Patterson's first moment for cubic Gauss sums

1 · The essentials

The manuscript studies quantities called cubic Gauss sums, attached to prime numbers in a particular setting. It adds their normalized values over all the relevant primes up to a size limit $X$ . The chosen family consists of all primary Eisenstein primes, including both conjugates. The result gives a formula describing the dominant behavior of this total as the limit grows. Its leading term is positive and proportional to $X^{5/6} / lo g X$ . The manuscript also states that weighting these values by certain fixed patterns in the primes’ angles makes the total negligible on that same scale. The main result has a Lean formalization, although the supplied information does not specify its exact scope.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine adding readings from many meters, each labeled with a direction on a dial. The total has a precise positive trend, while certain direction-weighted totals become negligible on the scale of that trend. This image does not describe Gauss sums or how they are constructed.

Why it matters

The result addresses Patterson’s first-moment conjecture in its all-primary-Eisenstein-prime form. It specifies not just the scale of the total but also its positive leading constant. The angular result distinguishes this overall contribution from those obtained using each fixed nonzero angular mode.

2 · Going further

The problem is to determine the first moment of normalized cubic Gauss sums over all primary Eisenstein primes of norm at most $X$ , counting both conjugates. Here, the first moment means summing the values under consideration without raising them to an additional power. The manuscript identifies this question as Patterson’s first-moment conjecture.

For this sum, the sharp-cutoff main term is $\frac{6}{5} c_{*} X^{5/6} / lo g X$ , where $c_{*} = (2 π)^{2/3} / (3Γ (2/3))$ . The asymptotic statement means that the ratio of the sum to this main term tends to $1$ as $X$ tends to infinity. The result is stated without additional conjectural assumptions.

The manuscript also shows cancellation at the first-moment scale for every fixed nonzero angular Fourier mode of the prime argument: its contribution is $o (X^{5/6} / lo g X)$ . This concerns each fixed mode and does not, by itself, give a uniform statement for modes that vary with $X$ .

The supplied excerpt states the conclusions but does not explain the argument, so its method cannot be described from this source. Only one manuscript is presented; no relationship to companion manuscripts is documented. The main result has a Lean formalization, with no supplied details identifying exactly which statements it covers.

Useful notions

Eisenstein primes, norm, and conjugation
Normalized cubic Gauss sums
Asymptotic equivalence
Complex argument and Fourier modes

What was known before / what is new

Before

The source identifies the problem as Patterson’s first-moment conjecture. It does not describe earlier partial results or previously available methods.

New

The manuscript shows an unconditional asymptotic formula for the all-primary-Eisenstein-prime version, including both conjugates and an explicit positive leading constant. It also shows that every fixed nonzero angular mode contributes negligibly at the first-moment scale. The main result has a Lean formalization.

Limits and assumptions

The statement concerns normalized sums over all primary Eisenstein primes of norm at most $X$ , including both conjugates. It describes a limit as $X$ grows, not an exact equality for every $X$ . The angular result concerns each fixed nonzero mode; no uniformity for a varying mode is specified. The source supplies no detailed error bound, argument, or remaining open question. A Lean formalization of the main result exists, but its exact scope is not stated, so it should not be assumed to cover every claim.

Mini glossary

First moment: In this statement, the sum of the normalized values over the chosen family of primes.
Sharp cutoff: The restriction to primes whose norm is at most $X$ .
Main term: An expression describing the dominant behavior of the sum as $X$ tends to infinity.
Angular Fourier mode: A component of a periodic weighting that depends on the primes’ angle, or argument.
Negligible contribution at a scale: A contribution whose ratio to that scale tends to zero.

Keywords : Cubic Gauss sums · First moment · Eisenstein primes

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Proves unconditionally the all-primary-prime form of Patterson’s first-moment asymptotic: normalized cubic Gauss sums over primary Eisenstein primes of norm at most X, including both conjugates, have an explicit positive main term of order $X^{5/6} / lo g X$ . Every fixed nonzero prime-angle Fourier mode has smaller order.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/023.md

Manuscripts in this family

★

An unconditional first moment for cubic Gauss sums

25/09/2026

To be verified

We prove Patterson's first-moment conjecture for normalized cubic Gauss sums over all primary Eisenstein primes, unconditionally. The sharp-cutoff main term is $(6/5) c_{*} X^{5/6} / lo g X$ , where $c_{*} = (2 π)^{2/3} / (3Γ (2/3))$ . For every fixed nonzero angular Fourier mode of the prime argument, we also prove cancellation at the first-moment scale.

PDF GitHub folder