How many times in a year is an analytic number theorist supposed to faint from admiration? We’ve learnt of the full three prime Vinogradov Theorem by Helfgott, then of Zhang’s proof of the bounded gap property for primes. Now, from Oberwolfach, comes the equally (or even more) amazing news that James Maynard has announced a proof of the bounded gap property that manages not only to ask merely for the Bombieri-Vinogradov theorem in terms of information concerning the distribution of primes in arithmetic progressions, but also obtains a gap smaller than 700 (in fact, even better when using optimal narrow k-tuples), where the efforts of the Polymath8 project only lead to 4680, using quite a bit of machinery.
(The preprint should be available soon, from what I understand, and thus a full independent verification of these results.)
Two remarks, one serious, one not (the reader can guess which is which):
(1) Again, from friends in Oberwolfach (teaching kept me, alas, from being able to attend the conference), I heard that Maynard’s method leads to the bounded gap property (with increasing bounds on the gaps) using as input any positive exponent of distribution for primes in arithmetic progressions (where Bombieri-Vinogradov means exponent 1/2; incidentally, this also means that the Generalized Riemann Hypothesis is strong enough to get bounded gaps, which did not follow from Zhang’s work). From the point of view of modern proofs, there is essentially no difference between positive exponent of distribution and exponent 1/2, since either property would be proved using the large sieve inequality and the Siegel-Walfisz theorem, and it makes little sense to prove a weaker large sieve inequality than the one that gives exponent 1/2. Question: could one conceivably even dispense with the large sieve inequality, i.e., prove the bounded gap property only using the Siegel-Walfisz theorem? This is a bit a rhetorical question, since the large sieve is nowadays rather easy, but maybe the following formulation is of some interest: do we know an example of an increasing sequence of integers $latex n_k$, not sparse, not weird, that satisfies the Siegel-Walfisz property, but has unbounded gaps, i.e., $latex \liminf (n_{k+1}-n_k)=+\infty?$
(2) There are still a bit more than two months to go before the end of the year; will a bright PhD student rise to the challenge, and prove the twin prime conjecture?
[P.S. Borgesian readers will understand the title of this post, although a spanish version might have been more appropriate…]
