In Slate today, I have a piece about Yitang Zhang’s amazing proof of the bounded gaps conjecture. Actually, very little of the article is about Zhang himself or his proof; I wanted instead to explain why mathematicians believed that bounded gaps (or twin primes) was true in the first place, via Cramér’s heuristic that primes behave like random numbers.
And a lot of twin primes is exactly what number theorists expect to find no matter how big the numbers get—not because we think there’s a deep, miraculous structure hidden in the primes, but precisely because we don’t think so. We expect the primes to be tossed around at random like dirt. If the twin primes conjecture were false, that would be a miracle, requiring that some hitherto unknown force be pushing the primes apart.
Quomodocumque 
A possibly amusing observation on Zhang’s result (which takes nothing from his accomplishment) is that the constant is so large that the theorem doesn’t improve on the average spacing until we reach primes with many millions of digits.
Nice story, but I think you may have misstated the prime number theorem? Or else I’m confused.
Nice to see the subjunctive used in the mass media!
Wait, where’s the subjunctive?
Were I to have used it, you’d think I’d remember it.
…that actually makes a lot of sense :)
> More classically minded analytic number theorists are already wondering whether Zhang’s proof can be modified to avoid such abstruse stuff.
Really?
In, e.g. Goldston-Pintz-Yildirim, all of this is conspicuously absent. No Kloostermann sums, no exponential sums over finite fields, no Weil-Deligne, etc. In their followup work with Graham, even the contour integrals go away, and the whole proof is quite elementary (although neither short nor easy).
It seems that, overgeneralizing massively, we prove the best error terms when we can find some kind of nice arithmetic structure in whatever sums we’re trying to bound. I don’t know so much about etale cohomology, Deligne’s (or Katz’s work), etc. but I do know the punchline — we can get as much cancellation in exponential sums as we might reasonably hope for, provided we can associate a “motive” in the right way. (Experts, please feel free to correct me or say this better!) Conversely, when we don’t have any good way of finding structure, we typically prove suboptimal bounds.
After skimming Zhang’s paper, it seems his technical achievement is to revise GPY so as to set up these nice exponential sums — in other words, to finesse the sum until it is sitting squarely under Deligne’s hammer, at which point the cord is released, and *squash*.
Do you (or others) think it’s possible or desirable that Zhang’s proof be modified in such a way, without making a tremendous mess of the proof?
You know, I think the way I wrote it makes it sound more like there are people who think it’s actively bad to use algebraic geometry, which clearly isn’t the case — I meant something a little more like, people are trying to understand to what extent the algebraic geometry is truly essential to the argument, because that is in itself an interesting mathematical question. At least one colleague of mine on FB expressed surprise (not displeasure!) to see the Weil bounds appearing.
For my own part, of course, I think it’s delightful to “cherchez la sheaf” in problems where the algebraic geometry isn’t immediately apparent.
Do you mean “Cherchez le faisceau”?
Pardon my French…
I went back and forth but finally decided I preferred the Frenglish.
“requiring that some hitherto unknown force be pushing the primes apart” is (should I not be mistaken) in the present subjunctive…
I think that’s a bare infinitive, actually…
Dear Frank,
The sentence,
>After skimming Zhang’s paper, it seems his technical achievement is to revise GPY so as to set up these nice exponential sums — in other words, to finesse the sum until it is sitting squarely under Deligne’s hammer, at which point the cord is released, and *squash*.
has increased my desire to deeply read this paper significantly. The more I can invoke Wile E. Coyote in a math talk, the better!
No, it’s not an infinitive! An infinitive is a noun form of a verb. In English, it is usually flagged with the word “to.” “To err is human; to forgive, divine.” The subjunctive is a mood. A verb in the subjunctive still plays the grammatical role of a verb, but with a suppositional or counterfactual quality. “If this be error, and upon me proved…” When you come across a construction like “should it be,” “that it be,” “lest it be,” you’re probably looking at a present subjunctive.
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