James Maynard, auteur du théorème de l’année

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 n_k, not sparse, not weird, that satisfies the Siegel-Walfisz property, but has unbounded gaps, i.e., \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…]

À la…

A few months ago, I wondered what could be the largest cluster of foreign words in the Oxford English Dictionary, citing the examples of femme something-or-other and sympathique and company. It turns out that there is a much larger one! Here is the à la cluster:

à la 1579
à la bonne heure 1750
à la broche 1806
à la brochette 1821
à la carte 1816
à la crème 1741
à la débandade 1779
à la fourchette 1817
à la Française 1589
à la modality 1753
à la mode 1637
à-la-modeness 1669
à la mort 1536
à la page 1930
à la roi 1852
à la royale 1853
à la Russe 1775
à la Turquie 1676

That’s no less than 18 items (the date on the right is the first OED citation). It’s interesting that so many have to do with food, and even more that three or four are basically synonyms of “in fashion” (this is what à la page basically means). I have to admit to being partial to à-la-modeness for its translanguage qualities, although I don’t know if I will be able to use it intelligently anytime soon (though one never knows; after all, I did manage to sneak ptarmigan in a recent paper…)

A missing word

From the blog of the rare books collection of the ETH Library, I just learnt that the word for the study and classification of grape species that I was looking for is “ampelography” (ampélographie in French).

(The relevance of this word to my daily life is that the computers on my home network are named after grapes; red grapes are reserved for desktops and white for laptops.)

OED clusters

Today’s “word of the day” from the OED was “femme incomprise”. The list of nearby words contains:

  • femme (first quote 1814, from a letter of Byron)
  • femme de chambre (first quote 1741)
  • femme de ménage (first quote 1826)
  • femme du monde (first quote 1849)
  • femme fatale (first quote 1879; one wouldn’t guess that this is taken from an article in that well-known journal of cosmopolitan sophisticates, the St Louis Globe Democrat)
  • femme incomprise (first quote 1841)

I wonder if there is a bigger cluster of foreign words with a common root?

The other one I know and like, though it is not in strictly alphabetic order, is also quite impressive:

  • simpatico, simpatica (first quote 1864, “The Frau Professorin was less ‘simpatica’”, from the memoir of a certain H. Sidgwick)
  • sympathique (first quote 1859, in a letter of Queen Victoria, “The sight of a professor or learned man alarms me, and is not sympathique to me”)
  • sympathisch (first quote 1911)