Irrationality of special values

Irrationality proofs by small linear forms and by determinants. The irrationality of ζ(2) and Apéry's theorem that ζ(3) is irrational, by Beukers' integrals and an involution of the unit interval; then the complete proof that Catalan's constant G = 1 - 1/9 + 1/25 - ... is irrational: determinants of mixed moments with Chebyshev rows whose Taylor contact cancels ζ(2), lower bounds for their denominators at the prime 2 and at the odd primes by base-p digit reductions, nonvanishing at prime scales by Frobenius and a finite certificate, and an upper bound at the real place by an integral formula, bounded interpolation, Chebyshev expansions of logarithmic kernels and an exact root certificate.

Seven lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Small linear forms and Apéry's theorem
2. Chebyshev rows and mixed moments
3. Denominators at the prime two
4. Denominators at the odd primes
5. Nonvanishing at prime scales
6. The real place: integral formula and two bounds
7. The energy bound and the irrationality of Catalan's constant

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Completions, the p-adic numbers and complete discretely valued fields · Local fields
• The prime number theorem · The Riemann zeta function
• Monotone convergence and Fubini's theorem (core course Measure and Integration), and Cauchy's integral formula and the maximum modulus principle (core course Complex Analysis).