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On the exponent of distribution for convolutions of $\mathrm{GL(2)}$ coefficients to smooth moduli

The paper establishes that the exponent of distribution for the convolution of Hecke eigenvalues of a holomorphic cusp form with the constant function 1 in arithmetic progressions reaches $\frac{1}{2}+\frac{1}{46}$ when the modulus is square-free.

Original authors: Rongjie Yin

Published 2026-05-13

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Rongjie Yin

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

Imagine you are a detective trying to find a specific pattern in a massive, chaotic crowd of numbers. In the world of mathematics, these numbers are called "integers," and they often hide in "arithmetic progressions"—which is just a fancy way of saying numbers that share a common remainder when divided by a specific number (like all numbers that leave a remainder of 1 when divided by 5).

The paper by Rongjie Yin is about how well we can predict the behavior of a very specific type of number pattern when we look at them through these "remainder filters."

Here is the breakdown of the paper using simple analogies:

1. The Main Character: The "Hecke Eigenvalue"

Think of the numbers in this study as a special kind of musical note generated by a complex instrument (a "holomorphic cusp form"). These notes are called Hecke eigenvalues ( $\lambda_f(n)$ ). They are not random; they follow strict, hidden rules.

The author is interested in a "convolution" of these notes. Imagine taking two streams of these musical notes and mixing them together to create a new, louder stream of sound. This new stream is called $\lambda_f \ast 1$ .

2. The Problem: The "Noise" in the Crowd

Mathematicians want to know: If I look at all the numbers up to a huge limit ( $X$ ) that share a specific remainder (say, remainder $a$ when divided by $q$ ), do the "notes" in that group balance out evenly?

The Expectation: If you pick a random group of numbers, the notes should be distributed fairly evenly among all possible remainders.
The Reality: Sometimes, the notes clump together in certain groups, creating "noise" or "bias."

The goal is to measure how large the "remainder filter" ( $q$ ) can be before the pattern breaks down. If the filter is too big, the noise becomes too loud to hear the pattern.

3. The "Exponent of Distribution": The Size of the Filter

The paper introduces a concept called the exponent of distribution. Think of this as a "zoom level" or a "filter size."

If the exponent is low, we can only look at small groups (small filters) and still see the pattern clearly.
If the exponent is high, we can look at massive groups (huge filters) and still see the pattern clearly.

The higher the exponent, the better the mathematician's prediction is.

4. The Previous State of Affairs

Before this paper, mathematicians had different "zoom levels" depending on the type of number they were studying:

For simple numbers (like the divisor function), they could zoom in quite far.
For the specific "GL(2) notes" (the Hecke eigenvalues) mentioned in this paper, the best previous record was set by Kowalski, Michel, and Sawin. They proved they could see the pattern clearly if the filter size was up to a certain limit (roughly the square root of the total numbers, plus a tiny bit more).

5. The Breakthrough: Smoother Filters

The author, Rongjie Yin, asks: "What if we only look at filters that are square-free?"

Analogy: Imagine a sieve. A "square-free" sieve has holes that don't repeat in a square pattern (like 4, 9, 16). It's a "cleaner" sieve.
The Result: By restricting the problem to these "cleaner" square-free filters, Yin manages to push the "zoom level" higher than ever before.

The New Record:
Yin proves that for these specific musical notes, the pattern holds true even when the filter is as large as $X^{1/2 + 1/46}$ .

The previous best was roughly $X^{1/2 + 1/102}$ .
The new result ( $1/46$ ) is a significant improvement over the old one ( $1/102$ ).

6. How Did They Do It? (The Tools)

To achieve this, the author used a "separation of oscillation" technique.

The Metaphor: Imagine trying to hear a specific instrument in a noisy orchestra. The notes are "oscillating" (vibrating) very fast.
The Strategy: Instead of trying to listen to the whole orchestra at once, the author uses a mathematical "separation" technique to isolate the vibrations.
The Tools: They used a "q-van der Corput method" (a way of smoothing out the rough edges of the numbers) and combined it with other classic tools like the "Poisson summation formula" (which is like translating a sound wave into a frequency spectrum to see what's really there).

Summary

In simple terms, this paper is a victory in the game of "Number Pattern Prediction."

The Goal: Predict how musical notes (Hecke eigenvalues) are distributed in groups.
The Obstacle: The groups were getting too big, and the predictions were getting fuzzy.
The Solution: By focusing on "clean" groups (square-free moduli) and using advanced separation techniques, the author sharpened the prediction.
The Result: We can now predict the pattern accurately for much larger groups than was previously possible.

The paper does not claim this will immediately change how we build bridges or cure diseases; it is a pure mathematical achievement that pushes the boundaries of our understanding of how numbers behave in the deep structure of the universe.

Technical Summary: On the Exponent of Distribution for Convolutions of GL(2) Coefficients to Smooth Moduli

Problem Statement
The paper addresses the problem of determining the exponent of distribution for the convolution of Hecke eigenvalues of a holomorphic cusp form with the constant function 1, denoted as $\lambda_f * 1$ . Specifically, the author investigates the asymptotic behavior of the sum
$\sum_{\substack{n \le X \\ n \equiv a \pmod q}} (\lambda_f * 1)(n)$
compared to the expected main term $\frac{1}{\phi(q)} \sum_{\substack{n \le X \\ (n,q)=1}} (\lambda_f * 1)(n)$ . The goal is to establish the largest possible exponent $\theta$ such that the error term is bounded by $X q^{-1} (\log X)^{-A}$ for all moduli $q \le X^{\theta - \varepsilon}$ . While the Generalized Riemann Hypothesis predicts $\theta = 1/2$ , and the Bombieri-Vinogradov theorem confirms this on average, the specific case for convolutions of GL(2) coefficients over square-free moduli has been a subject of recent refinement.

Methodology
The proof relies on a combination of analytic number theory techniques, specifically tailored to handle bilinear sums involving GL(2) coefficients and hyper-Kloosterman sums over square-free moduli.

Decomposition and Smoothing: The error term is decomposed using a smooth partition of unity. The problem is reduced to estimating bilinear sums of the form:
$S(M', L') = \sum_{m, l} \lambda_f(m) V^*(m) W^*(l) \text{Kl}_3(aml; q)$
where $\text{Kl}_3$ denotes the hyper-Kloosterman sum of order 3, and $V^*, W^*$ are smooth cutoff functions.
Separation of Oscillation: A key strategy employed is the "separation of oscillation" technique. This allows the authors to decouple the oscillatory components of the sum, facilitating the application of bounds for bilinear forms.
Summation Formulas: The proof utilizes the Poisson summation formula and the GL(2) Voronoi summation formula to transform the original sums. This transformation shifts the problem into the dual domain, introducing dual parameters $M^* = q^2/M$ and $L^* = q/L$ .
Bilinear Form Estimates: The core of the argument involves establishing non-trivial upper bounds for bilinear forms with GL(2) coefficients. The author proves three specific lemmas (Lemma 2.1, 2.2, and 2.3) that bound sums involving $\lambda_f(m)$ and $\text{Kl}_3(aml; q)$ .
- Lemma 2.1 provides bounds using the separation of oscillation technique.
- Lemma 2.2 applies the exponent pair method (specifically using the pair $(1/2, 1/2)$ ) to estimate sums with general coefficients.
- Lemma 2.3 handles the case where one variable is fixed.
Optimization: By analyzing the exponents of the moduli $q$ in the resulting bounds (denoted as $\sigma(\mu', \nu')$ ), the author determines the optimal range for the parameters $\mu'$ and $\nu'$ (where $M' \approx q^{\mu'}$ and $L' \approx q^{\nu'}$ ). The proof demonstrates that for specific ranges of these parameters, the exponent of the error term remains strictly less than 1, allowing for a saving over the trivial bound.

Key Contributions and Results
The primary result of the paper is Theorem 1.1, which establishes a new exponent of distribution for $\lambda_f * 1$ when the modulus $q$ is square-free.

Main Theorem: For any $\varepsilon > 0$ , $A > 0$ , and square-free integer $q$ satisfying
$q \le X^{\frac{12}{23} - \varepsilon} = X^{\frac{1}{2} + \frac{1}{46} - \varepsilon},$
the asymptotic formula holds with the error term:
$\sum_{\substack{n \le X \\ n \equiv a \pmod q}} (\lambda_f * 1)(n) - \frac{1}{\phi(q)} \sum_{\substack{n \le X \\ (n,q)=1}} (\lambda_f * 1)(n) \ll \frac{X}{q} (\log X)^{-A}.$
This improves upon the previous best result for prime moduli by Kowalski, Michel, and Sawin, which achieved an exponent of $1/2 + 1/102$ .
Technical Achievement: The paper successfully extends the methods used for prime moduli to the more general case of square-free moduli. The author notes that the algebraic geometry methods used in previous works for prime moduli do not directly extend to smooth moduli. Instead, this paper employs the $q$ -van der Corput method and specific bilinear form estimates to overcome this limitation.

Significance and Claims
The paper claims that the exponent $1/2 + 1/46$ is admissible for square-free moduli, representing a significant improvement over the $1/2 + 1/102$ bound previously known for prime moduli. The author emphasizes that the key input is the estimate for bilinear sums with GL(2) coefficients obtained via the separation of oscillation technique.

The work situates itself within the broader context of the distribution of arithmetic functions, drawing parallels to results for the divisor function $d_3(n)$ (where $1/2 + 1/46$ was previously achieved for prime moduli by Fouvry, Kowalski, and Michel). The paper asserts that by focusing on square-free moduli and utilizing the $q$ -van der Corput method, it achieves a larger exponent than the prime-modulus case treated in [8], thereby advancing the understanding of the distribution of GL(2) coefficients in arithmetic progressions.