we-solve (beta)

Math-2791 · mathematics · Number theory

Patterson first moment over Eisenstein primes

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
In Z[omega], $\omega =\exp (2\pi $ i/3), let primary primes pi satisfy $\pi =1$ modulo3, let $chi_\pi $ be the cubic residue character, and put $G(\pi )=N(\pi )^{-1/2}$ $sum_(v$ modulo pi) $chi_\pi (v)$ exp(2pi i Tr(v/pi)). Is $sum_(N(\pi )<=X)$ G(pi) asymptotic to $(6/5)(2\pi )^{2/3}/(3$ Gamma(2/3)) times $X^{5/6}/\log $ X, counting all primary prime elements including conjugate split primes?

Short English statement

In Z[omega], $\omega =\exp (2\pi $ i/3), let primary primes pi satisfy $\pi =1$ modulo3, let $chi_\pi $ be the cubic residue character, and put $G(\pi )=N(\pi )^{-1/2}$ $sum_(v$ modulo pi) $chi_\pi (v)$ exp(2pi i Tr(v/pi)). Is $sum_(N(\pi )<=X)$ G(pi) asymptotic to $(6/5)(2\pi )^{2/3}/(3$ Gamma(2/3)) times $X^{5/6}/\log $ X, counting all primary prime elements including conjugate split primes?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2791-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2791

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 023; 2026-09-25).

Patterson first moment over Eisenstein primes

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 023; 2026-09-25), the asserted result is that averaging cubic Gauss sums with the prescribed normalization over the entire primary Eisenstein-prime set gives Patterson's main term, and every fixed nonzero angular mode contributes a negligible amount.

Mathematical statement of the claimed result

Manuscript 1.
Set $\omega=e^{2\pi i/3}$, $\mathcal O=\mathbb Z[\omega]$, $N(b)=b\bar b$ and $\operatorname{Tr}(z)=z+\bar z$. For a primary element $b\equiv1\pmod3$, let $\chi_b$ be its cubic residue symbol, zero on nonunits, and define \[ G(b)=N(b)^{-1/2}\sum_{v\bmod b}\chi_b(v)e^{2\pi i\operatorname{Tr}(v/b)},\qquad c_*=(2\pi)^{2/3}/(3\Gamma(2/3)). \] The claimed unconditional sharp-cutoff first moment is \[ \sum_{\substack{N\pi\le X\\\pi\equiv1\pmod3\text{ prime}}}G(\pi) =\tfrac65c_*X^{5/6}/\log X+o(X^{5/6}/\log X)\quad(X\to\infty). \] All primary prime elements are included, in particular both conjugate primes above a split rational prime.

Outline of the argument

For manuscript 1 (family 023; 2026-09-25), angular Hecke characters and a Voronoi transformation give Type I estimates. Corrected dispersion and exceptional-moment bounds are joined to an exact prime decomposition, and smoothing is removed only after each fixed angular mode has been controlled.

Formal verification

For manuscript 1 (family 023; 2026-09-25), OpenAI lists Lean statements for this assertion: averaging cubic Gauss sums with the prescribed normalization over the entire primary Eisenstein-prime set gives Patterson's main term, and every fixed nonzero angular mode contributes a negligible amount.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

The linked family Comparator checks are PattersonFirstMoment.lean.

Their named declarations include OAI.CubicFirstMoment.patterson_angularCancellation, OAI.CubicFirstMoment.patterson_angularComparison, OAI.CubicFirstMoment.patterson_firstMoment.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

Go to contribution →