The first moment of cubic Gauss sums
Patterson's first-moment conjecture for cubic Gauss sums, proved unconditionally after OpenAI's 2026 manuscript: the normalized cubic Gauss sums over the primary primes of Z[omega] of norm at most X add up to (6/5) c_* X^(5/6)/log X, with c_* = (2 pi)^(2/3)/(3 Gamma(2/3)). Cubic Gauss sums and their cubes; large sieves for cubic characters; the level Voronoi formula; exceptional character moments; corrected dispersion; an exact decomposition of the prime sum; the sharp cutoff; fixed angular modes and the powers G^k with k not divisible by 3.
Eight lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
- Cubic Gauss sums and Patterson's conjecture
- Large sieves for cubic characters
- The level formula and Type I sums
- Character moments at two exceptional configurations
- Corrected dispersion
- An exact decomposition of the prime sum
- Bilinear sums and the sharp cutoff
- Angular modes and powers of Gauss sums
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- The Fourier inversion theorem and the dual Haar measure (Harmonic analysis on locally compact abelian groups)
- The Poisson summation formula (Harmonic analysis on locally compact abelian groups)
- Quasi-characters and Hecke characters
- Hecke L-functions and the Dedekind zeta function
- Power residue symbols and reciprocity laws · Class field theory
- Ray class fields, conductors and ideal reciprocity · Class field theory
- Gauss sums · Dirichlet L-functions and primes in progressions
- The prime number theorem for arithmetic progressions · Dirichlet L-functions and primes in progressions
- Counting primes by elementary means: Chebyshev and Mertens · The Riemann zeta function
- The Gamma function and Stirling's formula · The Riemann zeta function
- Mean values of Dirichlet polynomials and of zeta on the critical line · The Riemann zeta function
- Cubic reciprocity and ideal reciprocity (course Class field theory), Gauss sums and primes in progressions (course Dirichlet L-functions and primes in progressions), Stirling's formula and mean values of Dirichlet polynomials (course The Riemann zeta function), Hecke L-functions (course Adèles, idèles and Tate's thesis), and Poisson summation (course Harmonic analysis on locally compact abelian groups).