Uh, I should have interchanged “sufficient” and “necessary” there. Sorry for the confusion.

]]>Still, here’s the basic idea, The “natural,” “combinatorial” way to do the hard step in Dirichlet’s proof is to say, well, the primes are pseudorandom, and then apply quadratic reciprocity, and the result falls out pretty much immediately. (I *think.* At least I convinced myself that this is true earlier.) The problem is that (a), the primes aren’t (AFAIK) known to be pseudorandom in any sufficiently strong sense for this argument to go through, and (b) Any such notion of pseudorandomness would very likely imply immediately that the primes are uniformly distributed in !

I have a hunch that this isn’t just necessary, but is basically sufficient: in order to do the hard step, you have to show that the primes are (at least in a weak sense) pseudorandom. And this is certainly beyond the reach of counting arguments, and I don’t see an obvious way to do something like that without appealing to complex or Fourier analysis.

]]>Yeah, it’s Zagier’s, unless we’ve both seen the same misattribution.

I suspect the permutation isn’t really that motivated; it’s entirely likely that it’s an ad-hoc construction, “built to order” for this problem — finite group theory has all sorts of things like that. I haven’t sat down and tried to do it, but I have a hunch that if you sat down and tried to construct an involution with exactly one fixed point, that definition might just be the first one to pop out.

I have some thoughts on the main post, but they’re nebulous and would just be embarrassing if I posted them now. Will likely comment later.

]]>That proof is due to Zagier, if I’m not mistaken. But it seems like David’s suggestion requires much more careful counting than the technique can provide, although I agree that it’s very nice. (But until someone motivates the definition of the permutation, I won’t really believe the proof.)

]]>This is probably my favorite application of elementary group theory to number theory, so I think it would be really cool if it found a nice application. Unfortunately, I’ve never seen it used except for this one exercise.

]]>I would call this a rephrasing of Furstenburg’s proof, and while it is a nice intuition it doesn’t seem to generalize readily to any special cases of Dirichlet’s theorem.

]]>1 (pessimistic) Have you seen Keith Conrad’s expository note on which cases of Dirichlet’s theorem can be approached by elementary techniques? I find it very convincing. In particular, he argues that it should be very hard to distinguished primes which are 2 modulo 5 from those which are 3 modulo 5 by any elementary method. Unfortunately, Keith’s website seems to be down right now, but googling “Keith Conrad” should get it for you when it comes back up.

It is my (not very informed) belief that any proof of Dirichlet’s theorem has to involve Fourier analysis on .

2 (optimistic) Suppose you allow me Fourier analysis on finite groups, and elementary estimates for sums, but NOT complex analysis or algebraic number theory. Then the hardest step in Dirichlet’s proof is to establish that, for any nonsquare integer D, D is a quadratic residue for at least half the primes. It is easy to establish that D is a quadratic residue infinitely often, by looking at the polynomial x^2-D, but we need the stronger result.

There are several ways to prove this. One of them is to look at the values of the polynomial x^2 – D y^2 and apply exactly the sort of argument you are using. I’ve never seen the details carried out without complex analysis and algebraic number theory, but I wouldn’t be surprised if it could be done.

]]>