The small positive bias in cubic Gauss sums

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Family 023 · Number theory
Importance 3/5 · Provisional editorial rating
Sources consulted 2026-10-10

1. The question

Add many points on the unit circle in the complex plane. If their directions are balanced, their sum should be much smaller than the number of points. But “balanced in the limit” does not rule out a smaller systematic bias.

Cubic Gauss sums provide an arithmetic example. Their normalized values become equidistributed around the unit circle as the prime varies, yet computations suggest a persistent positive first moment. Patterson predicted the scale and constant of that bias.

Family 023 claims an unconditional asymptotic for the sum over all primary Eisenstein primes with a sharp upper cutoff. It also claims cancellation for every fixed nonzero angular Fourier mode and for certain higher powers of the Gauss sums. The precise indexing and normalization determine the leading constant, so they deserve explicit treatment.

2. Background and essential definitions

Cubes and exponential sums

Write $e(t)=\exp(2\pi i t)$. For a rational prime $p$, Kummer’s cubic sum is

$$S_p=\sum_{x\bmod p}e(x^3/p).$$

The summands have absolute value one, but their phases can cancel. If $p\equiv2\pmod3$, cubing permutes the residues modulo $p$, so the sum is zero. The interesting rational primes satisfy $p\equiv1\pmod3$, where nonzero cubes occupy only one third of the nonzero residue classes.

For the natural cubic characters, it is convenient to work with the Eisenstein integers

$$\mathcal O=\mathbb Z[\omega],\qquad\omega=e^{2\pi i/3}.$$

These form a triangular lattice in the complex plane. Complex conjugation sends $\omega$ to $\omega^2$. The norm and trace are

$$N(z)=z\bar z=|z|^2,\qquad\operatorname{Tr}(z)=z+\bar z.$$

In particular, $N(a+b\omega)=a^2-ab+b^2$.

An element prime to three is primary when it is congruent to one modulo three. This chooses one generator for each ideal prime to three. The sum in the manuscript counts prime ideals through these chosen generators. For a split rational prime $p\equiv1\pmod3$, both conjugate prime ideals are included.

Normalized Gauss sums

For a primary prime $\varpi$, the cubic residue character $\chi_{\varpi}$ takes values in $\{1,\omega,\omega^2\}$ on nonzero residues. It records the cubic residue class and is extended by zero at zero. Concretely, its value is the cube root of unity congruent to $v^{(N\varpi-1)/3}$ modulo $\varpi$.

The manuscript normalizes the Gauss sum as

$$G(\varpi)=\frac1{\sqrt{N\varpi}} \sum_{v\bmod\varpi}\chi_{\varpi}(v) e\!\left(\operatorname{Tr}(v/\varpi)\right).$$

For a prime argument its absolute value is one. A first moment is the sum of these complex values, rather than the sum of their absolute values. It detects a preferred direction that absolute values cannot see.

An angular Fourier mode weights a prime by

$$\left(\frac{\varpi}{|\varpi|}\right)^\ell, \qquad\ell\in\mathbb Z.$$

This probes how the bias correlates with the direction of the prime in the complex plane. The mode $\ell=0$ is the ordinary first moment.

3. Development history and previous partial results

Kummer’s nineteenth-century computations suggested a positive bias in cubic sums. Heath-Brown and Patterson proved equidistribution of normalized cubic Gauss sums on the unit circle. That says the distribution is balanced on the prime-counting scale $X/\log X$, but leaves room for a smaller average of order $X^{5/6}/\log X$.

Patterson proposed this finer first-moment scale in 1978. Heath-Brown’s cubic large sieve supplied an unconditional upper bound $O_\varepsilon(X^{5/6+\varepsilon})$. An upper bound close to the expected power is still different from an asymptotic: it does not identify the sign, leading constant or a remainder smaller than the main term.

The analytic background comes from cubic theta series and metaplectic forms, which encode Gauss sums in automorphic objects with functional equations. Patterson and Yoshimoto developed the relevant coefficient and twist calculations.

Dunn and Radziwiłł proved a smoothed first-moment asymptotic conditional on the Generalized Riemann Hypothesis for Hecke $L$-functions. Their 2024 Annals paper introduced an explicit level-aspect Voronoi summation formula and a corrected dispersion argument. Some nonzero cube frequencies contribute to the main term, so treating all nonzero frequencies as errors would lose the bias itself.

The current manuscript imports their unconditional summation formula and residue calculations, together with established large-sieve and mean-square estimates. It states that its new cancellation arguments do not use their GRH-dependent estimates. It also removes smoothing to obtain the ordinary sharp-cutoff sum.

4. What the new manuscripts claim

Define

$$c_* =\frac{(2\pi)^{2/3}}{3\Gamma(2/3)},$$

where $\Gamma(s)=\int_0^\infty t^{s-1}e^{-t}\,dt$ for $s>0$ is the gamma function. Theorem 1.1 claims

$$\sum_{\substack{N\varpi\le X\\\varpi\equiv1\pmod3\ \mathrm{prime}}} G(\varpi) =\frac65c_*\frac{X^{5/6}}{\log X} +o\!\left(\frac{X^{5/6}}{\log X}\right).$$

The sum is over all primary Eisenstein primes. The little-oh remainder is smaller than the first-moment scale. There is no smoothing function suppressing the endpoint near $X$.

Partial summation gives the corresponding logarithmically weighted statement:

$$\sum_{N\varpi\le X}G(\varpi)\log N\varpi =\frac65c_*X^{5/6}+o(X^{5/6}),$$

with the same primary-prime convention.

Converting to rational primes

For a split rational prime $p=\varpi\bar\varpi$, the manuscript records

$$G(\varpi)+G(\bar\varpi)=\frac{S_p}{\sqrt p}.$$

Inert prime ideals have norm $p^2$, and their total contribution below norm $X$ is $O(\sqrt X)$, negligible here. Therefore the rational-prime consequence is

$$\sum_{\substack{p\le X\\p\equiv1\pmod3}} \frac{S_p}{2\sqrt p} \sim\frac35c_*\frac{X^{5/6}}{\log X} =\frac{(2\pi)^{2/3}}{5\Gamma(2/3)} \frac{X^{5/6}}{\log X}.$$

The factor of two comes from counting both conjugate Eisenstein primes versus the rational-prime summand normalized by $2\sqrt p$. The manuscript explicitly notes conflicting printed constants in two historical formulations and identifies its theorem with the all-primary-prime formulation. The formulas above preserve its stated conversion rather than silently identifying those conventions.

Fixed angular modes and higher powers

Appendix A claims that, for each fixed nonzero integer $\ell$,

$$\sum_{N\varpi\le X}G(\varpi) \left(\frac{\varpi}{|\varpi|}\right)^\ell =o_\ell\!\left(\frac{X^{5/6}}{\log X}\right).$$

Together with the cubic Gauss-sum identity, this gives cancellation at the same scale for every fixed power $G(\varpi)^k$ with $3\nmid k$ and $|k|>1$. The first powers $k=\pm1$ retain the bias. Nonzero multiples of three are not included in this higher-power conclusion.

5. Why the advance matters

Provisional importance: 3/5. Taken as stated, this removes a major conditional hypothesis from a classical first-moment prediction and proves it at a sharp cutoff. It identifies a subtle arithmetic bias that ordinary equidistribution cannot detect.

There is no contradiction between the two scales. The main term divided by the number of prime ideals is of order $X^{-1/6}$, so the average still tends to zero. Yet the accumulated sum has a positive and precisely predicted asymptotic.

The methodological contribution is also focused: the paper obtains stronger cancellation for structured prime sequences without claiming a generally stronger cubic large sieve for arbitrary coefficients. Its rating reflects a substantial advance in exponential sums and analytic number theory, with several complementary moments still outside the theorem.

6. The main idea

Compare with a simple arithmetic model

The analytic target is to compare $G(\varpi)$, after summation, with

$$c_*(N\varpi)^{-1/6}.$$

This is not a pointwise approximation: $|G(\varpi)|=1$, while the model tends to zero. It is an average model for the small directional bias.

The prime ideal theorem and partial summation give

$$\sum_{N\varpi\le X}(N\varpi)^{-1/6} \sim\frac65\frac{X^{5/6}}{\log X}.$$

The exponent $5/6$ and the factor $6/5$ thus arise naturally from integrating a weight of degree $-1/6$ against prime density. The difficult task is proving that the Gauss-sum average matches that model closely enough.

Preserve the cube-frequency main term

Prime sums are decomposed into sums with one main variable and bilinear sums involving products. Voronoi summation transforms the first kind. For bilinear sums, Poisson summation introduces dual frequencies.

The nonzero frequencies that are cubes yield the model term. The corrected dispersion method subtracts it explicitly before estimating the remaining frequencies. Existing large sieves handle most ranges, but two configurations of conductor and length require additional work.

The manuscript uses moments at several lengths to move a hypothetical large contribution into a range where a Gram-matrix estimate is stronger. A Gram matrix records pairwise inner products, so control of its off-diagonal entries limits how coherently a family can align. These estimates apply to smooth convolutions of prime sequences with independent coefficients, not arbitrary coefficient arrays.

Keep independence through the prime decomposition

A decomposition chosen by a least-prime condition can couple the two variables, invalidating an estimate that requires independent coefficients. The paper instead orders prime norm bins and stops according to counts in the final bin. A binomial weight separates the two sides exactly.

The remaining narrow transition involves three primes of nearly equal norm. Their sparsity supplies the extra saving needed there. This is one reason the structure of prime coefficients matters.

Remove the smoothing

A smoothed cutoff is analytically convenient because its Mellin transform decays quickly. Approximating a sharp cutoff more finely introduces larger imaginary parts of the Mellin variable, often called heights. The paper must control heights up to slightly beyond $X^{1/6}$.

At low heights it compares the Gauss sum with its model. At large heights it estimates the Gauss-sum term directly. Localizing both norm variables improves the diagonal contribution near a product boundary; away from that boundary, oscillation in the Mellin height produces the saving.

Combining these ranges permits removal of the smooth weight. Appendix A keeps the additional fixed angular factors through the same framework, with constants allowed to depend on the mode.

7. Scope, limitations, and open questions

Every angular mode is fixed before $X$ grows. The theorem does not give uniformity for a mode that increases with $X$, nor automatically imply equally strong results in angular sectors shrinking with $X$.

The complementary higher-moment prediction is only partly covered. Nonzero powers divisible by three require stronger information about angular Hecke prime sums. The little-oh error does not specify a power-saving rate.

The main term is a collective bias; it says nothing like $G(\varpi)\approx c_*(N\varpi)^{-1/6}$ for each prime. It also does not prove the Generalized Riemann Hypothesis or a stronger general cubic large sieve. The unconditional claim comes from estimates tailored to the structured families in this first-moment problem.

The normalization caveat is mathematical rather than cosmetic. Counting one rational prime, both conjugate prime ideals, or a differently scaled real cubic sum changes the constant. Any comparison with another formulation must carry those factors explicitly.

8. Sources and further reading

Theorem 1.1 states the primary-prime first moment. Section 1.2 gives the rational-prime conversion and its normalization caveat. Theorem A.1 and equation (A.14) specify the fixed-angular and higher-power consequences.

Dunn–Radziwiłł’s paper linked in section 3 is the principal analytic predecessor. The manuscript’s references identify Kummer, Patterson, Heath-Brown and Yoshimoto for the historical and metaplectic inputs. Optional background on Gauss sums and Eisenstein integers helps visualize the characters and complex prime lattice.

Source manuscripts

• M1. OpenAI, An unconditional first moment for cubic Gauss sums (September 25, 2026).

Sources consulted on 10 October 2026 at repository revision fd4aeeb. Download the manuscript citations.