Apéry's theorem

In mathematics, Apéry's theorem is a result in number theory that states the number ζ(3) is irrational. That is, the number:$zeta(3)=sum\_\{n=1\}^infty\; frac\{1\}\{n^3\}=frac\{1\}\{1^3\}+frac\{1\}\{2^3\}+frac\{1\}\{3^3\}+ldots\; =\; 1.2020569ldots$ cannot be written as a fraction "p"/"q".

History

Euler proved in the eighteenth century that if "n" is a positive integer then we have:$frac\{1\}\{1^\{2n+frac\{1\}\{2^\{2n+frac\{1\}\{3^\{2n+frac\{1\}\{4^\{2n+ldots\; =\; frac\{p\}\{q\}pi^\{2n\}$for some rational number "p"/"q". Specifically, writing the infinite series on the left as ζ(2"n") he showed:$zeta(2n)\; =\; (-1)^\{n+1\}frac\{B\_\{2n\}(2pi)^\{2n\{2(2n)!\}$where the "B_n" are the rational Bernoulli numbers. Once it was proved that π^"n" is always irrational this showed that ζ(2"n") is irrational for all positive integers "n".

No such representation in terms of π is known for the so-called odd zeta constants, the values ζ(2"n"+1) for positive integers "n", so it is not known if there are rational (or even algebraic) numbers ξ_"n" such that:$displaystyle\{zeta(2n+1)=xi\_npi^\{2n+1.$

Because of this, no proof could be found to show that the odd zeta constants were irrational, even though they were - and still are - all believed to be transcendental. However, in June 1978 Roger Apéry gave a talk entitled "Sur l'irrationalité de ζ(3)." During the course of the talk he outlined proofs that ζ(3) and ζ(2) were irrational, the latter using methods simplified from those used to tackle the former rather than relying on the expression in terms of π. Due to the wholly unexpected nature of the result and Apéry's blasé and very sketchy approach to the subject many of the mathematicians in the audience dismissed the proof as flawed. Three of the audience members suspected Apéry was onto something, though, and set out to fill in the gaps in his proof.

Two months later these three - Henri Cohen, Hendrik Lenstra, and Alfred van der Poorten - finished their work, and on August 18 Cohen delivered a lecture fleshing out the proof given by Apéry. Following the talk Apéry himself took to the podium to explain the source of some of his ideas. [A. van der Poorten, "A proof that Euler missed", Math. Intelligencer 1 (1979), pp. 195-203.]

Apéry's Proof

Apéry's original proof [R. Apéry, "Irrationalité de ζ(2) et ζ(3)", Astérisque 61 (1979), pp. 11-13.] was based on the well known irrationality criterion from Dirichlet, which states that a number ξ is irrational if there are infinitely many coprime integers "p" and "q" such that:for some fixed "c",δ>0.

The starting point for Apéry was the series representation of ζ(3) as:Roughly speaking, Apéry then defined a sequence "c_n,k" which converges to ζ(3) about as fast as the above series, specifically:He then defined two more sequences "a_n" and "b_n" that, roughly, have the quotient "c_n,k". These sequences were:and:The sequence "a_n"/"b_n" converges to ζ(3) fast enough to apply the criterion, but unfortunately "a_n" is not an integer after "n"=2. Nevertheless, Apéry showed that even after multiplying "a_n" and "b_n" by a suitable integer to cure this problem the convergence was still fast enough to guarantee irrationality.

Later Proofs

Within a year of Apéry's result an alternative proof was found by Frits Beukers [F. Beukers, "A note on the irrationality of ζ(2) and ζ(3)", Bull. London Math. Soc. 11 (1979),pp. 268-272.] . His proof replaced Apéry's series with integrals involving the Legendre polynomials. Using a representation that would later be generalised to Hadjicostas's formula, Beukers showed that:$int\_\{0\}^\{1\}int\_\{0\}^\{1\}frac\{-log(xy)\}\{1-xy\}P\_\{n\}(x)P\_\{n\}(y)dxdy=frac\{A\_\{n\}+B\_\{n\}zeta(3)\}\{operatorname\{lcm\}left\; [1,ldots,n\; ight]\; ^\{3$for some integers "A_n" and "B_n" and where "P_n"("x") is the "n"^th Legendre polynomial. Using partial integration and the assumption that ζ(3) was rational and equal to "a"/"b", Beukers eventually derived the inequality:

A more recent proof by Wadim Zudilin [W. Zudilin, "An Elementary Proof of Apéry's Theorem", (2002),http://arxiv.org/abs/math/0202159] is more reminiscent of Apéry's original proof, and also has similarities to a fourth proof by Yuri Nesterenko [Y.V. Nesterenko, "A Few Remarks on ζ(3)", Mat. Zametki 59 (1996), pp. 865-880. Englishtranslation in Math. Notes 59 (1996), pp. 625-636.] . These later proofs again derive a contradiction from the assumption that ζ(3) is rational by constructing sequences that tend to zero but are bounded below by some positive constant. They are somewhat less transparent that the earlier proofs, relying as they do on hypergeometric series.

Higher Zeta Constants

Apéry and Beukers could simplify their proofs to work on ζ(2) as well thanks to the series representation:Due to the success of Apéry's method a search was undertaken for a number ξ₅ with the property that:If such a ξ₅ were found then the methods used to prove Apéry's theorem would be expected to work on a proof that ζ(5) is irrational. Unfortunately, extensive computer searching [D. H. Bailey, J. Borwein, N. Calkin, R. Girgensohn, R. Luke, and V. Moll, "Experimental Mathematics in Action", 2007.] has failed to find such a constant, and in fact it is now known that if ξ₅ exists and if it is an algebraic number of degree at most 25, then the coefficients in its minimal polynomial must be enormous, at least 10³⁸³, so extending Apéry's proof to work on the higher odd zeta constants doesn't seem likely to work.

Despite this, many mathematicians working in this area expect a breakthrough sometime soon [Jörn Steuding, "Diophantine Analysis", 2005.] . Indeed, recent work by Wadim Zudilin and Tanguy Rivoal has shown that infinitely many of the numbers ζ(2"n"+1) must be irrational [T. Rivoal, "La fonction zeta de Riemann prend une infnité de valuers irrationnelles aux entiers impairs", Comptes Rendus Acad. Sci. Paris Sér. I Math. 331 (2000), pp. 267-270.] , and even that at least one of the numbers ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational [W. Zudilin, "One of the numbers ζ(5); ζ(7); ζ(9); ζ(11) is irrational", Uspekhi Mat. Nauk 56:4 (2001), pp. 149-150.] . Their work uses linear forms in values of the zeta function and estimates upon them to bound the dimension of a vector space spanned by values of the zeta function at odd integers. Hopes that Zudilin could cut his list further to just one number did not materialise, but work on this problem is still an active area of research.

ee also

*Apéry's constant, another name for ζ(3)

References