Worked solution: Maynard's multidimensional sieve: bounded gaps between primes (2013)

Frequently asked questions

Are

M_5>2

and

M_{105}>4

found by trial and error, or is there a system?

Maynard uses a structured family of polynomial test functions (products of a low-degree polynomial in each variable, symmetrized appropriately) and then numerically optimizes the finitely many free parameters — closer to a guided search than pure trial and error, and systematic enough to also yield the general asymptotic bound for large $k$ .

Source

James Maynard, 2013

Small gaps between primes

arXiv:1311.4600v3

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Condensed summary

Step 6 of 8: Getting numbers: exhibiting

F

's that make

M_{105}>4

Previous step Next step

In plain words

The abstract variational bound $M_k$ becomes useful only once someone plugs in explicit candidate functions $F$ and checks, sometimes with real computation, that the resulting ratio is large. Maynard does this for two key cases: a modest $k=5$ , and a much larger $k=105$ .

Surprisingly, using $F$ built from products of low-degree polynomials in the $k$ variables (rather than trying to guess the true optimizer), one can already push $M_5$ above $2$ and $M_{105}$ above $4$ — numbers that, plugged into the machinery of the previous step, are exactly strong enough to finish the proof.

M_5>2,\qquad M_{105}>4

Detailed analysis

Maynard's Proposition 4.3 states two explicit numerical facts obtained by testing polynomial candidate functions $F$ against the definition of $M_k$ : $M_5>2$ , and $M_{105}>4$ (Maynard 2013, §4, Proposition 4.3, parts 1–2). Neither number is claimed to be the true supremum — they are simply lower bounds, obtained by direct (if lengthy) calculation with specific, explicitly written-down $F$ , sufficient for what comes next.

Maynard also proves an asymptotic statement for large $k$ : $M_k > \log k - 2\log\log k - 2$ once $k$ is large enough (Maynard 2013, §4, Proposition 4.3, part 3), obtained from a more systematic family of test functions rather than an ad hoc choice; this is what ultimately drives Theorem 1.1's general bound for arbitrarily many primes in bounded windows.

These numerical facts are the one place in the whole argument where explicit computation — rather than pure structural reasoning — is doing real work; the rest of the proof is 'if $M_k$ is at least this large, then...'. The final step assembles everything.