Deligne to estimate Ramanujan sums, but we can now confidently play complicated games with the trace functions without being afraid of having lost a very special property of these particular sums…)

The working technique also shows that one can argue without the Weyl-type shifts in the original paper — this was not entirely surprising since Heath-Brown’s improvement of the Friedlander-Iwaniec exponent for the bare $latex d_3(n)$ (and also ours) do not involve such shifts.

Finally, all our current attempts to play games to avoid Deligne-level estimates for exponential sums have hit barriers…

There is of course still much to be done.

I just remembered a point I had intended to make concerning the exponential sum of Friedlander and Iwaniec that is crucial in Zhang’s work on gaps between primes, but which slipped my mind. This may present the argument in my note with Fouvry and Michel in a more enlightening way, although it does not simplify the proof (I’ve now added this as a remark in the PDF file.)

We view the sum as
$latex S(\alpha,\beta)=\sum_{x\in \mathbf{F}_p} a_1(x)\overline{a_2(x)}$
where $latex p$ is a prime and the coefficients $latex a_1(x)$ and $latex a_2(x)$ are values of Kloosterman sums:
$latex a_1(x)=K_2(x),\quad\quad a_2(x)=K_2(\beta x/(x+\alpha))$
for some parameters $latex \alpha$ and $latex \beta$, non-zero elements of $latex \mathbf{F}_p$, putting
$latex K_2(x)=-\frac{1}{\sqrt{p}}\sum_y e\Bigl(\frac{xy+1/y}{p}\Bigr).$

Now the point is that, from the “automorphic” view of trace functions (as discussed in one of my previous posts), both $latex a_1(x)$ and $latex a_2(x)$ can be seen as the Hecke eigenvalues, at the prime $latex T-x$, of a cusp form on $latex GL_2(\mathbf{F}_p(T))$, i.e., of the analogue of a classical cusp form, living however over a function field instead of $latex \mathbf{Q}$ — so the polynomial ring $latex \mathbf{F}_p[T]$ plays its role of cousin of $latex \mathbf{Z}$. This result (the existence of these cusp forms) is by no means obvious, but it follows from Deligne’s construction of Kloosterman sheaves and Drinfeld’s proof of the Langlands correspondance for $latex GL(2)$ over function fields.

In any case, if one admits this, the sum above is clearly an exact analogue of a sum of Hecke eigenvalues at primes of a Rankin-Selberg $latex L$-function associated to two cusp forms! If we know the Riemann Hypothesis for this Rankin-Selberg $latex L$-function, we can then very classically establish a bound
$latex S(a,b)\ll \sqrt{p},$
in full analogy with conditional results for Rankin-Selberg $latex L$-functions over number fields, provided the two cusp forms are not the same (otherwise there is a pole with a larger contribution). But here the two cusp forms are not the same simply because their “conductor” (in the arithmetic sense: in fact, just the location of ramified primes matters, i.e., the analogue of the divisors of the level of a primitive cusp form) are not equal!

Finally, the implied constant in applying the Riemann Hypothesis is uniformly bounded in terms of the conductor of the two cusp forms, and these are bounded independently of the prime and parameters, “explaining” the Birch-Bombieri result.

I emphasize again that this is not really a good way of writing a proof, because showing that the Rankin-Selberg $latex L$-functions over function fields satisfy GRH is harder than proving Deligne’s Riemann Hypothesis in the more geometric language of sheaves and their trace functions, simply because the argument reduces to Deligne’s result (this was, I think, first done by Lafforgue, though maybe Drinfeld had proved it for $GL(2)$). But this interpretation should make the argument very readable and natural to an analytic number theorist.