Artin's primitive root conjecture

Every integer other than -1 that is not a square is a primitive root modulo at least c x/(log x)^2 primes in (x, 2x), after OpenAI's 2026 paper. A zero-free strip Re s > 1 - 10^-6 for the Hecke L-functions of every cyclotomic field containing the twelfth roots of unity, from cubic Gauss sums: S-integers, sextic reciprocity, Kummer characters, the cubic theta function of Kazhdan and Patterson, a Weyl reflection, a quadratic large sieve, a Poisson probe and its Euler product, zero detection; uniform complete splitting in Kummer fields; primes p = crQ + 1 from a block sieve, Bombieri-Vinogradov, marked bilinear estimates and Buchstab's densities.

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

1. Artin's conjecture and the index of a prime
2. S-integers and sixth-power classes
3. Sextic reciprocity and Kummer characters
4. Gauss sums and the correction characters
5. The coefficient family and the plan of the zero-free argument
6. Cubic theta and its Whittaker coefficients
7. Norm sheets and exact Gaussian smoothing
8. A theta expression for the coefficient sums
9. The Weyl reflection
10. The quadratic large sieve and the reflected moment
11. The Poisson probe
12. The Euler product of the probe
13. Zero detection for the frequency characters
14. The zero-free strip
15. Uniform complete splitting in Kummer fields
16. A block sieve and rough-number densities
17. Marked Type II estimates and rough factors
18. Weights, mass and distribution in a progression
19. Primes with controlled predecessors

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Lebesgue integration and smoothing — selected programme proofs (Distributions, kernels and analytic singularities)
• Schwartz functions and Fourier inversion (Distributions, kernels and analytic singularities)
• Bernoulli series and Poisson summation (Distributions, kernels and analytic singularities)
• The adèle ring of a number field
• Idèles and the idèle class group
• Additive characters, self-dual measures and Poisson summation on the adèles
• Quasi-characters and Hecke characters
• Tate's local theory at the finite places
• Tate's local theory at the infinite places
• Tate's global theory: continuation and functional equation
• Hecke L-functions and the Dedekind zeta function
• Counting ideals of bounded norm · Number fields
• Cyclotomic fields · Number fields
• Decomposition of primes in extensions · Number fields
• Dirichlet's unit theorem · Number fields
• Discriminants and integral bases · Number fields
• Hilbert's ramification theory in Galois extensions · Number fields
• The different and the discriminant · Number fields
• Artin L-functions, conductors and discriminants · Class field theory
• Hilbert and ring class fields, and quadratic prime forms · Class field theory
• Hilbert symbols and local conics · Class field theory
• Hilbert's Theorem 90 and Kummer theory · Class field theory
• Power residue symbols and reciprocity laws · Class field theory
• The global reciprocity law · Class field theory
• The Siegel–Walfisz theorem · Dirichlet L-functions and primes in progressions
• The Bombieri–Vinogradov theorem · Dirichlet L-functions and primes in progressions
• Explicit formulas for Dirichlet L-functions and primes in progressions · Explicit formulas and positivity
• Hensel's lemma, squares and roots of unity in p-adic fields · Local fields
• Places of number fields in extensions and the product formula · Local fields
• The multiplicative group of a local field · Local fields
• 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
• Entire functions of order one and the Hadamard product of ξ · The Riemann zeta function
• Growth in the critical strip: convexity and the Lindelöf hypothesis · The Riemann zeta function
• The prime number theorem · The Riemann zeta function
• The Riemann hypothesis implies the Lindelöf hypothesis · The Riemann zeta function
• A mean value theorem and logarithmic phases at large height (Marked Type II estimates for prime predecessors)
• Long prime polynomials (Marked Type II estimates for prime predecessors)
• The marked Type II estimate (Marked Type II estimates for prime predecessors)
• Number fields and class field theory: units, ideal classes, discriminants, cyclotomic fields, Kummer theory, Hilbert symbols and power reciprocity laws (courses Number fields and Class field theory); local fields (course Local fields); adeles, idele class characters and Hecke L-functions (course Adèles, idèles and Tate's thesis); the Riemann zeta function and Dirichlet L-functions up to the Siegel-Walfisz and Bombieri-Vinogradov theorems (courses The Riemann zeta function and Dirichlet L-functions and primes in progressions).