A Note on the Irrationality of ζ(2) and ζ(3)

  title={A Note on the Irrationality of $\zeta$(2) and $\zeta$(3)},
  author={Frits Beukers},
  journal={Bulletin of The London Mathematical Society},
  • F. Beukers
  • Published 1979
  • Mathematics
  • Bulletin of The London Mathematical Society
At the “Journees Arithmetiques” held at Marseille-Luminy in June 1978, R. Apery confronted his audience with a miraculous proof for the irrationality of ζ(3) = l-3+ 2-3+ 3-3 + .... The proof was elementary but the complexity and the unexpected nature of Apery’s formulas divided the audience into believers and disbelievers. Everything turned out to be correct however. Two months later a complete exposition of the proof was presented at the International Congress of Mathematicians in Helsinki in… Expand
Consequences of Apéry’s work on ζ(3)
At the end of the 1970’s it seemed that my fate as a young beginning research mathematician was closely linked with work of Roger Apery, but through the diophantine equation x2 + D = pn in the unknown integers x, n, where D, p are given integers with p prime. Expand
Similarities in Irrationality Proofs for π, ln2, ζ(2), and ζ(3)
The irrationality of π dominated a good 2000 years of mathematical history, starting with the closely related circle-squaring problem of the ancient Greeks. In 1761 Lambert proved the irrationalityExpand
The group structure for ζ(3)
1. Introduction. In his proof of the irrationality of ζ(3), Apéry [1] gave sequences of rational approximations to ζ(2) = π 2 /6 and to ζ(3) yielding the irrationality measures µ(ζ(2)) < 11.85078. ..Expand
Arithmetic of linear forms involving odd zeta values
The story exposed in this paper starts in 1978, when R. Apery [Ap] gave a surprising sequence of exercises demonstrating the irrationality of ζ(2) and ζ(3). (For a nice explanation of Apery’sExpand
An Elementary Proof of Ramanujan's Formula for $ \zeta(2m+1) $
It’s well known that many number theoretic properties of ζ(2n + 1) are nowadays still unsolved mysteries, such as the rationality, transcendence and existence of some closed-form functional equationExpand
A Computer-Algebra-Based Formal Proof of the Irrationality of ζ(3)
The formal verification of an irrationality proof of ζ(3), the evaluation of the Riemann zeta function, using the Coq proof assistant is described, complete up to a weak corollary of the Prime Number Theorem. Expand
A Formal Proof of the Irrationality of ζ(3)
This paper presents a complete formal verification of a proof that the evaluation of the Riemann zeta function at 3 is irrational, using the Coq proof assistant, and establishes that some sequences satisfy a common recurrence. Expand
Criteria for irrationality of Euler’s constant
By modifying Beukers' proof of Apery's theorem that ζ(3) is irrational, we derive criteria for irrationality of Euler's constant, γ. For n > 0, we define a double integral In and a positive integer SExpand
Irrationality of certain p-adic periods for small p
Following Apery's proof of the irrationality of zeta(3), Beukers found an elegant reinterpretation of Apery's arguments using modular forms. We show how Beukers arguments can be adapted to a p-adicExpand
A new formula for ζ(2n + 1) (and how not to prove that ζ(5) is irrational)
Using a new polylogarithmic identity, we express the values of ζ at odd integers 2n + 1 as integrals over unit n−dimensional hypercubes of simple functions involving products of logarithms. We thenExpand