At a conference in 1978 Roger Apéry outlined a proof that

is irrational. He published a brief note on his results in the journal ‘Astérisque’ (Apery 1979) and a fuller, but still cryptic, account two years later (Apery (1981)). Olivier and Batut (1980) provided a detailed explanation of Apéry’s method of generating rapidly converging continued fractions. Because none of these papers is readily accessible I have provided links to pdf copies. van der Poorten’s Mathematical Intelligencer article is widely cited and indeed sometimes given as the primary reference for Apéry’s proof. Certainly, being in English, it served to publicise Apéry’s result and to highlight connections with other areas of mathematics. But it provides virtually no insight into the ideas underlying Apéry’s approach.

In the pdf here, I attempt to provide a full explanation of Apéry’s method of proof. This is outlined below.

The proof relies on the criterion that α is irrational if there exist integers p_{n} and q_{n }such that 0<|q_{n}α – p_{n}|=T_{n}<ε for any given 1>ε>0

For example it can be easily shown that e is irrational by taking

and

as T_{n} is positive and decreases with 1/n.

In 1761 Lambert showed that π is irrational by finding a continued fraction whose partial numerators and partial denominators, p_{n} and q_{n} respectively, have the desired property (see earlier post). Apéry’s method is also based on use of continued fractions. These are essentially recurrences of the form

(n ≥2)

(n ≥2)

where the a_{n} and b_{n} are rational functions of n. p_{n} and q_{n} are referred to as partial numerators and partial denominators respectively and their ratio as partial convergents. If α can be represented as a series whose terms are rational functions of n, then it can also, quite easily, be represented as a continued fraction. The connection with the irrationality criterion arises because

Apéry developed a method by means of which, starting with partial numerators and denominators p_{n,0} and q_{n,0} derived from a series representation of a number, a hierarchy of continued fractions p_{n,1}, q_{n,1}; p_{n,2}, q_{n,2} and so on can be derived and in a variety of cases, including that of ζ(3), these converge to the same value as the original continued fraction and the T_{n,k} increase less rapidly with increasing n as k increases. Moreover the diagonal elements p_{n,n} and q_{n,n} are also partial numerators and denominators of a continued fraction having the same value as the original and the corresponding T_{n} is even more strongly constrained.

For ζ(3) this ‘diagonal’ continued fraction has the form

(n > 1)

It is not too difficult to show from the recurrence that q_{n} increases with e^{3.4n} and T_{n} is non-zero and decreases with e^{-3.4n}. p_{n} and q_{n} are rational so to prove that ζ(3) is irrational it remains to show that they can be made integers by multiplying by some integer function of n whose product with T_{n} is still a decreasing function of n.

Apéry astounded his audience by claiming that all q_{n} are integers and all p_{n} become integers if multiplied by 2(L_{n})^{3} where L_{n} is the smallest positive integer divisible by each of the integers between 1 and n. L_{n} is the exponent of the Ψ function used by Chebyshev in his investigations on the Prime Number Theorem and it is well known as a consequence that L_{n} increases with e^{n}. Therefore Apéry’s claim, if true, immediately proves the irrationality of ζ(3).

Apéry used a generating function method to solve the recurrence and obtain ‘closed form’ representations of q_{n} and p_{n}. However it is more straightforward to work with the algorith Apéry used to generate his hierarchy of continued fractions. This is a form of partial difference equation, and closed form solutions can be found in terms of shifted factorials. A shifted factorial (r)_{s} is the product of s successive positive integers, the largest of which is r. The rationale for searching for solutions in this form is that (r)_{s} in discrete calculus is the analogue of r^{s} in the calculus of the real variable r.

Using this approach we find that

where Σ_{j} is the sum of the first j inverse cubes and Σ_{0} and (r)_{0} are defined to be zero and one respectively.

Since the product of any n consecutive positive integers is divisible by n! the truth of Apéry’s claim is immediately evident.