Approaches to Riemann hypothesis using methods outside number theory - MathOverflow [closed]

Approaches to Riemann hypothesis using methods outside number theory

2010-08-05T22:50:11Z

Background: Once an analytic number theorist remarked to me that all attempts to prove the Riemann hypothesis using number theoretic methods have failed. Since then that remark stuck in my mind.

The veracity of the above alluded number theorist's opinion does not really matter for the question to make sense; I just included it for background.

Question:

What are some promising methods from outside number theory to approach Riemann hypothesis?

I know two:

The geometric approach of Artin, Hasse, Weil and Deligne, the most important result being the proof of the Weil Conjectures.
The Bost-Connes approach. This is outlined by Lieven Le Bruyn for instance and has a hint of thermodynamics .

I imagine that both of the above are cited by some people as the basis for the hopes that the theory of the field with one element will prove the Riemann hypothesis. Again, this question formally has no need to be connected the theory of field with one element to make sense. Other than just mentioning the above, let us not get into that aspect.

I am interested in other possible and promising methods. I am not interested in an equivalent formulation of Riemann hypothesis which is no better than the original. Both the above are very promising in terms of undiscovered things and might give a much better "big picture".

An approach I am ambivalent about, is that of Baez-Duarte. Though it does provide some evidence. I do not know whether it is any easier to prove Riemann hypothesis that way, rather than the original statement.

Please give me examples of any other methods; preferably very "promising" ones.

Edit 1: The meaning of "methods outside number theory" is the following: Nothing in the book of Ivic or Titchmarsch and Heath-Brown. More precisely, methods outside the traditional sybjects of elementary number theory and analytic number theory. I have given two examples above. One with algebraic geometry and one with thermodynamics.

Answer by Joseph O'Rourke for Approaches to Riemann hypothesis using methods outside number theory

2010-08-05T23:00:12Z

This is well below the technical level of the replies you seek and deserve (and which others more knowledgeable will no doubt supply), but I can't resist mentioning Freeman Dyson's idea, which I encountered in his "Birds and frogs" article in the Notices of the American Mathematical Society [56 (2): 212–223, 2009]. Here it is from a Wikipedia entry:

"The Riemann hypothesis implies that the zeros of the zeta function form a quasicrystal, meaning a distribution with discrete support whose Fourier transform also has discrete support. Dyson suggested trying to prove the Riemann hypothesis by classifying, or at least studying, 1-dimensional quasicrystals."

Update (9Nov12). See Nick S's recent comments.

Answer by Micah Milinovich for Approaches to Riemann hypothesis using methods outside number theory

2010-08-06T01:08:17Z

I heard it said by a number of mathematicians that the thing that sets the Riemann Hypothesis apart from (almost all) other famous unsolved problems is that no one has ever suggested a reasonable first step toward a proof.

Answer by Andrew L for Approaches to Riemann hypothesis using methods outside number theory

2010-08-06T01:57:40Z

Not to give anything away,but I'm pursuing this fall a set of pont set topological methods of analyzing the integers that I'm hoping will yield some fruit. But it's too early to tell.

That being said-I think a topological approach may yield systems of open sets of integers that may bring us closer to a solution-particularly on Riemann surfaces with exotic topologies.

Hope I didn't give away any magic tricks just now.........

Answer by Felipe Voloch for Approaches to Riemann hypothesis using methods outside number theory

2010-08-06T02:18:30Z

It's impossible to say whether an approach is promising. In my view, the most interesting approach is via constructing $\mathbb{F}_1$. The wikipedia page has links to the various attempts. One hopes that after a suitable theory is constructed, one of the proofs of the function field analogue of RH will be translatable to the number field case.

http://en.wikipedia.org/wiki/Field_with_one_element

p.s. I think this is number theory and I don't agree with your analytic number theorist.

Answer by Pete L. Clark for Approaches to Riemann hypothesis using methods outside number theory

2010-08-06T03:11:24Z

So far as I know, there is no approach to the Riemann Hypothesis which has been fleshed out far enough to get an even moderately skeptical expert to back it, with any odds whatsoever. I think this situation should be contrasted with that of Fermat's Last Theorem [FLT]: a lot of number theorists, had they known in say 1990 that Wiles was working on FLT via Taniyama-Shimura, would have found that plausible and encouraging. Wiles' work was absolutely a tour de force, but at the ground level it used preexisting tools in the number-theoretic community, tools (e.g. Mazur's theory of Galois deformations) whose power and relevance to the problems at hand were appreciated and known not to have been fully exploited. Similarly, the proof of Serre's Conjecture by Khare-Wintenberger represents some of the best number-theoretic work in the last decade, and if you were an expert in the field in 2000 (again, not me -- but I have friends), then unless you could somehow predict the powerful techniques that Mark Kisin would develop over the course of the next several years, your estimate of when Serre's Conjecture would be proven would probably be off by as much as a decade. But people knew (or felt they knew, correctly as it turns out) that it was just a matter of time.

In contrast, despite the existence of several "programmes" by leading mathematicians to prove RH, if it were actually proved in, say, 2012, there would be the mathematical equivalent of worldwide rioting. It's just not at all clear that we can get there from here: most of the work which has been done on RH in the last 150 years has led (only) to our having a suitably healthy respect for the problem and its importance in mathematics as a whole.

That said, I think that approaches to RH should not be evaluated on whether they are likely to culminate in a proof of RH -- who knows? -- but whether they are interesting and seem likely to lead to interesting mathematics along the way. A lot of people seem to like the $\mathbb{F}_1$ approach for this reason, as in Felipe's answer. (And indeed, this answer began as a comment to his.)

Answer by Charles Matthews for Approaches to Riemann hypothesis using methods outside number theory

2010-08-06T08:01:24Z

I think, tautologously, any method proving the Riemann Hypothesis (or even seriously improving our knowledge on the zeroes) becomes "number theory" immediately. That said, I know what the question means: is there a dinky way of looking at the heat equation, say, that reduces what we need to prove to some piece of mathematics that is within reach?

Well, I don't know, and it is decades since I thought about these things seriously. I once thought topological entropy was a good start (and never heard the starter's pistol for that).

The most hopeful thing I have heard about this recently has been the work of Akiyama and Tanigawa (see http://www.ams.org/journals/mcom/1999-68-227/S0025-5718-99-01051-0/) which begins the process of connecting Sato-Tate with RH. I know no details, but Sato-Tate has come within reach. http://people.math.jussieu.fr/~harris/SatoTate/notes/equidistribution.pdf knows much more about this than I do, which wouldn't be hard. Assuming Extended Sato-Tate has some sort of traction on GRH at all, this is hopeful to me not because this is going to be the answer, but it might be "within one really new idea" of the answer. GRH is to be fitted into the general automorphic representation theory as the Artin conjecture has been for many years (?!). So there may emerge a conditional proof that is only (?) dependent on some things in Lie theory fitting together combinatorially or functorially as they should (?). I.e. we prove RH for the rationals by Harish-Chandra's technique over induction over all reductive groups and the way they fit into each other (?). People can tell where this is going by now. Weil's positivity derived from the explicit formulae doesn't come softly from commutative harmonic analysis, but perhaps there is a "hidden secret" in non-commutative harmonic analysis that will do.

Answer by Peter Arndt for Approaches to Riemann hypothesis using methods outside number theory

2010-08-06T08:30:16Z

There is Deninger's approach in which one hopes to produce the Riemann Zeta function via the trace of an endomorphism of an infinite dimensional cohomology group, related to foliated spaces: Read his own account or this talk by Eric Leichtnam.

It is sort of what you call the geometric approach in the question, but spiced up with dynamical systems and foliations.

Answer by doetoe for Approaches to Riemann hypothesis using methods outside number theory

2010-08-06T16:09:44Z

I have no idea to what extent the idea of Saharon Shelah, about which I read in David Ruelle's popular account the mathematician's brain that uses mathematical logic to prove the RH is promising, but certainly it is different. For as far as I can understand (from Ruelle), it basically comes down to proving that RH is undecidable in Peano arithmetic, in which case the consistency of Peano arithmetic would imply its truth (also in ZFC).

EDIT: Here is the quote from Shelah's paper:

2.3 Dream: Prove that the Riemann Hypothesis is unprovable in PA, but is provable in some higher theory.

What basis does my hope for this dream have? First, the solution of Hilbert’s 10th problem tells us that each problem of the form “is the theory ZFC +φ consistent” can be translated to a (specific) Diophantine equation being unsolvable in the integers, moreover the translation is uniform (this works for any reasonable (defined) theory, where consistent means that no contradiction can be proved from it). Second, we may look at parallel development “higher up”; as the world is quite ordered and reasonable.

Note that there is a significant difference between $\Pi_2$ sentences (which say, e.g., for a given polynomial $f$, the sentence $\varphi_f$ saying that for all natural numbers $x_0 , \ldots , x_{n−1}$ there are natural numbers $y_0 , \ldots , y_m$ such that $f (x_0 , \ldots , y_0 , \ldots ) = 0$) and $\Pi_1$ sentences saying just that, e.g., a certain Diophantine equation is unsolvable. The first ones can be proved not to follow from PA by restricting ourselves to a proper initial “segment” of a nonstandard model of PA. For $\Pi_1$ sentences, in some sense proving their consistency show they are true (as otherwise PA is inconsistent). Naturally, concerning statements in set theory, models of ZFC are more malleable, as the method of forcing shows.

Answer by anon for Approaches to Riemann hypothesis using methods outside number theory

2010-08-22T04:29:27Z

Hey,

I remember that in an interview, Selberg said that what Conne did was essentially getting a "different access" to the explicit formula, but that his proof did not yield new "hard facts" about the riemann zeta function. However take me with a grain of salt: I am not an expert in algebraic fields.

At the same time analytic methods give us information that are of a statistical nature. So for example they are quite well suited to prove that there is a positive proportion of the zeros on the half-line (40%, which is impressive enough I think). They can also show that almost all zeroes are concentrated in a box of height T and width $10^{10}/\log T$ around the line $\sigma = 1/2$. So those results are very useful in their own rights: For example they allow us to prove that the prime number theorem holds in intervals of length $x^{1/2+1/10}$ something you would expect to know only assuming the RH or a quasi-RH. In fact for many arithmetic applications the results we already know allow us (often/sometimes?) to by-pass the Riemann Hypothesis.

So I wouldn't say that all approaches failed hopelessly. We are armed to deal with the problem, and paraphrasing Montgomery: "Sometimes I have the impression that we are missing just a fundamental insight to prove RH" .