Generalized proofs for the irrationality of ζ(2) and ζ(3)

Abstract

Euler’s famous formula for even zeta values immediately points out that they are irrational. Nevertheless, the arithmetic nature of odd zeta values remains a mystery. Roger Ap´ery [1], in 1978, made a breakthrough by proving that ζ(3) is irrational. Over the last four decades, many mathematicians have given different proofs of Ap´ery’s theorem. The proof that Ap´ery presented was quite intricate however, Frits Beukers [4] gave an elementary proof for the same using the definite integrals. In this thesis, motivated by the elementary proof of irrationality of ζ(2) and ζ(3) due to Frits Beukers, we generalize some of the important lemmas, which played a crucial role in the Beukers’ proof. We also investigate some expressions (multiple integrals) that seemingly look quite promising but lack the ability to prove the irrationality of some zeta-value. In this thesis, we generalize Beukers’ proof to present a new proof of the irrationality of ζ(s) at s = 2,3. We also mention a conjecture in which the integral expression is actually promising, proof of which may lead to the conclusion that all positive integer zeta-values are irrational.