Maynard–Tao and Polymath
In 2013 Yitang Zhang stunned the world by proving that infinitely many pairs of primes lie within
70\,000\,000 of each other — the first ever bounded gap. His proof
(see Zhang's theorem) was a heroic, intricate
extension of the GPY sieve: to make the numbers work
he had to push the "level of distribution" of the primes past the Bombieri–Vinogradov barrier
of \theta = \tfrac12, using deep algebraic-geometry estimates for primes in
arithmetic progressions to smooth moduli.
A few months later, two people independently found a completely different route that was
simpler and far stronger. A young postdoc, James Maynard, and — at
essentially the same moment — Terence Tao replaced Zhang's one-dimensional weight with
a multidimensional sieve. The payoff was spectacular: bounded gaps came out using only the
classical level \theta = \tfrac12, with no Zhang-style extension needed, and
the bound dropped from 70 million to a few hundred almost overnight. This page
is the story of that method, why extra dimensions capture more, and how far the record has since fallen.
The setup: admissible tuples
Fix a set of k distinct integer offsets
\mathcal{H} = \{h_1, h_2, \dots, h_k\}. We want infinitely many
n for which at least two of the shifted values
n+h_1, \dots, n+h_k are simultaneously prime. For that even to be possible,
\mathcal{H} must be admissible: for every prime
p, the offsets must not cover all residue classes mod
p — otherwise one of the n+h_i is always divisible
by p, killing the primes. This is exactly the hypothesis of the
Hardy–Littlewood k-tuple conjectures,
which predict such tuples are prime all together infinitely often.
The strategy of GPY, Zhang and Maynard is the same in spirit: attach a non-negative
weight w(n) \ge 0 to each n that
is large exactly when the tuple looks "prime-rich", and compare two weighted sums,
S_1 = \sum_{N \le n < 2N} w(n), \qquad S_2 = \sum_{N \le n < 2N} \Big(\sum_{i=1}^{k}\mathbf{1}_{\,n+h_i \text{ prime}}\Big)\, w(n).
If we can show S_2 > m\, S_1, then some n
in the range has more than m of its n+h_i prime.
Everything comes down to choosing the weight so that this ratio beats the target
m.
The old weight versus the new weight
GPY (one-dimensional). Selberg's sieve suggests weighting by the square of a divisor
sum over the single product P(n) = (n+h_1)(n+h_2)\cdots(n+h_k):
w_{\mathrm{GPY}}(n) = \Bigg(\sum_{d \,\mid\, P(n)} \lambda_d\Bigg)^{2}, \qquad \lambda_d = \mu(d)\, F\!\left(\frac{\log d}{\log R}\right).
The coefficients \lambda_d are governed by one smooth function
F of the single variable
\tfrac{\log d}{\log R}. One variable is all the freedom you get — and it is
not enough: GPY alone proved primes come infinitely often much closer than the average gap, but could
not reach a fixed bound.
Maynard–Tao (multidimensional). Do not lump the factors together. Let each factor
n + h_i carry its own divisor d_i, and
weight by
w_{\mathrm{MT}}(n) = \Bigg(\ \sum_{\substack{d_1 \mid n+h_1,\ \dots,\ d_k \mid n+h_k \\ d_1 d_2 \cdots d_k \,<\, R}} \lambda_{d_1, \dots, d_k}\Bigg)^{2}.
Now the coefficients are indexed by a whole vector
(d_1, \dots, d_k), and they are read off from a smooth function
F of k variables,
\lambda_{d_1,\dots,d_k} \;\approx\; \Bigg(\prod_i \mu(d_i)\Bigg)\, F\!\left(\frac{\log d_1}{\log R}, \dots, \frac{\log d_k}{\log R}\right).
The GPY weight is the special case where F only depends on
x_1 + \cdots + x_k — a thin slice of the full space of functions. By allowing
F to vary in each coordinate independently, Maynard and Tao unlocked a
vastly larger search space of weights, and the best one in that larger space is dramatically more
efficient.
Why the extra dimensions capture more
Think of it as an optimisation. Both methods maximise the ratio
S_2 / S_1 over a family of weights; the family is parametrised by the choice
of F. In GPY, F lives on a one-dimensional line
(only the total \log(d_1\cdots d_k)/\log R matters). In Maynard–Tao,
F lives on the whole k-dimensional simplex
\{x_i \ge 0,\ \sum x_i \le 1\}. A maximisation over a bigger set can only do
better — and it does, enormously.
Concretely, the ratio to beat is the quotient of two integrals of F over the
simplex. Maynard showed the relevant eigenvalue-like quantity
M_k \;=\; \sup_{F} \frac{\sum_{i=1}^{k}\displaystyle\int F_i^2}{\displaystyle\int F^2}, \qquad F_i = \int_0^{1} F\,dx_i,
grows like M_k \sim \log k. If M_k > 4m, then
(on Bombieri–Vinogradov, \theta = \tfrac12) infinitely many admissible
k-tuples contain m+1 primes. Because
\log k \to \infty, every fixed target
m is eventually reachable just by taking k large
enough — the one-dimensional method, stuck near a constant, never could.
-
Assuming only Bombieri–Vinogradov (level \theta = \tfrac12), for every
m \ge 1 there are infinitely many n with
n, n+1, \dots containing at least m+1 primes
in a bounded-length window; hence
\liminf_{n\to\infty}\, (p_{n+m} - p_n) \;<\; \infty \quad\text{for every fixed } m.
-
In particular m=1 gives bounded prime gaps
\liminf (p_{n+1}-p_n) < \infty, with an explicit bound of
600 obtained quickly.
Two things GPY and Zhang could not do
The multidimensional sieve is not just a shorter proof of Zhang's theorem — it is genuinely stronger in
two ways.
1. Only \theta = \tfrac12 is needed. Zhang had to
painstakingly break the Bombieri–Vinogradov barrier, extending distribution estimates to smooth moduli
with a level \theta = \tfrac12 + \tfrac{1}{584}. Maynard and Tao get bounded
gaps using the classical Bombieri–Vinogradov theorem exactly as it stands — no barrier-breaking. This
is why the method is so much simpler and why it was reproduced and refined within weeks.
2. Clusters of m primes, not just pairs. Zhang's argument
produced pairs of nearby primes. Maynard–Tao produces, for every fixed m,
infinitely many bounded-length windows each containing m primes — infinitely
many tight clusters. For instance there are infinitely many intervals of bounded length
containing three primes, or ten primes, or a million primes.
Take the admissible triple \mathcal{H} = \{0, 2, 6\} (check: mod
2 the offsets are 0,0,0 — one class missed; mod
3 they are 0,2,0 — class 1
missed; larger primes are automatically fine). A one-dimensional GPY weight forces
F(x_1,x_2,x_3) to depend only on x_1+x_2+x_3, so
every direction in the simplex is treated identically.
The multidimensional weight instead picks, say, a symmetric polynomial like
F = (1 - x_1 - x_2 - x_3)\,(a + b(x_1+x_2+x_3) + \cdots) and tunes the
coefficients to maximise M_3. Numerically
M_3 \approx 1.6 already, climbing steadily; by
k = 105 one has M_k > 4, enough for
m = 1 and the first unconditional bounded gap. The lesson: give the sieve more
knobs (one per coordinate) and the optimiser finds a far better weight than the single-knob version ever
could.
The shrinking record
Once the method was public, an online collaboration — the Polymath Project, coordinated
by Terence Tao as Polymath8 — optimised every ingredient in the open. The bound on
\liminf (p_{n+1}-p_n) collapsed by more than five orders of magnitude in
barely a year:
| Date | Who | Bound on the gap |
|---|
| May 2013 | Zhang | 70\,000\,000 |
| 2013 (Polymath8a) | Polymath | 4\,680 |
| Nov 2013 | Maynard / Tao | 600 |
| 2014 (Polymath8b) | Polymath | 246 |
246 is the current unconditional record: there are infinitely
many pairs of primes differing by at most 246. Twin primes would be gap
2 — still tantalisingly out of reach.
What a stronger distribution hypothesis would buy
The bound 246 uses the proven level \theta = \tfrac12.
If the primes are actually as well-distributed as the Elliott–Halberstam conjecture
(level \theta arbitrarily close to 1) predicts, the
Maynard–Tao machine tightens dramatically.
- On the Elliott–Halberstam conjecture, \liminf (p_{n+1}-p_n) \le 12.
- On a suitable generalised Elliott–Halberstam conjecture, \liminf (p_{n+1}-p_n) \le 6.
So the admissible triples \{0,4,6\} or \{0,2,6\}
would each contain two primes infinitely often. But even the full strength of Elliott–Halberstam does
not deliver gap 2 — this style of sieve provably cannot reach the twin
prime conjecture. That final step from 6 to 2 needs a
genuinely new idea.
Two things students routinely garble.
(a) The record is 246, and it is not going to 2
by more effort of the same kind. Even assuming the strongest reasonable distribution conjectures,
the multidimensional sieve stalls at gap 6. There is a hard theoretical
obstruction (the "parity problem") that blocks these sieve methods from ever certifying a gap of
2. Twin primes will need fundamentally new mathematics — the current record is a
milestone, not a near-miss.
(b) Maynard–Tao needs only \theta = \tfrac12; Zhang needed more.
Do not say "Maynard reproved Zhang the same way." Zhang's whole difficulty was pushing past the
Bombieri–Vinogradov level \theta = \tfrac12 to smooth moduli. Maynard and Tao
avoid that entirely — their strength comes from the weight (extra dimensions), not from a better
distribution estimate. Same conclusion, opposite source of power.
Polymath is a form of massively collaborative mathematics started by Timothy Gowers in
2009: a problem is posted on a blog and anyone may contribute in the open comments, with the write-up
published under the collective pseudonym "D. H. J. Polymath". Zhang's and Maynard's breakthroughs were the
perfect fuel — the method had many separately optimisable pieces (the choice of F,
the size of k, the narrowest admissible tuple, the distribution input).
Polymath8a squeezed Zhang's 70 million down to
4\,680; after Maynard's paper, Polymath8b combined the multidimensional
sieve with the earlier optimisations to reach 246. It is one of the clearest
success stories of open, crowd-sourced research mathematics — dozens of contributors, a live-updated
record, and a genuinely new theorem at the end.
The state of play
Put the pieces together and the landscape is remarkably clean. Unconditionally, primes come within
246 infinitely often, and — for any fixed
m — infinitely many bounded windows hold m primes.
Conditionally on Elliott–Halberstam the gap shrinks to 12, and to
6 on its generalisation. And the twin prime conjecture — gap exactly
2 — remains open, walled off from these methods by the parity problem. A
problem that had no bounded-gap result at all before 2013 now has an entire ladder of them; only the top
rung is still missing.