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

άπειρος άπειρος

Here άπειρος is the (classical) Greek word for infinity or eternity, and the graffito means…

Apéry forever!

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.

Kepler would have loved it!

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!

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.