The special values of the Riemann zeta function at even integers
) 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
Leonhard Euler proved that if n is a positive integer then for some rational number
Specifically, writing the infinite series on the left as
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
[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)."
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 original proof[3][4] was based on the well-known irrationality criterion from Peter Gustav Lejeune Dirichlet, which states that a number
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
as Roughly speaking, Apéry then defined a sequence
by a suitable integer to cure this problem the convergence was still fast enough to guarantee irrationality.
Within a year of Apéry's result an alternative proof was found by Frits Beukers,[5] who replaced Apéry's series with integrals involving the shifted Legendre polynomials
, Beukers eventually derived the inequality which is a contradiction since the right-most expression tends to zero as
is rational by constructing sequences that tend to zero but are bounded below by some positive constant.
They are somewhat less transparent than the earlier proofs, since they rely upon hypergeometric series.
Apéry and Beukers could simplify their proofs to work on
as well thanks to the series representation Due to the success of Apéry's method a search was undertaken for a number
were found then the methods used to prove Apéry's theorem would be expected to work on a proof that
Unfortunately, extensive computer searching[8] has failed to find such a constant, and in fact it is now known that if
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
Work by Wadim Zudilin and Tanguy Rivoal has shown that infinitely many of the numbers
[10] 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.
Higher zeta constants have application to physics: they describe correlation functions in quantum spin chains.