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On the Density of Low Lying Zeros of a Large Family of Automorphic $L$ -functions

Assuming the Generalized Riemann Hypothesis, this paper generalizes recent techniques to establish the Katz-Sarnak density conjecture for higher centered moments and the two-level density of a large family of automorphic $L$ -functions over significantly extended ranges of test function supports, revealing that employing distinct test functions can further extend these validity ranges.

Original authors: Timothy Cheek, Pico Gilman, Kareem Jaber, Steven J. Miller, Marie-Hélène Tomé

Published 2026-08-18

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CC BY 4.0

Original authors: Timothy Cheek, Pico Gilman, Kareem Jaber, Steven J. Miller, Marie-Hélène Tomé

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

In the vast landscape of mathematics, there is a hidden order governing the distribution of prime numbers, the building blocks of arithmetic. For over a century, mathematicians have suspected that these numbers do not appear randomly, but rather follow a deep, rhythmic pattern that can be heard if one listens to the right frequency. This pattern is encoded in a special family of complex functions known as L-functions. These functions act like musical scores, where the "notes" are specific points on a graph called zeros. The most important of these zeros lie near the center of the score, and their arrangement holds the key to understanding how primes are spaced. A major conjecture in the field suggests that the way these zeros cluster is not unique to numbers, but mirrors the behavior of eigenvalues—essentially, the natural frequencies—of large, classical groups of matrices. This idea, known as the Katz-Sarnak philosophy, proposes a profound unity between number theory and the geometry of symmetry. While this connection has been observed in many cases, proving it for large families of these functions has remained a formidable challenge, particularly when trying to see the pattern clearly across a wide range of frequencies.

A team of researchers has now pushed the boundaries of this understanding, extending the range in which this connection is known to hold true. By developing a new method to analyze the density of these low-lying zeros, they have shown that the predicted patterns emerge more clearly than ever before. The team focused on a large family of automorphic L-functions, which are sophisticated generalizations of the functions associated with prime numbers. Their goal was to verify that the statistical distribution of the zeros near the center of these functions matches the predictions made by the Katz-Sarnak philosophy. Previous work had confirmed this match for a limited range of frequencies, but the new study significantly widens the window through which this agreement can be observed. The researchers achieved this by generalizing techniques used in recent years, allowing them to handle more complex scenarios involving multiple zeros simultaneously.

The core of their achievement lies in how they measured the "density" of these zeros. Instead of looking at a single zero at a time, the team analyzed groups of zeros together, studying what are called moments, which describe how these groups fluctuate around their average position. They proved that for a specific, expanded range of test functions, the statistical behavior of these groups aligns perfectly with the theoretical predictions. This is a significant step forward because it confirms that the underlying symmetry of these mathematical objects is robust, even when examined under more stringent conditions. The researchers found that for certain configurations, the range of agreement nearly doubles what was previously known. For instance, in one specific case involving three zeros, the range of frequencies where the pattern holds was extended from a limit of two-thirds to three-quarters. This improvement is not merely a matter of stretching a number; it represents a deeper penetration into the structure of these functions, revealing that the predicted order is more pervasive than earlier methods could detect.

To reach this conclusion, the team had to navigate a series of intricate technical hurdles. They began by converting a sum over the zeros of the functions into a sum over prime numbers, a standard technique that translates the problem into a language where powerful tools can be applied. However, this translation introduced a complication: the primes involved had to be distinct to avoid mathematical singularities. The researchers devised a clever combinatorial argument to reduce the problem to this case of distinct primes, effectively filtering out the messy overlaps that would otherwise obscure the signal. Once this was done, they applied a trace formula, a tool that connects sums over primes to sums over the spectrum of the functions themselves. This step allowed them to isolate the contributions from different parts of the mathematical landscape, separating the "holomorphic" and "discrete" parts from the more difficult "Eisenstein" series components.

The most challenging part of the work involved handling the Eisenstein series, which arise from a different type of symmetry and introduce competing signs that can cancel out or amplify each other in unpredictable ways. In previous studies, these terms were manageable only under strict limitations. The new team had to develop a way to handle these terms when multiple primes were involved, a situation that creates a complex interplay of signs. They discovered that by carefully analyzing the divisibility properties of the primes relative to the level of the functions, they could bound the contribution of these difficult terms. This allowed them to extend the range of validity for their results. They found that while the method hits a natural limit when trying to go even further—due to the inability to shift certain mathematical contours past a critical line—the range they achieved is a substantial improvement over the state of the art.

The implications of this work are twofold. First, it provides strong evidence that the Katz-Sarnak philosophy is a fundamental truth about these families of functions, holding true even when the test functions used to probe them are different from one another. This is the first time such a benefit has been demonstrated by choosing distinct test functions, suggesting that the tools used to explore these patterns can be more flexible than previously thought. Second, the techniques developed to handle the sums over multiple primes and the Eisenstein series have applications to other quantities in number theory that involve sums over primes. The researchers note that while their method has reached a natural limit for the current approach, the ability to extend the range of agreement by using different test functions opens a new avenue for future exploration. The work stands as a testament to the power of refining existing tools to uncover deeper layers of mathematical truth, confirming that the hidden symmetries of the number system are indeed as elegant and universal as the theory predicts.

Technical Summary: On the Density of Low Lying Zeros of a Large Family of Automorphic L-Functions

Problem Statement
The paper investigates the distribution of low-lying zeros (zeros near the central point $s=1/2$ ) of a large family of automorphic $L$ -functions associated with holomorphic cuspforms of weight $k$ and level $q$ . Specifically, it addresses the Katz-Sarnak density conjecture, which predicts that the statistical distribution of these zeros, when scaled by the analytic conductor, agrees with the distribution of eigenvalues near 1 of a classical compact group (in this case, the orthogonal group $O(N)$ ).

While previous work by Iwaniec, Luo, and Sarnak established this agreement for test functions with Fourier support in $(-2, 2)$ , and Baluyot, Chandee, and Li (BCL) extended this to $(-4, 4)$ for a family averaged over levels $q \asymp Q$ , the range of validity for higher-order statistics remained limited. The authors aim to generalize the BCL results to the $n$ -th centered moments of the one-level density and the two-level density, determining the maximal support $\sigma$ for the test functions' Fourier transforms for which the Katz-Sarnak predictions hold.

Methodology
The proof relies on a combination of analytic number theory techniques, spectral theory, and combinatorial arguments:

Explicit Formulas and Averaging: The authors utilize an explicit formula to convert sums over zeros into sums over primes. They employ a harmonic average over the family of forms $H_k(q)$ and an additional average over levels $q \asymp Q$ using a smooth weight function $\Psi$ .
Reduction to Distinct Primes: A critical step involves reducing the computation of the $n$ -th centered moment to a sum over $n$ distinct primes not dividing the level. This is achieved through a delicate combinatorial argument and induction on $n$ . The authors show that terms involving repeated primes or primes dividing the level contribute negligibly or can be bounded using specific decay properties, provided the support of the test functions is restricted.
Trace Formulae: To handle the sums over distinct primes, the authors apply the Petersson trace formula to remove the dependence on the level $q$ and transition to a spectral sum. Subsequently, the Kuznetsov trace formula is applied to convert the sum over primes into spectral terms (holomorphic forms, Maass forms, and Eisenstein series).
Spectral Bounding:
- Holomorphic and Maass Terms: The contributions from the discrete spectrum (holomorphic and Maass cuspforms) are bounded using the Kuznetsov trace formula and properties of the Hecke basis. The authors generalize techniques from BCL to handle higher moments, utilizing bounds on Bessel transforms and the quasi-orthogonality of Hecke eigenvalues.
- Eisenstein Series: The continuous spectrum (Eisenstein series) presents the most significant technical obstruction. The authors analyze the Fourier coefficients of Eisenstein series, which introduce competing sign influences in the exponents when $n \geq 2$ . Unlike the $n=1$ case, where contour shifting is straightforward, the presence of multiple primes with distinct divisibility conditions relative to the level prevents a uniform reduction of the exponent size. The authors split the primes into sets based on their divisibility by the level and the cusp parameters, analyzing principal and non-principal characters separately.
Handling Level Divisors: A novel technical challenge arises when $n \geq 2$ : the necessity to "add back" primes dividing the level $q$ after applying the trace formula. The authors demonstrate that for test functions with sufficiently restricted support, these terms can be bounded, though this imposes a limit on the achievable support $\sigma$ .

Key Contributions and Results
The paper establishes the following main results under the Generalized Riemann Hypothesis (GRH) for $L(s, f)$ , $L(s, \text{sym}^2 f)$ , and Dirichlet $L$ -functions:

$n$ -th Centered Moments: The Katz-Sarnak density predictions hold for the $n$ -th centered moment of the one-level density for test functions whose Fourier transform is supported in $(-\sigma, \sigma)$ , where:
$\sigma = \min\left\{ \frac{3}{2(n-1)}, \frac{4}{2n - 12 \nmid n} \right\}$
For the specific case of $n=3$ , this improves the previously best-known support from $\sigma = 2/3$ to $\sigma = 3/4$ .
Two-Level Density: The two-level density agrees with the Katz-Sarnak prediction for test functions with supports $\sigma_1 = 3/2$ and $\sigma_2 = 5/6$ . The sum of these supports is $7/3$ , which exceeds the previous best-known sum of supports ($2$).
New Phenomenon: The paper identifies a new phenomenon where choosing different test functions (with different supports) allows for an extension of the range in which the density conjecture is known to hold, beyond what is possible with identical test functions.

Significance and Limitations
The authors claim this work provides the first evidence that varying test functions can improve density results. The significance lies in extending the range of validity for the Katz-Sarnak conjecture in the context of large families of automorphic $L$ -functions, pushing the boundaries of current analytic techniques.

However, the paper acknowledges specific limitations:

Support Limit: The restriction $\sigma \leq \frac{3}{2(n-1)}$ is necessitated by the difficulty of bounding the terms arising from primes dividing the level when $n \geq 2$ . Removing this restriction would require fundamentally different techniques.
Eisenstein Series Obstruction: The limit on the support is also dictated by the behavior of Eisenstein series coefficients. The "competing sign influences" in the exponents of these terms prevent the contour from being shifted past the critical line, establishing a natural limit for the current methodology.
Recent Developments: The authors note that while their results were novel upon posting, subsequent work (CLL25) has improved upon several of these results, though differences remain, particularly regarding the benefit of non-identical test functions.

In conclusion, the paper advances the understanding of low-lying zeros by generalizing the methods of Baluyot, Chandee, and Li to higher moments and two-level densities, overcoming significant combinatorial and spectral obstacles to extend the known range of the Katz-Sarnak density conjecture.

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