Why the Riemann Hypothesis?
January 20th, 2008 by WaltWay back when, I had a post about explaining the Riemann hypothesis in elementary terms. I thought I’d go into some more detail.
The Riemann hypothesis is regarded as one of the outstanding open problems in mathematics. Part of the reason is that it has a certain mystique, since Riemann conjectured it back in 1859, and it has withstood many attempts to prove it since then. A bigger reason is that it solution (either positive or negative) is the main obstacle to answering the question “How many primes are there?”
The fact that there are infinitely many primes goes back to Euclid. The next most logical question is to ask how many primes there are less than a given number. Thanks to the Prime Number Theorem, we know that there are approximately n / ln n primes less than a given number. But this is only an approximation. How good or bad of an approximation is it? We don’t know. That is the question the Riemann hypothesis is trying to answer.
January 21st, 2008 at 3:29 am
Another reason, at least from my point of view, for the importance of the Riemann hypothesis, is that it is the first and simplest of a long series of conjectures, all widely believed to be true and none proved. Just as with Langlands’ program and the proofs of Wiles and Taylor-Wiles, when this happens, one can expect the proof of the first result to open interesting gates. Then again, I have Eric Saias on the record saying that he is convinced the proof might be short and classical (fifteen pages long, in his words).
January 21st, 2008 at 3:49 pm
The rise of interlocking conjectures in number theory is a very interesting development. The only parallel that I know of is complexity theory with P != NP, etc. I’m not really sure what weight to put on “believed to be true”, though. The fact that they are hard to refute has to mean something, but I don’t know what the significance of expert opinion is for mathematics.
January 22nd, 2008 at 5:33 pm
Indeed, Paul Koosis, in his analysis class, told us that until Carleson settled the question of pointwise convergence of L^2 functions in 1965-6, many experts, including Zygmund believed the (true, as it turned out) result to be false.
January 23rd, 2008 at 9:45 am
It is not only a question of expert opinion, but also of analogous statements being true. For the Riemann hypothesis, I guess an important fact is that it is true for algebraic varieties, leaving at least the hope that there is some cohomological theories in the style of Denninger or Lichtenbaum that proves the results for arithmetic schemes following the same lines as Deligne’s proof.
That said, my personal inclinations are towards prudence, and I don’t believe in conjectures I don’t understand (in the sense that I don’t see a reason for why they should be true) so I certainly believe that the L functions of motives over Q satisfy functionnal equation but I wouldn’t bet 10$ on the truth of GRH.
January 23rd, 2008 at 2:54 pm
I read somewhere that Carleson thought it was false, and he had an idea for a class of couterexamples, but was surprised when he could show that class of counterexamples couldn’t exist. That was that he thought it might be true.
January 23rd, 2008 at 3:13 pm
Interesting about Carleson — such experiences might make interesting case studies for Lakatosian analyses.
January 23rd, 2008 at 7:25 pm
Walt, Todd Trimble: I thought I read about the same thing in an AMM or Fields Institute publication.
I agree with Todd Trimble. This is a fascinating example in the Philosophy of Mathematics. What would Imre Lakatos (1922-1974) have made of it? What can his successors do with it?
This is the sort of discussion that is so much erudite fun on n-Category Cafe. I love it when Ars Mathematica reaches this level of dscourse, too. Good thread!
January 26th, 2008 at 10:21 pm
My favorite interpretation of the zeta function comes out of the so-called “explicit formula” (in the context of L-functions), where the zeroes appear as harmonic frequencies in the distribution of primes.
Kepler would have loved it!
February 11th, 2008 at 12:52 pm
The zeta function is a little world unto itself, full of apparently paradoxical intersections of different styles and eras out of the history of math. For example…
In 1978, the focus of research on the Riemann Hypothesis had long since shifted to generalizations over algebraic number fields, where prime ideals replace prime integers, and it’s probably fair to say that virtually nobody expected significant progress to emerge out of “old” math manipulated by a rather old mathematician. So of course that’s exactly what happened.
Euler had summed zeta(2) to (pi^2)/6, and proved that for even s, zeta(s) is a rational multiple of pi^s. About the rationality, irrationality, or transcendence of zeta(s) for odd s, nothing had been established for 200 years after Euler, until the highly political, often arrested, and generally obnoxious Roger Apéry of the University of Caen appeared at the Journées Arithmétiques de Marseille-Luminy in 1978, claiming to prove the irrationality of zeta(3) with a series of unlikely-looking assertions involving bizarre formulae that none of the mathematicians present could understand.
Out of this phantasmagorical hodge-podge Apéry extracted a series of rational approximations to zeta(3) converging to it so rapidly that it could not be rational, according to the classical theorem of Liouville.
Incredulity reigned in Marseille-Luminy, but Apéry was eventually vindicated, although his miraculous formulae were never extended beyond s=3, and 30 years later we aren’t much further along. In 2001 Zudilin proved that at least one of the 4 values z(5), z(7), z(9) and z(11) is irrational, and this peculiar odd-man-out result is typical of the always peculiar progress of research into the zeta function, and the odd men who pursue it.
Apéry achieved a measure of fame for his brilliant result, produced at the advanced age of 61, but the last word in his story was written by an anonymous graffito artist on a wall at the University of Caen after the announcement of Apéry’s death in 1994.
Apéry a péri.
Further reading: For Mathematica nerds, there’s a fun demonstration of “Apéry’s Rational Approximation to His Constant” on Wolfram’s site, and a good biographical sketch of Apéry by his son here.
February 11th, 2008 at 3:46 pm
On further reflection, I can’t resist one-upping the anonymous prankster from Caen, and writing a different memorial graffito on the (virtual) wall, in accord with Apéry’s Greek ancestry:
Here άπειρος is the (classical) Greek word for infinity or eternity, and the graffito means…
February 12th, 2008 at 4:21 am
The Apery discovery and puns are very pretty.
Apéry’s constant is defined by zeta(3)=1.2020569…,
(Sloane’s A002117) where zeta(z) is the Riemann zeta function. Apéry (1979) proved that zeta(3) is irrational, although it is not known if it is transcendental. Sorokin (1994) and Nesterenko (1996) subsequently constructed independent proofs for the irrationality of zeta(3) (Hata 2000). zeta(3) arises naturally in a number of physical problems, including in the second- and third-order terms of the electron’s gyromagnetic ratio, computed using quantum electrodynamics.
It is not known if zeta(3) is normal (Bailey and Crandall 2003).
See the nice survey at:
Weisstein, Eric W. “Apéry’s Constant.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/AperysConstant.html
“If I were to awaken after having slept for a thousand years, my first question would be: has the Riemann hypothesis been proven?”
– David Hilbert