# Bounded gaps between primes: some grittier details

Because I was teaching a course on prime numbers this semester and had just finished a chapter on Vinogradov’s method when his paper appeared, I promptly switched my plans for the last classes in order to present some aspects of Yitang Zhang’s theorem on bounded gaps between primes. In addition, one of the speakers of this week’s conference celebrating 25 years of the “Zahlentheorie Seminar” of ETH had to cancel at short notice, and I replaced her and gave yesterday another survey-style talk. The notes for the latter (such as they are…) can be downloaded in scanned form.

My insights to Zhang’s work remain clearly superficial, but here are some remarks going a bit beyond what I mentioned in the previous post, coming after these lectures, and some discussions with Ph. Michel and P. Nelson.

(1) The most delicate estimates seem to be those for the “Type III” sums. These concern the “good” distribution in invertible residue classes of an arithmetic function $f(n)$ for integers $N, modulo a "large modulus" $q$, where $f(n)$ is of the very special type
$f(n)=\sum_{m_1n_1n_2n_3=n}\alpha(m_1),$
and the variables $m_1$, $n_1$, $n_2$ and $n_3$ are, roughly, of size $M_1$, $N_i$ with $M_1N_1N_2N_3=N$, and (crucially) $q$ is a bit larger than $N^{1/2}$: one needs to handle these for $q$ up to $N^{1/2+\delta}$ for some $\delta>0$ in order to obtain bounded gaps between primes.

The lengths $M_1$ and $N_i$ are constrained in various ways, and the most critical case seems to be when $M_1\approx q^{1/8}$, $N_i\approx q^{5/8}$ (the $\approx$ means that one must be able to go a bit beyond such a case, since $N$ is a bit beyond $q^2$).

(2) Another point is that Zhang manages to bound these sums for each individual residue class $a\pmod q$ (coprime to $q$). In other words, denoting
$\Delta_f(N,q,a)=\sum_{N
he proves individual bounds for $\Delta_f(N,q,a)$, instead of average bounds over $q$ (as in the other main part of this argument).

Also, he does not need to use the variable $m_1$ at all (but since the $\alpha(m_1)$ are mostly unknown coefficients, and the sum is rather short, exploiting it does not seem easy). Hence the result looks enormously like controlling the distribution in residue classes of the ternary divisor function. This is exactly the question that Friedlander and Iwaniec had studied in the famous paper where they proved that the exponent of distribution is at least $1/2+1/230$, but their argument is not quite sufficient for Zhang's purpose.

(3) One of the last tricks is Zhang's second use of the structure of the moduli $q$ that are involved in his argument: these were chosen to have only prime factors $\leq z=N^{\delta}$ for some small positive $\delta$), and Zhang exploits the "granular" (or friable) structure of such moduli in order to obtain flexibility in the possibility of factoring them as $q=rs$ with $r$ of size determined up to a factor at most $z$. This is particularly important for the “other” sums (it gives bilinear structure and makes it possible to use the dispersion method of Linnik, as already done by Fouvry-Iwaniec and Bombieri-Friedlander-Iwaniec in their works on primes in arithmetic progressions). For the "type III" case, it does not seem to be so much of the essence, but Zhang needs to gain a very small amount compared with Friedlander-Iwaniec, and does so by factoring $q=rs$ with $r$ rather small. He then gains from $r$ a factor $r^{1/2}$ which is essential, by exploiting the fact that a Ramanujan sum modulo $r$ is bounded (so he gets more than square-root cancellation from $r$…)! This is an extremely special situation, and right now, it is what seems the most “miraculous” about this proof (at least to me).

It is for the contribution of the complementary divisor $s$ that Zhang manages to position himself into applying the estimate of Birch and Bombieri for the exponential sums which Friedlander-Iwaniec had also encountered.

(4) This use of Deligne’s work is also very delicate: one can not relax the requirement of square-root cancellation, except by very tiny amounts. For instance, obtaining a bound of size $p^2$ for the three-variable sum modulo $p$ is useless; in fact, the bound $p^2$ can be considered here as the trivial estimate, since the sum can be written as an average of one-variable Kloosterman sums. With Zhang’s parameters, one needs an estimate of for the sum which is no larger than $p^{3/2+1/2000}$ (or so) in order to get the desired gain. However, as I explained in the last five minutes of my talk today (and as is explained in this note with Fouvry and Michel) the Birch-Bombieri bound is very well understood from a conceptual point of view.

(5) I was very curious, when first looking at the paper, to see how Zhang would handle the residue classes in the Goldston, Pintz, Yıldırım method, since the most uniform results on primes in arithmetic progressions (those of Fouvry-Iwaniec and Bombieri-Friedlander-Iwaniec) are constrained to use (essentially) a single residue class. What happens is that Zhang detects these classes by inserting a factor corresponding to their characteristic function, and by avoiding the Kloostermania approaches that rely on spectral theory of automorphic forms. The important properties of these residue classes are their multiplicative structure (coming from the Chinese Remainder Theorem) and the fact that, on average over moduli, there are not too many of them (the average is bounded by a power of $\log N$). In particular, his use of the dispersion method is in fact closer in spirit to some of its earliest uses by Fouvry and Iwaniec (for integers without small prime factors instead of primes), which also involved, in the final steps, the classical Weil bound for Kloosterman sums instead of (seemingly stronger) results on sums of Kloosterman sums.

### Kowalski

I am a professor of mathematics at ETH Zürich since 2008.

## 6 thoughts on “Bounded gaps between primes: some grittier details”

1. v08ltu says:

As far as I know, Zhang should take $r$ in Section 14 as large as possible that allows the argument to work (namely, running $n\sim N_3$ modulo $r$ must not be too big, even when multipled by $K^2$).

Instead, he seems to take it as small as possible (13.12) to beat the losses incurred from $H$ introduction and/or $N_1$ exceeding of $x^{3/8}$ and $d_1$ exceeding of $x^{1/2}$ (namely $8\omega+2\omega$, which is $40\omega$ as you win by $\sqrt r$ and apply Cauchy, these $N_1$ and $d_1$ are linked to $H$ selection so you can see it either way).

Unless I am wrong, this is an inefficiency. I think $r$ can be something like $x^{1/12-6\omega}$ or $x^{1/10-8\omega}$ for 14.8 and 14.5, also 14.4 to work.

I think a philosophical reason partially as to why it works is that his Weyl shift was by $hkr$, notably a multiple of $r$ so that the Ramanujan sum pops out of the factorization $e_d()$ down to $e_r()e_s()$. But I agree that is mysterious.

2. bengreen says:

Emmanuel, this is an excellent post and, from my reading of the paper so far (and also my position of considerably less expertise than you) I think I agree with everything you say. Concerning the factorisation d = qr, it seems that by the Chinese remainder theorem one gets to factor exponentials e_d(.) as e_q(.) e_r(.), and then some incomplete Kloosterman sums mod d become a bit less incomplete mod q (though still incomplete – they are completed by the Fourier expansion of the cutoff n ~ N_3/r, the Fourier coefficients being the kappa(m_i)). Is this splitting of e_d as e_q e_r, the e_r part getting hit by Cauchy-Schwarz, the “dispersion”? As will be clear, I have only a vague sense of what goes on, though. Another question is how much would simplify if one had the conjectured best estimates on incomplete Kloosterman sums (Hooley’s hypothesis R or something).

3. v08ltu says:

My understanding is that (up to epsilon) Hypothesis R achieves sqrt(Length) rather than sqrt(Modulus). If this can be applied to both n-variables in (13.7), the bound there would be HN_1 * sqrt(N_2N_3) rather than the N_1d^(5/4) as FI achieve via Deligne. Note HN_1 is just the modulus d itself, so the comparsion when N_i are equal is simply dN vs d^(5/4)N, an easy win. In the attenuated case you have x^(1/2+5/16) and needed to do better than x^(15/16) and this wins.

But if Hooley R can only be applied to one n-variable, the result is HN_1 * N_2*sqrt(N_3), and this is x^(1/2+5/16+5/32) in the border case, not beating x^(15/16).