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Critical Zeros and Unconditional Mean Value Theorems for twisted $\hbox{PGL}(2)$ and $\hbox{PGL}(3)$ $\mathrm{L}$ -functions

This paper establishes unconditional mean value theorems and proves that at least $1/9$ of the zeros of twisted $\mathrm{PGL}(2)$ and $\mathrm{PGL}(3)$ $L$ -functions lie on the critical line by developing a refined, uniform Asymptotic Large Sieve that avoids reliance on the Generalized Ramanujan Conjecture.

Original authors: Brian Conrey, Chung-Hang Kwan, Yongxiao Lin, Caroline L. Turnage-Butterbaugh

Published 2026-07-02

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CC BY 4.0

Original authors: Brian Conrey, Chung-Hang Kwan, Yongxiao Lin, Caroline L. Turnage-Butterbaugh

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: Finding Hidden Patterns in a Cosmic Symphony

Imagine the universe of numbers is a massive, complex symphony. In this symphony, there are specific musical notes called L-functions. These aren't just random sounds; they encode deep secrets about how prime numbers (the building blocks of arithmetic) are distributed.

Mathematicians have long suspected that all the "critical" notes in this symphony (the zeros of these functions) lie on a specific, perfect line called the critical line. This is known as the Riemann Hypothesis. However, proving that every note lies on this line is like trying to prove a perfect melody exists without being able to hear the whole song at once. It's currently impossible.

So, instead of trying to prove all notes are perfect, the authors of this paper asked a more modest question: "What percentage of these notes are actually on the perfect line?"

The Challenge: A Musical Wall

For a long time, mathematicians could easily count the perfect notes for simple instruments (like the Riemann zeta function, which is like a solo violin). They found that at least 40% of the notes were perfect.

But when they tried to apply the same counting method to more complex instruments (specifically, PGL(3) functions, which are like a full orchestra playing a chaotic, multi-layered piece), they hit a wall. The math required to count these notes relied on assumptions about the orchestra's behavior that hadn't been proven yet. It was like trying to count the notes in a symphony by assuming the musicians were playing perfectly, without actually listening to them.

The Solution: The "Asymptotic Large Sieve"

The authors developed a new, refined tool called the Asymptotic Large Sieve.

The Analogy:
Imagine you have a giant bucket of mixed sand and gold flakes (the zeros of the L-functions). You want to separate the gold (the zeros on the critical line) from the sand.

Old Method: You tried to pick out the gold flakes one by one. This worked for small buckets (simple functions) but failed for the massive, chaotic buckets (complex functions) because the sand was too fine and the gold too hidden.
New Method: The authors built a super-fine, intelligent sieve. Instead of picking flakes out, they shook the whole bucket in a very specific, rhythmic way (averaging over a family of functions). This shaking causes the gold flakes to clump together in a predictable pattern, while the sand settles differently.

By using this "shaking" technique, they could count the gold flakes without needing to know the exact behavior of every single grain of sand.

The Key Breakthroughs

1. The "Unconditional" Victory
The most exciting part of this paper is that they did this without relying on unproven guesses.

The Problem: Previous attempts to count these notes for the "orchestra" (PGL(3)) required a "mild assumption" that the musicians were playing perfectly (the Generalized Ramanujan Conjecture).
The Fix: The authors realized that by averaging over a huge family of different "versions" of the symphony (twisting the functions with different characters), the messy parts canceled each other out naturally. They didn't need to assume the musicians were perfect; the math worked out on its own.

2. The Results: How Many Notes are Perfect?

For the Solo Violin (PGL(2)): They confirmed that at least 1/3 of the notes are on the critical line. This matches the best-known result for the simplest functions, but now it's proven for this specific family of complex functions.
For the Full Orchestra (PGL(3)): They proved that at least 1/9 of the notes are on the critical line.
- Note: 1/9 might sound small, but in the world of these complex functions, finding any guaranteed percentage is a massive breakthrough. It proves that the "perfect line" isn't just a myth; it actually holds a significant portion of the music.

3. The "Mollifier" (The Noise-Canceling Headphones)
To make this work, they used a mathematical device called a mollifier.

Analogy: Imagine trying to hear a whisper in a noisy room. You put on noise-canceling headphones that are tuned to cancel out the specific background noise.
In the paper, the mollifier is a special polynomial that "cancels out" the messy, unpredictable parts of the L-functions, leaving only the clean, countable parts. The authors had to design a very specific, delicate version of these headphones to handle the complexity of the PGL(3) orchestra.

Why This Matters (According to the Paper)

The paper doesn't claim to solve the Riemann Hypothesis or predict stock markets. Instead, it achieves two main things:

It breaks a barrier: It shows that the powerful "Levinson's method" (the technique for counting zeros) can be applied to complex, degree-3 functions without needing unproven assumptions.
It provides a new tool: The "Asymptotic Large Sieve" they refined is a flexible tool. The authors suggest it can be used to solve other difficult problems in number theory, acting as a new kind of "mathematical microscope" for looking at the distribution of numbers.

Summary

Think of this paper as a group of engineers who figured out how to count the perfect notes in a chaotic, 100-instrument orchestra without needing to know if every musician was playing perfectly. They built a special "shaking machine" (the sieve) that sorted the music by itself. They proved that even in this chaos, at least 1 out of every 9 notes is perfectly in tune, and for smaller ensembles, 1 out of every 3. This is a major step forward in understanding the hidden order of the universe of numbers.

Technical Summary: Critical Zeros and Unconditional Mean Value Theorems for Twisted PGL(2) and PGL(3) L-functions

Problem Statement
The paper addresses two central problems in analytic number theory: establishing the proportion of zeros of automorphic $L$ -functions lying on the critical line and deriving unconditional asymptotic formulas for their mean values. Specifically, the authors investigate the family of $L$ -functions $L(s, \Pi \times \chi)$ , where $\Pi$ is a cuspidal automorphic representation of $PGL_r(\mathbb{A}_{\mathbb{Q}})$ for $r \in \{2, 3\}$ , and $\chi$ ranges over primitive Dirichlet characters of conductor up to $Q$ .

A primary obstacle in extending Levinson's method to higher-degree $L$ -functions (specifically degree 3) is the reliance on unproven bounds toward the Generalized Ramanujan Conjecture (GRC). Previous applications of the Asymptotic Large Sieve (ALS) to automorphic forms often required the bound $O_\epsilon(n^\epsilon)$ for Dirichlet coefficients, which is only known under GRC. For degree 3, obtaining an asymptotic formula for the sixth moment of the Riemann zeta function is widely considered inaccessible by current methods, making the direct application of Levinson's method to individual degree-3 $L$ -functions infeasible. This work seeks to overcome these barriers by studying the family of twisted $L$ -functions, exploiting averaging to achieve unconditional results without assuming GRC.

Methodology
The authors develop a refined, flexible, and uniform version of the Asymptotic Large Sieve (ALS) tailored for automorphic $L$ -functions. The core of the argument involves analyzing the shifted twisted mean square:
$M_{\alpha, \beta}(h, k; \Pi) = \sum_{(q, q_\Pi)=1} W\left(\frac{q}{Q}\right) \sum_{\chi}^* \chi(h)\chi(k) \int_{\mathbb{R}} \eta(t) L(1/2+\alpha+it, \Pi \times \chi) L(1/2+\beta-it, \tilde{\Pi} \times \chi) \left(\frac{h}{k}\right)^{-it} dt$
where the sum is over primitive characters $\chi \pmod q$ and the integral is weighted by a smooth test function $\eta$ .

The technical execution involves several sophisticated steps:

Divisor Switching and Orthogonality: The authors apply the Hybrid Large Sieve (HLS) and orthogonality relations for Dirichlet characters three times. They employ a "divisor switching" technique to separate variables and reduce the conductor size, splitting the sum into diagonal ( $D$ ), off-diagonal ( $L$ ), and off-diagonal ( $U$ ) components based on the size of a parameter $C$ .
Handling Arithmetic Complexity: A significant portion of the work is dedicated to managing the arithmetic of the automorphic coefficients $\lambda_\Pi(n)$ $λ_{Π} (n)$ without assuming GRC. The authors introduce a "Condition ( $\Lambda$ $Λ$ )" on the Dirichlet polynomial coefficients and a "Hypothesis ( $\Pi_4$ $Π_{4}$ )" regarding the fourth moment of the coefficients of $\Pi$ $Π$ .
- For $PGL(2)$, Hypothesis ( $\Pi_4$ ) is known unconditionally (Moreno–Shahidi).
- For $PGL(3)$, Hypothesis ( $\Pi_4$ ) is assumed for non-self-dual cases but is known unconditionally for self-dual representations (Jiang–Lü).
GCD Sums and Bilinear Estimates: To bound the off-diagonal terms without the full GRC, the authors utilize deep estimates for GCD sums (specifically the Dyer–Harman bound) and bilinear forms. This allows them to bypass the need for sharp pointwise bounds on $\lambda_\Pi(n)$ that would otherwise be required.
Hecke Algebra Computations: The paper extensively uses symbolic computation (Mathematica) to handle elaborate Hecke algebra relations and Euler product expansions. This is necessary to derive explicit formulas for local factors and to verify holomorphy of arithmetic factors in regions where GRC is not assumed.
Levinson's Method: Once the mean value theorem is established, the authors apply Levinson's method using a mollifier constructed from the coefficients of the inverse $L$ -function. This involves calculating the proportion of zeros on the critical line based on the asymptotic behavior of the mean square.

Key Contributions and Results

Unconditional Positive Proportion for PGL(3): The paper proves that for a self-dual cuspidal automorphic representation $\Pi_0$ of $PGL(3)$, at least $1/9 - O(\epsilon)$ of the zeros of $L(s, \Pi_0 \times \chi)$ lie on the critical line as $Q \to \infty$ . This is the first unconditional result of its kind for a family of $PGL(3)$ $L$ -functions. For non-self-dual $\Pi$ , the result holds under Hypothesis ( $\Pi_4$ ).
Improved Results for PGL(2): The authors obtain corresponding results for $PGL(2)$, showing that at least $1/3 - O(\epsilon)$ of the zeros lie on the critical line. These results are fully unconditional, quantitatively stronger than previous attempts, and appear to be new in this specific context of twisted families.
New Asymptotic Formula: The paper establishes a new asymptotic formula with a power-saving error term for the mean square of $L(s, \Pi \times \chi)$ $L (s, Π \times χ)$ multiplied by a Dirichlet polynomial with arbitrary coefficients. This holds in the hybrid range $Q^\epsilon \le T \le Q^{1/3-\epsilon}$ $Q^{ϵ} \leq T \leq Q^{1/3 - ϵ}$ .
- When $T = Q^\epsilon$ , the Dirichlet polynomial can have length $\theta < 1/2 - \epsilon$ .
- When $\theta = 0$ , the error term is $O_\epsilon(Q^{7/4+\epsilon})$ .
Evidence for CFKRS Conjectures: The derived asymptotic formula provides evidence for the Conrey–Farmer–Keating–Rubinstein–Snaith (CFKRS) conjectures regarding large twists and large vertical shifts.
Refined Asymptotic Large Sieve: The work develops a version of the ALS that does not require unproven progress toward GRC. It demonstrates that by carefully analyzing the arithmetic structures of the sums (specifically using GCD sums and bilinear estimates), one can obtain unconditional results for degree 3 $L$ -functions.

Significance
The authors claim that this work resolves a long-standing difficulty in applying Levinson's method to higher-degree automorphic $L$ -functions. By moving from individual $L$ -functions to families and utilizing the specific arithmetic of the twists, they bypass the need for the Generalized Ramanujan Conjecture. The paper highlights that the "diagonal" terms in the moment calculation are sufficient to determine the numerical proportion of zeros ( $1/9$ ), while the "off-diagonal" analysis, though subtle, can be controlled unconditionally using the refined sieve techniques and GCD sum estimates.

The use of symbolic computation to handle the intricate Hecke algebra relations for $PGL(3)$ is noted as a crucial technical enabler, allowing the authors to verify the holomorphy of complex arithmetic factors that would be intractable by hand. The results provide a robust framework for future applications in analytic number theory, including simultaneous large values of $L$ -functions and zero-density estimates, without relying on unproven conjectures.