The irrationality exponent of π
The theorem that the irrationality exponent of π is 2: for every ν > 2, |π − p/q| > q^(−ν) for all large q. Irrationality exponents, Dirichlet's and Liouville's theorems; a weighted multiplicity estimate along commuting vector fields, proved with formal flow coordinates and Bézout's inequality; a curve inequality for logarithmic curves with separated weights, by a zero estimate and logarithmic residues; interpolation of jets on logarithmic curves through a blow-up, Kleiman's theorem and Serre vanishing; an interpolation determinant at Gaussian rational centres with an arithmetic lower bound and an analytic upper bound from Taylor collisions; and the convergence of the Flint–Hills series, with the exact convergence region of the sums of n^(−a)|sin n|^(−b).
Six lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
- Irrationality exponents
- Weighted multiplicity estimates
- The curve inequality
- Interpolation on logarithmic curves
- An interpolation determinant for π
- The irrationality exponent of π is two
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- Coefficient rings and the Cohen structure theorem (Commutative algebra for geometry)
- Dimension theory of Noetherian local rings (Commutative algebra for geometry)
- Discrete valuation rings, normal rings and Serre's criterion (Commutative algebra for geometry)
- Smooth algebras over a field and the Jacobian criterion (Commutative algebra for geometry)
- Blowing up · Morphisms of schemes
- Normalization · Morphisms of schemes
- Very ample invertible sheaves, Segre and Veronese embeddings · Morphisms of schemes
- Cohomology of sheaves on ringed spaces · Cohomology of quasi-coherent sheaves
- Coherent sheaves on projective schemes: Serre's theorems · Cohomology of quasi-coherent sheaves
- Bézout's inequality for isolated zeros (Intersection numbers and positivity)
- Intersection numbers of line bundles (Intersection numbers and positivity)
- Nef line bundles and Kleiman's theorem (Intersection numbers and positivity)
- Intersection numbers, Kleiman's theorem and Bézout's inequality (course Intersection numbers and positivity); blow-ups and Serre vanishing (courses Morphisms of schemes and Cohomology of quasi-coherent sheaves); Cauchy's estimates (core course Complex Analysis).