🔢 mathematics

Quartic Gauss sums over primes and metaplectic theta functions

This paper improves upon Patterson's 1987 estimates for quartic Gauss sums over primes by introducing new Type-I and Type-II bounds that leverage the quadratic large sieve over $\mathbb{Q}(i)$ and Suzuki's evaluation of Fourier-Whittaker coefficients, while also proposing conjectures for the asymptotic behavior of related moments.

Original authors: Chantal David, Alexander Dunn, Alia Hamieh, Hua Lin

Published 2026-02-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Chantal David, Alexander Dunn, Alia Hamieh, Hua Lin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A Cosmic Game of Hide-and-Seek

Imagine the world of numbers as a vast, infinite ocean. In this ocean, there are special islands called primes (numbers like 2, 3, 5, 7, 11 that can't be divided evenly by anything else).

Mathematicians have long been fascinated by a specific game played on these islands called Gauss Sums. Think of a Gauss Sum as a "magic score" assigned to each prime number. This score isn't just a random number; it's a complex value that spins around a circle, like a hand on a clock.

For a long time, mathematicians knew how to calculate these scores for simple games (called "quadratic" sums). But when they tried to play a more complex version of the game (called "quartic" sums, involving the fourth power), the scores started behaving in a very mysterious way. They didn't seem to distribute evenly; instead, they seemed to have a slight "bias," clustering in certain spots on the clock face more often than others.

The Problem: The Old Map Was Outdated

In 1987, a mathematician named Patterson drew a map of this ocean. He estimated how "noisy" or "large" these magic scores could get when you added them up over many primes. His map said the noise could be as loud as $X^{19/20}$ (where $X$ is the size of the ocean you are looking at).

The authors of this paper (Chantal David, Alexander Dunn, Alia Hamieh, and Hua Lin) looked at Patterson's map and said, "We can do better." They wanted to prove that the noise is actually much quieter than previously thought.

The Solution: New Tools for a New Journey

To improve the map, the authors had to navigate through two different types of terrain, which they call Type-I and Type-II sums. Think of these as two different hiking trails leading to the same destination.

1. The Type-II Trail: The "Quadratic Sieve"

On this trail, the authors used a tool called the Quadratic Large Sieve.

The Analogy: Imagine you are trying to count how many fish are in a net. The net has holes of different sizes. The "Large Sieve" is a mathematical net that filters out the "noise" (the irrelevant numbers) so you can see the "fish" (the important patterns) clearly.
The Breakthrough: They used a sophisticated version of this sieve (developed by Onodera and Goldmakher-Louvel) specifically designed for the Gaussian numbers (a special type of number system involving $\sqrt{-1}$ ). This allowed them to filter out the noise much more efficiently than before, proving that the noise on this trail is only about $X^{5/6}$ .

2. The Type-I Trail: The "Metaplectic Telescope"

This trail was much trickier. It involved looking at the "residue" of a very complex mathematical object called a Metaplectic Theta Function.

The Analogy: Imagine trying to hear a whisper in a storm. The "Theta Function" is the storm, and the "residue" is the whisper. In simpler versions of this problem (the "cubic" case), mathematicians knew exactly what the whisper sounded like. But in this "quartic" case, the whisper was hidden.
The Breakthrough: The authors used a discovery by a mathematician named Suzuki. Suzuki found a secret rule that tells you how the whisper changes when you look at "square" numbers. By using this rule, the authors realized that the whisper (the mathematical term causing the bias) is actually much smaller than they thought. Instead of being loud, it's a soft hum of size $X^{3/4}$ .

The Result: A Sharper Map

By combining these two improved trails, the authors proved that the total noise in the system is bounded by $X^{5/6}$ (dominated by the Type-II trail) and specifically $X^{3/4}$ for the bias term.

Old Map (1987): Noise could be up to $X^{0.95}$ .
New Map (2026): Noise is only up to $X^{0.83}$ (and the bias is even smaller at $X^{0.75}$ ).

This is a significant improvement. It means our understanding of how these numbers behave is much more precise.

The Big Guess (Conjecture)

The paper ends with a bold guess (a conjecture). The authors believe that the "bias" they found (the slight clustering of the scores) isn't just a fluke; it's a fundamental law of the universe of numbers. They propose that the bias is exactly $X^{3/4}$ .

They admit that proving this exact number is incredibly difficult—so difficult that even the most powerful assumptions in mathematics (like the Generalized Riemann Hypothesis) might not be enough to solve it right now. But they have laid the groundwork for future explorers to try.

Summary in One Sentence

The authors used advanced mathematical nets and telescopes to prove that the mysterious "scores" assigned to prime numbers are much more orderly and less chaotic than we thought in 1987, bringing us closer to understanding a deep, hidden pattern in the fabric of numbers.

Technical Summary: Quartic Gauss Sums Over Primes and Metaplectic Theta Functions

Problem Statement
The paper addresses the asymptotic behavior of sums involving normalized quartic Gauss sums over prime moduli in the Gaussian quadratic field $\mathbb{Q}(i)$ . Specifically, it seeks to improve upon the 1987 estimates of Patterson regarding the sums:
$\sum_{c \in \mathbb{Z}[i]} \text{rg}_4(c) \left(\frac{c}{|c|}\right)^\ell \Lambda(c) R\left(\frac{N(c)}{X}\right)$
where $\text{rg}_4(c)$ is the normalized quartic Gauss sum, $\Lambda(c)$ is the von Mangoldt function, and $R$ is a smooth test function.

The authors note a significant distinction between the cubic and quartic cases. While Heath-Brown and Patterson established equidistribution for cubic Gauss sums and Patterson conjectured a main term of size $X^{5/6}$ for the first moment of cubic sums (a bias later confirmed under GRH), the quartic case presents unique analytic obstructions. The authors aim to refine the error term for the quartic case, moving beyond Patterson's bound of $O(X^{19/20})$ .

Methodology
The proof employs a combination of analytic number theory techniques adapted to the Gaussian integers and the theory of metaplectic forms. The core strategy involves the following steps:

Vaughan's Identity: The sum over primes is decomposed into Type-I and Type-II sums (multilinear sums) to isolate the arithmetic structure of the von Mangoldt function.
Type-II Estimates: For Type-II sums, the authors utilize the twisted multiplicativity of quartic Gauss sums, specifically the relation $\text{rg}_4(ab) \approx \text{rg}_4(a)\text{rg}_4(b)(\frac{a}{b})_2$ . This reduces the problem to estimating quadratic character sums. They apply the quadratic large sieve over $\mathbb{Q}(i)$ (due to Onodera and Goldmakher–Louvel) to bound these sums, achieving a bound of $O(X^{5/6+\varepsilon})$ .
Type-I Estimates and Metaplectic Forms: Type-I sums are more intricate. They involve Dirichlet series of the form $\psi^{(4)}_\beta(s, \nu, \ell)$ $ψ_{β}^{(4)} (s, ν, ℓ)$ , which are identified as Fourier-Whittaker coefficients of Kubota's metaplectic Eisenstein series on the 4-fold cover of $SL_2(\mathbb{Z}[i])$ $S L_{2} (Z [i])$ .
- The authors analyze the analytic properties of these series, including their meromorphic continuation and functional equations.
- A critical component is the evaluation of residues (poles) of these series. Unlike the cubic case where Patterson provided closed forms for all Fourier coefficients, the quartic case has undetermined coefficients. The authors rely on Suzuki's evaluation of coefficients at square indices, $\psi^{(4)}(a^2)$ , which provides a saving of $N(a)^{-1/4}$ .
- To handle the critical line, they employ an approximate functional equation and an average bound (Voronoi-type formula) combined with the quadratic large sieve. This yields an average bound that saves a factor of $A^{1/2}$ over the convexity bound for the level of distribution $A$ .

Key Contributions

Improved Bounds: The paper establishes a new upper bound for the sum of quartic Gauss sums over primes. The error term is improved from Patterson's $O(X^{19/20})$ to $O(X^{5/6+\varepsilon})$ for Type-II sums and $O(X^{3/4+\varepsilon})$ for Type-I sums. The final result for the total sum is $O(X^{5/6+\varepsilon})$ (dominated by the Type-II bottleneck) for sharp cutoffs, or $O(X^{3/4+\varepsilon})$ with smoothing in specific contexts, though the main theorem states a bound of $O(X^{5/6+\varepsilon})$ generally, with the Type-I contribution specifically noted as $O(X^{3/4+\varepsilon})$ .
Application of Suzuki's Results: The work explicitly leverages Suzuki's partial determination of Fourier-Whittaker coefficients for quartic theta functions (specifically at square indices) to derive the $X^{3/4}$ contribution from the pole term in Type-I sums.
Conjecture on Bias: The authors propose Conjecture 1.2, suggesting that the first moment of quartic Gauss sums over primes exhibits a bias of size $X^{3/4}$ . This is significantly smaller than the $X^{5/6}$ bias observed in the cubic case. The conjecture posits that for $\ell=0$ , the sum is asymptotic to $b_\beta X^{3/4}$ , while for $\ell \neq 0$ , the sum is $O(X^{1/2+\varepsilon})$ .

Results
Theorem 1.1 states that for a smooth function $R$ supported on $[1, 2]$ , and for any $\ell \in \mathbb{Z}$ , $\beta \in \{1, 1+\lambda^3\} \pmod 4$ , and $\varepsilon > 0$ :
$\sum_{\substack{c \in \mathbb{Z}[i] \\ c \equiv \beta \pmod 4}} \text{rg}_4(c) \left(\frac{c}{|c|}\right)^\ell \Lambda(c) R\left(\frac{N(c)}{X}\right) \ll_{\varepsilon, R} X^{5/6+\varepsilon} + X^{3/4+\varepsilon}|\ell|^{3/2+\varepsilon}$
as $X \to \infty$ . This represents a substantial improvement over the previous $X^{19/20}$ bound.

Significance and Limitations
The paper claims that the $X^{3/4}$ term arises from the pole of the metaplectic Eisenstein series, a phenomenon specific to the quartic case. The authors note that proving the conjectured asymptotic $X^{3/4}$ main term is currently out of reach, even under the Generalized Riemann Hypothesis (GRH), due to the difficulty in controlling the error terms in the Type-II sums (the $X^{5/6}$ bottleneck).

The authors highlight that extending these methods to $n$ -th order Gauss sums for $n \geq 5$ is obstructed by the fact that the relevant Dirichlet series in Type-I sums behave like $GL_{n-2}$ objects, leading to increased conductors and difficulty in error control. The paper refrains from attempting to remove the smoothing function $R$ to achieve a power-saving loss, opting instead to present the best possible exponent achievable with current technology and smoothing.

The work connects the distribution of Gauss sums to the spectral theory of metaplectic forms, specifically the residues of Eisenstein series on the 4-fold cover of $GL_2$ , reinforcing the deep link between number theoretic sums and automorphic forms.