# Apéry's theorem

Apéry's theorem In mathematics, Apéry's theorem is a result in number theory that states the Apéry's constant ζ(3) is irrational. That is, the number {displaystyle 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 where p and q are integers. The theorem is named after Roger Apéry.

The special values of the Riemann zeta function at even integers 2n (n > 0) can be shown in terms of Bernoulli numbers to be irrational, while it remains open whether the function's values are in general rational or not at the odd integers 2n + 1 (n > 1) (though they are conjectured to be irrational).

Contents 1 History 2 Apéry's proof 3 Later proofs 4 Higher zeta constants 5 References 6 External links History Euler proved that if n is a positive integer then {displaystyle {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(2n) he showed {displaystyle zeta (2n)=(-1)^{n+1}{frac {B_{2n}(2pi )^{2n}}{2(2n)!}}} where the Bn are the rational Bernoulli numbers. Once it was proved that π n is always irrational this showed that ζ(2n) is irrational for all positive integers n.

No such representation in terms of π is known for the so-called zeta constants for odd arguments, the values ζ(2n + 1) for positive integers n. It has been conjectured that the ratios of these quantities {displaystyle {frac {zeta (2n+1)}{pi ^{2n+1}}},} are transcendental for every integer n ≥ 1.[1] Because of this, no proof could be found to show that the zeta constants with odd arguments were irrational, even though they were (and still are) all believed to be transcendental. However, in June 1978, Roger Apéry gave a talk titled "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 proof and Apéry's blasé and very sketchy approach to the subject, many of the mathematicians in the audience dismissed the proof as flawed. However Henri Cohen, Hendrik Lenstra, and Alfred van der Poorten suspected Apéry was on to something and set out to confirm his proof. Two months later they finished verification of Apéry's proof, and on August 18 Cohen delivered a lecture giving full details of the proof. After the lecture Apéry himself took to the podium to explain the source of some of his ideas.[2] Apéry's proof Apéry's original proof[3][4] was based on the well known irrationality criterion from Peter Gustav Lejeune Dirichlet, which states that a number ξ is irrational if there are infinitely many coprime integers p and q such that {displaystyle left|xi -{frac {p}{q}}right|<{frac {c}{q^{1+delta }}}} for some fixed c, δ > 0.

The starting point for Apéry was the series representation of ζ(3) as {displaystyle zeta (3)={frac {5}{2}}sum _{n=1}^{infty }{frac {(-1)^{n-1}}{n^{3}{binom {2n}{n}}}}.} Roughly speaking, Apéry then defined a sequence cn,k which converges to ζ(3) about as fast as the above series, specifically {displaystyle c_{n,k}=sum _{m=1}^{n}{frac {1}{m^{3}}}+sum _{m=1}^{k}{frac {(-1)^{m-1}}{2m^{3}{binom {n}{m}}{binom {n+m}{m}}}}.} He then defined two more sequences an and bn that, roughly, have the quotient cn,k. These sequences were {displaystyle a_{n}=sum _{k=0}^{n}c_{n,k}{binom {n}{k}}^{2}{binom {n+k}{k}}^{2}} and {displaystyle b_{n}=sum _{k=0}^{n}{binom {n}{k}}^{2}{binom {n+k}{k}}^{2}.} The sequence an /bn converges to ζ(3) fast enough to apply the criterion, but unfortunately an is not an integer after n = 2. Nevertheless, Apéry showed that even after multiplying an and bn by a suitable integer to cure this problem the convergence was still fast enough to guarantee irrationality.