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.
