Non-vanishing for cubic Hecke $L$-functions

This paper proves unconditionally that a positive proportion of cubic Hecke $L$-functions over the Eisenstein field do not vanish at the central point by establishing a new asymptotic formula for the mollified second moment, a result that overcomes the lack of a perfectly orthogonal large sieve bound in this unitary family through novel applications of Patterson's theta function, Heath-Brown's sieve, and a new Lindelöf-on-average bound.

The Gist

Imagine you are a detective trying to solve a mystery about a vast, invisible city called the Number Field. In this city, there are special buildings called L-functions. These buildings are like complex musical instruments; when you play a note (evaluate the function at a specific point), it might produce a sound (a non-zero value) or it might be completely silent (zero).

For a long time, mathematicians have had a hunch (a famous conjecture by Chowla) that these instruments never go silent at a specific, critical spot called the "central point." They believe every single one of these instruments plays a note. However, proving this for every single instrument is incredibly hard. So, instead of checking every building, mathematicians try to prove that at least a certain percentage of them are making noise.

This paper, written by Chantal David and her team, is a breakthrough in solving this mystery for a specific type of instrument: the Cubic Hecke L-functions.

Here is the story of how they did it, broken down into simple concepts:

1. The Challenge: A Noisy City with a Twist

The authors are looking at a specific neighborhood in their city (the Eisenstein integers, a grid of numbers involving complex roots of unity). They are interested in "cubic" characters.

• The Problem: In simpler neighborhoods (like the "quadratic" ones), the math behaves nicely, like a perfectly organized choir. But in this "cubic" neighborhood, the math is messy. The values of these instruments (called Gauss sums) spin around in a circle like a chaotic dance, making it very hard to predict if they will be silent or loud.
• The Goal: They wanted to prove that at least 14% of these cubic instruments are definitely playing a note (non-zero) at the central point. Before this paper, we only knew this was true for a tiny, almost invisible fraction of them, or only if we assumed some unproven theories (like the Generalized Riemann Hypothesis).

2. The Tool: The "Mollifier" (The Noise-Canceling Headphones)

To prove that an instrument is playing, you can't just listen to the raw sound; it's too chaotic. The authors use a mathematical tool called a mollifier.

• The Analogy: Imagine trying to hear a whisper in a hurricane. You can't just listen harder. Instead, you put on noise-canceling headphones (the mollifier) that are tuned to cancel out the background chaos and amplify the specific signal you care about.
• In math terms, the mollifier is a carefully constructed "filter" that, when multiplied by the L-function, smooths out the wild fluctuations. If the filtered result is large, it proves the original instrument wasn't silent.

3. The Big Breakthrough: The Second Moment

The authors didn't just look at the average sound (the "first moment"); they had to look at the variance or the "energy" of the sound (the "second moment").

• The Old Way: In previous studies of similar problems, mathematicians used a "perfect sieve" (a mathematical filter) that worked beautifully because the numbers were perfectly organized.
• The New Reality: In this cubic neighborhood, the "sieve" is imperfect. The numbers don't line up perfectly. The authors realized that trying to use the old, perfect methods would fail.
• The Innovation: They developed a brand-new way to calculate this "second moment" with incredible precision. They used:
  • Patterson's Theta Function: A deep, ancient map of the city's hidden geometry.
  • Heath-Brown's Large Sieve: A powerful net for catching numbers, adapted for this messy cubic dance.
  • A New Bound: They proved a new rule about how much "noise" (error) is allowed in their calculations, ensuring their answer is solid.

4. The Result: 14% of the City is Alive

By combining their new "noise-canceling" filter with their new way of calculating the energy of the system, they proved:

• Unconditionally: They didn't need to assume any unproven theories. The math stands on its own.
• The Proportion: At least 14% (specifically, $1/7$) of the cubic L-functions in their family are non-zero.

Why Does This Matter?

Think of it like this: For years, we thought this chaotic cubic neighborhood might be mostly silent (or we just couldn't prove it wasn't). This paper turns on the lights and shows us that a significant chunk of the city is actually alive and singing.

It's a major step forward because:

1. It breaks a barrier: It works for a family of numbers that behaves differently (and more chaotically) than the ones we've studied before.
2. It's unconditional: The proof is solid, not based on "what if" scenarios.
3. It opens the door: The new mathematical techniques they invented (especially how they handled the messy "second moment") can now be used by other mathematicians to solve similar mysteries in other chaotic number systems.

In summary: The authors built a new, high-tech filter to cut through the noise of a chaotic mathematical world. They proved that, contrary to the fear that these numbers might be silent, a healthy, positive percentage of them are definitely making music.

Technical Summary

1. Problem Statement

The paper addresses the non-vanishing problem for the family of Hecke $L$-functions associated with primitive cubic characters over the Eisenstein quadratic number field $\mathbb{Q}(\omega)$, where $\omega = e^{2\pi i/3}$.

Specifically, the authors investigate the central values $L(1/2, \chi_q)$ for characters $\chi_q$ defined by cubic residue symbols $\left(\frac{\cdot}{q}\right)_3$, where $q \in \mathbb{Z}[\omega]$ is squarefree and $q \equiv 1 \pmod 9$.

Context and Difficulty:

• Folklore Conjecture: It is conjectured (Chowla) that $L(1/2, \chi) \neq 0$ for all primitive Dirichlet characters.
• Previous Status: For cubic characters, unconditional results were previously limited to proving non-vanishing for a set of density zero (infinite subsets). Conditional results (assuming GRH) had established non-vanishing for a positive proportion.
• The Obstacle: Unlike quadratic families (symplectic) or general Dirichlet families (unitary), the cubic family does not satisfy a "perfectly orthogonal" large sieve bound. The presence of Gauss sums (which are equidistributed on the unit circle for orders $>2$) and the lack of a perfect large sieve bound make the standard mollification methods significantly harder to apply.

2. Methodology

The authors employ the method of mollified moments, specifically utilizing the first and second mollified moments to derive a lower bound for the proportion of non-vanishing $L$-values.

Key Technical Components:

1. Mollification: They introduce a mollifier $M(q)$ (a short Dirichlet polynomial) designed to dampen the large values of $L(1/2, \chi_q)$ and isolate the non-vanishing behavior.
2. Approximate Functional Equations: They use approximate functional equations to express $L(1/2, \chi_q)$ and $|L(1/2, \chi_q)|^2$ as sums involving cubic Gauss sums.
3. Poisson Summation: Applied to sums over the modulus $q$ to transform the problem into dual sums involving Gauss sums.
4. Patterson's Cubic Theta Function: A crucial innovation is the use of Patterson's evaluation of the Fourier coefficients of the cubic metaplectic theta function. This allows the authors to relate sums of cubic Gauss sums to the coefficients $\tau_3(\mu)$, which vanish unless $\mu$ is a specific type of cube.
5. Cubic Large Sieve: They utilize Heath-Brown's cubic large sieve. However, since the cubic sieve is not perfectly orthogonal (unlike the quadratic case), they cannot achieve the optimal fourth-moment bounds used in previous quadratic works. Instead, they derive new Lindelöf-on-average upper bounds for the second moment of cubic Dirichlet series.
6. Balanced vs. Unbalanced Regimes: The error term analysis requires splitting the sums into "balanced" and "unbalanced" regimes based on the lengths of the Dirichlet polynomials. The authors develop distinct strategies for each to optimize the error terms.

3. Key Contributions

The paper's primary contribution is the first unconditional asymptotic evaluation of the mollified second moment for a cubic family of $L$-functions with a power-saving error term.

• Asymptotic Formula for the Second Moment:
  They prove an asymptotic formula for the smoothed second moment:
  $$\sum_{q} \mu^2(q) |L(1/2, \chi_q)|^2 F\left(\frac{N(q)}{X}\right) = 2D \check{F}(0) X (\log X + C_1 + C_2 \frac{\check{F}'}{\check{F}}(0)) + O_{F,\epsilon}(X^{5/6+\epsilon})$$
  where $D$ is an explicit constant.

  • Significance: Prior to this work, no asymptotic formula for the mollified second moment of a cubic family was known (even over function fields). The error term $O(X^{5/6+\epsilon})$ is a power saving over the main term $X \log X$.

• Handling the Non-Orthogonal Sieve:
  The authors demonstrate how to overcome the lack of a perfectly orthogonal large sieve bound. They establish a new upper bound for the second moment of cubic Dirichlet series indexed by the full family $F_3$, which is then used to control the error terms in the mollified second moment calculation.

• Evaluation of Twisted Sums:
  They provide a detailed evaluation of twisted sums of cubic Gauss sums, utilizing the meromorphic continuation and convexity bounds of the associated Dirichlet series $\psi(a, s)$ (related to Patterson's theta function).

4. Main Results

Theorem 1.1 (Non-vanishing):
For at least 14% of squarefree $q \in \mathbb{Z}[\omega]$ with $q \equiv 1 \pmod 9$, the central value $L(1/2, \chi_q)$ does not vanish.
More precisely, for sufficiently large $X$:
$$\sum_{\substack{q \in \mathbb{Z}[\omega] \\ q \equiv 1 (9) \\ N(q) \le X \\ L(1/2, \chi_q) \neq 0}} \mu^2(q) \ge \left(\frac{1}{7} - \epsilon\right) \sum_{\substack{q \in \mathbb{Z}[\omega] \\ q \equiv 1 (9) \\ N(q) \le X}} \mu^2(q)$$
(Note: The authors note that $1/7 \approx 14.28\%$. The abstract mentions 14%, while the proof yields $1/(1+6) = 1/7$ based on the optimal mollifier length $M \approx X^{1/6}$.)

Corollary 1.7 (Second Moment Asymptotic):
Establishes the asymptotic formula for the un-mollified second moment with an error term of $O(X^{5/6+\epsilon})$. This is a significant improvement over previous conditional results or results with larger error terms.

Corollary 1.8 (First Moment Asymptotic):
Establishes the asymptotic formula for the first moment with an error term of $O(X^{11/12+\epsilon})$.

5. Significance

• Breakthrough in Cubic Families: This is the first unconditional proof of a positive proportion of non-vanishing for cubic Hecke $L$-functions over a number field. Previous unconditional results were restricted to density-zero subsets.
• Methodological Innovation: The paper fundamentally differs from the works of Soundararajan (quadratic) and Iwaniec-Sarnak (Dirichlet) because the cubic family lacks the perfect orthogonality required for those methods. The authors' ability to derive a power-saving error term without a perfect large sieve bound opens the door for similar problems in higher-order characters.
• Connection to Diaconu's Conjecture: The error term $X^{5/6}$ is very close to the conjectured size of the secondary main term ($X^{5/6} Q(\log X)$) proposed by Diaconu. The authors speculate that with further refinement (e.g., unbalanced approximate functional equations), they could capture this secondary term.
• Function Field vs. Number Field: While non-vanishing results for function fields are often easier due to the Riemann Hypothesis being proven, this work bridges the gap for number fields, showing that unconditional positive proportion results are achievable even without GRH.

In summary, this paper resolves a long-standing open problem in analytic number theory by combining deep tools from the theory of metaplectic theta functions, large sieves, and moment calculations to prove that a positive proportion of cubic $L$-functions do not vanish at the central point.