Roger Apéry’s proof that zeta(3) is irrational

At a conference in 1978 Roger Apéry outlined a proof that

\displaystyle \zeta(3) =\sum_{n=1}^\infty\frac{1}{n^3}

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 pn and qsuch that 0<|qnα – pn|=Tn<ε  for any given 1>ε>0

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

\displaystyle q_n=n!  and  \displaystyle p_n=n!\sum_{k=0}^{n}\frac{1}{k!}

as Tn 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, pn and qn 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

p_1=a_1  p_2=b_2p_1  p_{n+1}=a_{n+1}p_{n-1}+b_{n+1}p_n (n ≥2)

q_1=b_1  q_2=a_2+b_2q_1  q_{n+1}=a_{n+1}q_{n-1}+b_{n+1}q_n (n ≥2)

where the an and bn are rational functions of n.  pn and qn 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

\displaystyle\left \vert q_{n}\alpha-p_{n}\right\vert=\left \vert q_{n}\sum_{j=n+1}^{\infty}\frac{(-1)^{j}a_j \dots a_1}{q_{j}q_{j-1}}\right\vert =T_{n}

Apéry developed a method by means of which, starting with partial numerators and denominators pn,0 and qn,0 derived from a series representation of a number, a hierarchy of continued fractions pn,1, qn,1; pn,2, qn,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 Tn,k increase less rapidly with increasing n as k increases. Moreover the diagonal elements pn,n and qn,n are also partial numerators and denominators of a continued fraction having the same value as the original and the corresponding Tn is even more strongly constrained.

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

a_{1}=6   b_1={5}

\displaystyle a_n=- \frac{(n-1)^3}{n^3}   \displaystyle b_n=34-\frac {51}{n}+\frac{27}{n^2}-\frac{5}{n^3}  (n > 1)

It is not too difficult to show from the recurrence that qn increases with e3.4n and Tn is non-zero and decreases with e-3.4n. pn and qn 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 Tn is still a decreasing function of n.

Apéry astounded his audience by claiming that all qn are integers and all pn become integers if multiplied by 2(Ln)3 where Ln is the smallest positive integer divisible by each of the integers between 1 and n. Ln 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 Ln increases with en.  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 qn and pn.  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 rs in the calculus of the real variable r.

Using this approach we find that

\displaystyle q_n=\sum_{j=0}^n \left [ \frac {(n+j)_{2j}}{(j!)^2} \right ]^2

\displaystyle p_n=q_{n}\Sigma_n+\sum_{j=0}^{n-1} \left [\frac{(n+j)_{2j}}{(j!)^2} \right ] ^2\left [\Sigma_n-\Sigma_j\right ]

where Σj is the sum of the first j inverse cubes and Σ0 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.

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