Uniform exclusion of Landau–Siegel zeros
OpenAI's theorem (2026) that (1 - beta) log q >= c for an absolute constant c > 0 and every real zero beta of every primitive real Dirichlet L-function of conductor q >= 3, the logarithmic form of the Landau–Siegel problem. The prime bias forced by a real zero near 1, an interpolation theorem for polynomials with separately bounded degrees on a projected integer box, a weighted determinant of conjugates in Q(sqrt d, sqrt 2), and its two bounds: Hadamard's inequality, and divisibility by the Frobenius congruence at the primes with chi(p) = -1. A zero-free region with no exceptional zero follows.
Four lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
- Real zeros and the prime bias
- Interpolation on a projected integer box
- A weighted determinant in a biquadratic field
- Two bounds for the determinant
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- Dirichlet characters · Dirichlet L-functions and primes in progressions
- Dirichlet's theorem on primes in arithmetic progressions · Dirichlet L-functions and primes in progressions
- The functional equation of Dirichlet L-functions · Dirichlet L-functions and primes in progressions
- Zero-free regions and the exceptional zero · Dirichlet L-functions and primes in progressions
- Siegel's theorem · Dirichlet L-functions and primes in progressions
- Dirichlet series and Euler products · The Riemann zeta function
- Counting primes by elementary means: Chebyshev and Mertens · The Riemann zeta function
- The Gamma function and Stirling's formula · The Riemann zeta function
- The real place: integral formula and two bounds (Irrationality of special values)
- Complex analysis and Dirichlet L-functions as in the programme courses The Riemann zeta function and Dirichlet L-functions and primes in progressions, cited at each use.
- Linear algebra over a field: determinants, rank, Hadamard's inequality.