🔢 mathematics

On the second moment and non-vanishing of central values of Hecke $L$ -functions of $r$ -th order characters

This paper establishes asymptotic formulas for the first and second twisted moments of $r$ -th order Hecke $L$ -functions over global fields containing the $2r$ -th roots of unity using multiple Dirichlet series, thereby proving that a positive proportion of their central values are non-vanishing for families of square-free and $r$ -th power-free ideals.

Original authors: Adrian Diaconu, Bogdan Ion, Vicenţiu Paşol, Alexandru A. Popa

Published 2026-07-30

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Adrian Diaconu, Bogdan Ion, Vicenţiu Paşol, Alexandru A. Popa

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 the universe of numbers not as a static list of 1, 2, 3, but as a vast, humming orchestra. In this orchestra, certain patterns called "L-functions" act like the sheet music for specific families of notes. Mathematicians have long been obsessed with the "central values" of these songs—the specific pitch played right in the middle of the scale. Why? Because whether a note is silent (zero) or loud (non-zero) at this exact spot tells us deep secrets about the structure of numbers themselves, much like how a single missing instrument can change the entire feeling of a symphony. For decades, we've known how to predict the volume of these songs for simple, "quadratic" families (like the square roots of numbers), but when the music gets more complex—requiring "r-th order" roots, like cube roots or fourth roots—the sheet music becomes a tangled mess of invisible threads. The big question has been: even in this chaotic, high-order music, do we ever hear a note? Or do the songs just fade into silence?

This paper, written by Adrian Diaconu, Bogdan Ion, Vicențiu Pașol, and Alexandru A. Popa, steps into this chaotic concert hall to answer that question for a wide range of complex musical families. The authors focus on "Hecke L-functions," which are the musical scores for numbers that live in fields containing special "roots of unity" (think of these as the geometric points that make a circle complete, like the corners of a perfect hexagon or octagon). Specifically, they look at families of these functions where the complexity is defined by an integer $r$ (where $r$ is 3 or higher). The team's main goal was to figure out the "second moment" of these songs—a statistical measure of their average loudness—and use that to prove that a significant number of these songs are actually playing a note at the center, rather than being silent.

To do this, the authors didn't just listen to the music; they built a new kind of "super-microscope" called a Multiple Dirichlet Series (MDS). Imagine trying to understand a single drop of water by looking at the whole ocean; that's what previous methods were like. The authors' MDS approach is like zooming in to see the individual molecules, allowing them to track how the "loudness" of these complex L-functions behaves as the numbers get bigger. They found that for these high-order families, the average loudness follows a predictable, smooth curve, and crucially, this curve never touches the floor.

The paper's findings are a mathematical proof, not just a guess. They established that for these families of numbers, a "positive proportion" of the central values are non-zero. In plain English, this means that if you pick a random song from this massive, complex family, there is a guaranteed, non-zero chance that it is playing a note at the center. It's not a rare fluke; it's a rule. For the specific case of cubic roots (where $r=3$ ), they proved that at least $1/12$ (minus a tiny error) of these songs are non-zero. For higher complexities ( $r \ge 4$ ), the guaranteed non-zero proportion is at least $1/(4r + 2)$ .

The authors also tackled a slightly different version of the problem involving "r-th power-free" ideals (numbers that don't have any factor repeated $r$ times). They found that even in this stricter category, a positive proportion of the songs are non-zero, and in fact, their method suggests an even higher proportion of non-zero notes than the simpler "square-free" version.

What makes this work particularly clever is how they handled the "noise." In the world of these complex numbers, there are "large sieve" estimates—mathematical tools that act like noise-canceling headphones—to filter out the static. The authors realized that their new MDS framework could see deeper into the structure of the music than previous methods, allowing them to capture a "second-order term" in their formulas. This is like hearing not just the main melody, but the subtle harmony underneath it. While they couldn't quite reach the theoretical limit of how quiet the music could get (a conjecture known as the "large sieve bound"), their method is a significant upgrade over the old ways, which were stuck in the noise.

In summary, this paper proves that for a vast family of complex number songs, the center is never silent. By using a sophisticated new mathematical lens, the authors showed that a guaranteed chunk of these musical scores always have a non-zero value, confirming that the deep structure of these numbers is robust and active, even in the most complicated scenarios. They didn't just guess; they calculated the exact formulas for the average volume and proved that the silence is impossible for a significant portion of the family.

Technical Summary: On the Second Moment and Non-Vanishing of Central Values of Hecke L-Functions of $r$ -th Order Characters

Problem Statement
The paper addresses the analytic behavior of Hecke $L$ -functions associated with primitive $r$ -th order residue symbols over global fields (specifically algebraic number fields) containing the $2r$ -th roots of unity, for integers $r \geq 3$ . The primary objectives are:

To establish asymptotic formulas for the first and second twisted moments of these $L$ -functions at the central point $s=1/2$ .
To deduce that a positive proportion of these central values are non-zero.
To extend these results to families of both square-free ideals and $r$ -th power-free ideals.

Prior to this work, unconditional positive-proportion non-vanishing results were limited to the quadratic ( $r=2$ ), cubic ( $r=3$ ), and quartic ( $r=4$ ) cases. Higher-order cases ( $r \geq 5$ ) lacked asymptotic formulas for the second moment in the square-free family, and existing methods for $r=3,4$ relied on specific large sieve estimates or case-by-case analyses that did not generalize uniformly.

Methodology
The authors employ the machinery of Multiple Dirichlet Series (MDS), a framework developed in previous works by the first author and collaborators (e.g., [Dia04], [Dia19], [DW21]). The approach differs fundamentally from the mollification techniques used in [Sou00] for the quadratic case or the specific large sieve bounds used in [DdFDS24] and [CdFD26] for cubic and quartic cases.

The core strategy involves:

Construction of MDS: The authors define a multiple Dirichlet series $Z_S(s; \psi_1, \psi_2, \rho)$ involving products of two Hecke $L$ -functions and a twisting character. This series is constructed via a "perfect" MDS framework, ensuring meromorphic continuation and a group of functional equations.
Decomposition via Sieving: To handle the restriction to square-free or $r$ -th power-free ideals, the authors utilize Möbius inversion. They express the restricted sums as linear combinations of "building block" MDS functions, denoted $\widetilde{Z}_{S \cup Sh}$ , which are constructed from twisted Kubota series (Dirichlet series of Gauss sums).
Analytic Continuation and Functional Equations: The building blocks are shown to satisfy a finite group of functional equations involving the Weyl group. The authors leverage the meromorphic continuation of Kubota's Dirichlet series (established in [BB06]) and the functional equations of the associated MDS to extend the domain of convergence.
Convexity Bounds and Large Sieve: To obtain asymptotic formulas, the authors derive convexity bounds for the MDS in the critical strip. A crucial step involves applying the large sieve inequality for $r$ -th order residue symbols (from [BGL14]) to bound the error terms. For the cubic case ( $r=3$ ), they utilize a refined bound from [DdFDS24] to improve the error term.
Contour Integration: The asymptotic formulas are derived by shifting the contour of integration in the complex plane, capturing the residues at the poles (specifically at $s_3=1$ and $s_3=1/2+1/r$ ).

Key Contributions and Results

Asymptotic Formulas for Moments:
- Second Moment (Theorem A): The paper establishes an asymptotic formula for the twisted second moment of $r$ -th order Hecke $L$ -functions over square-free ideals. The formula includes a main term of order $X$ and an error term of $O(X^{\frac{r+A_r}{r+1} + \epsilon})$ , where $A_r = 1/2$ for $r > 3$ and $A_3 = 1/3$ . For $r=3$ and $r=4$ , this recovers and refines previous results; for $r \geq 5$ , it provides the first such asymptotic formula.
- First Moment (Theorem B): An asymptotic formula for the twisted first moment is derived, capturing the second-order term $X^{1/2 + 1/r}$ specifically for $r=3$ . The constant for this term involves Whittaker-Fourier coefficients of theta functions on the $r$ -fold cover of $GL_2$ .
- Power-Free Families (Theorems D & E): The authors extend these results to $r$ -th power-free ideals. These formulas feature sharper error terms than the square-free case and include secondary terms for small $r$ (e.g., $r=3,4$ for the second moment).
Non-Vanishing Results:
- Square-Free Ideals (Theorem C): Using the mollification method applied to the first and second moments, the authors prove that a positive proportion of central values $L(1/2, \chi_a)$ are non-zero for square-free ideals. The proportion is at least $\frac{1}{12} - \epsilon$ for $r=3$ and $\frac{1}{4r+2} - \epsilon$ for $r \geq 4$ .
- Power-Free Ideals (Theorem F): A positive proportion of non-vanishing is established for the family of $r$ -th power-free ideals. The proportion is given by explicit constants involving Euler products, which are shown to be strictly better than the trivial bound derived from the square-free case.
Uniformity and Generality:
The method applies uniformly to any global field containing the $2r$ -th roots of unity. While the exposition focuses on number fields, the authors note that the framework admits direct analogues for function fields, where the Riemann Hypothesis could further sharpen the results.

Significance and Claims
The paper claims to provide the first unconditional asymptotic formulas for the second moment of $r$ -th order Hecke $L$ -functions for $r \geq 5$ . The significance lies in the uniformity of the approach: unlike previous methods that relied on specific large sieve estimates for low $r$ , this method utilizes the structural properties of Weyl group multiple Dirichlet series.

The authors highlight that their approach captures "deeper analytic information" invisible at the level of the fully sieved object. This is evidenced by the ability to detect second-order terms in the asymptotic expansion (e.g., for $r=3$ ) and the derivation of sharper error terms for $r$ -th power-free families through a delicate inductive argument.

The paper acknowledges limitations:

For $r=3$ , the error term in the second moment asymptotic formula falls just short of capturing the conjectured second-order term $X^{5/6}$ , a barrier linked to the current state of the cubic large sieve inequality.
The proportion of non-vanishing for $r=3$ is slightly smaller than that in [DdFDS24] because the authors did not optimize the mollifier parameter using a truncated Möbius sum to avoid increasing technical complexity.
For $r \geq 4$ , the coefficients of the secondary terms in the first moment involve Whittaker coefficients that are not well understood, limiting the explicit description of these terms.

In summary, the work extends the landscape of moment calculations for higher-order $L$ -functions, providing a robust framework for $r \geq 3$ and establishing unconditional non-vanishing results for families previously out of reach.

Technical Summary: On the Second Moment and Non-Vanishing of Central Values of Hecke L-Functions of rrr-th Order Characters

More like this

Technical Summary: On the Second Moment and Non-Vanishing of Central Values of Hecke L-Functions of $r$ -th Order Characters