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Simultaneous non-vanishing of Dirichlet $L$ --functions, II: Weighted central limit theorem

Assuming the Generalized Riemann Hypothesis, this paper establishes a weighted central limit theorem for the joint distribution of four Dirichlet $L$ -functions at the central point twisted by primitive characters modulo a large prime, thereby proving that a positive proportion of these characters yield central values that are simultaneously large and another positive proportion yield values that are simultaneously small and nonzero.

Original authors: Hung M. Bui, Alexandra Florea, Micah B. Milinovich

Published 2026-07-24

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CC BY 4.0

Original authors: Hung M. Bui, Alexandra Florea, Micah B. Milinovich

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 as a vast, cosmic orchestra. In this orchestra, there are special instruments called L-functions. You can think of these not as physical objects, but as complex musical scores that encode deep secrets about how numbers behave. Just as a musician might look for a specific note to see if a song is in tune, mathematicians look at a very specific spot on these scores—the "center" of the musical range—to see if the value is zero or not.

Why does this matter? In the world of number theory, if a score hits a "zero" at this center, it's usually just a boring, expected silence (like a rest in the music). But if the score is not zero, it's a loud, meaningful note that tells us something profound about the structure of numbers, often revealing hidden patterns or relationships that we can't see otherwise. For decades, mathematicians have been trying to prove that these scores almost never hit that silent zero by accident. They want to know: if we play a huge collection of these musical pieces, how often do we get a loud, non-zero note?

Now, imagine trying to listen to four different instruments playing at the exact same time. It's a chaotic mess of sound. The question becomes: Is it possible to find a moment where all four instruments are playing loud, non-zero notes simultaneously? This is the puzzle that Hung M. Bui, Alexandra Florea, and Micah B. Milinovich tackle in their paper. They don't just ask if the notes exist; they ask how the volume of these notes behaves when you look at a massive crowd of them. They are essentially trying to understand the "statistical weather" of these musical scores.

The Big Discovery: A Weighted Forecast

The authors of this paper have proven a new, sophisticated rule about how these four musical scores behave together. They call it a Weighted Central Limit Theorem. To understand this, let's use a metaphor.

Imagine you are a weather forecaster trying to predict the temperature in four different cities at the same time. Usually, if you look at enough days, the temperatures in these cities will follow a predictable bell curve (the "Central Limit Theorem"): most days are average, some are very hot, and some are very cold, but the extremes are rare.

However, there's a catch. Sometimes, one of the cities might be "broken"—maybe the thermometer is stuck at zero, or the city is in a state of silence. If you try to measure the temperature of a broken city, your data is useless. In the world of L-functions, this "broken" state is when the value is exactly zero. If you try to take the logarithm of zero (which is what the math requires to measure the "volume"), you get a mathematical disaster.

The authors' breakthrough is a clever trick called weighting. Instead of trying to measure every single city (or character), they decide to ignore the days where the thermometer is broken. They create a special "weight" that says, "If the value is zero, pretend this day doesn't exist in our calculation." If the value is non-zero, they give it a normal weight.

By using this "weighted" approach, they managed to prove something amazing: When you look at a huge family of these four L-functions, their "volumes" (once you ignore the zeros) behave exactly like four independent, random variables following a perfect bell curve.

Here is what they found, broken down:

The Four-Way Independence: They proved that the loudness of the first L-function has nothing to do with the loudness of the second, third, or fourth. They are like four dice being rolled at the same time; the result of one doesn't influence the others. This is a huge deal because it means the "music" of these four functions is completely uncorrelated.
The "Big" and "Small" Moments: Because they know the distribution is a perfect bell curve, they can predict exactly how often you will find extreme values.
- They showed that a positive proportion of these characters will have all four L-functions simultaneously very large. Specifically, the size of these values can be as big as $e^{c\sqrt{\log \log q}}$ (where $q$ is a large prime number and $c$ is a fixed constant).
- Conversely, they also showed that a positive proportion will have all four values simultaneously very small (but still non-zero), specifically smaller than $e^{-c\sqrt{\log \log q}}$ .
The Assumption: It is important to note that this entire proof relies on a famous, unproven hypothesis called the Generalized Riemann Hypothesis (GRH). Think of GRH as a "trust me" clause that mathematicians use to make the math work. The paper says, "If we assume GRH is true, then this beautiful, predictable pattern of four independent bell curves is definitely true."

Why This is a Big Deal

Before this paper, mathematicians could only prove that some of these values were non-zero, or that they were non-zero in pairs. Getting to four at once was a much harder mountain to climb. Previous attempts were like trying to guess the weather in four cities by looking at them one by one; you might get lucky with one or two, but predicting all four together was nearly impossible.

This paper changes the game by showing that if you filter out the "broken" days (the zeros), the remaining days follow a perfect, predictable statistical law. This allows them to say with certainty that you can find huge crowds of these characters where all four values are simultaneously huge, or simultaneously tiny.

The authors didn't just guess this; they built a rigorous mathematical machine. They used a "mollifier," which is a fancy mathematical tool that acts like a noise-canceling headphone. It dampens the wild, extreme spikes in the data so the underlying pattern (the bell curve) can be heard clearly. They combined this with a "random model," imagining the numbers as if they were generated by a random number generator, and proved that the real numbers behave almost exactly like this random model.

In short, this paper tells us that the chaotic world of these four L-functions is actually governed by a very orderly, predictable rhythm. As long as we assume the Generalized Riemann Hypothesis holds, we can confidently say that there is a vast, positive number of instances where these four mathematical instruments are playing a loud, non-zero symphony together, and just as many where they are playing a whisper-quiet, non-zero tune. It's a victory for understanding the hidden order in the noise of numbers.

Technical Summary: Simultaneous Non-Vanishing of Dirichlet L–Functions, II: Weighted Central Limit Theorem

Problem Statement
The paper addresses the simultaneous non-vanishing of central values of Dirichlet $L$ -functions, specifically $L(1/2, \chi\chi_j)$ for $j=1, \dots, 4$ , where $\chi$ varies over primitive characters modulo a large prime $q$ , and $\chi_j$ are fixed primitive characters modulo pairwise coprime integers $D_j$ . While previous work established that a positive proportion of characters yield non-vanishing central values (e.g., for one, two, or three twists), and that a positive proportion yield values bounded below by $1/\log q$ , this work seeks to establish a stronger simultaneous growth result. Specifically, the authors aim to prove that a positive proportion of characters satisfy $|L(1/2, \chi\chi_j)| \gg \exp(c\sqrt{\log \log q})$ simultaneously for all four $j$ , under the Generalized Riemann Hypothesis (GRH).

Methodology
The authors employ a probabilistic approach centered on a weighted central limit theorem (CLT) for the joint distribution of the four $L$ -values. The methodology builds upon techniques introduced in [3] (Bui, Evans, Lester, and Pratt) and [4] (Bui, Florea, and Milinovich).

Weighted Measure: To circumvent the obstruction that central values may vanish (rendering $\log |L(1/2, \chi)|$ undefined), the authors define a weighted measure $\mu_F$ . The weight function $F(\chi)$ is constructed using a mollifier $M(1/2, \chi)$ , which dampens extreme values of the $L$ -functions. Specifically, $F(\chi)$ is chosen such that it vanishes if any of the four central values vanish, but approximates 1 for "typical" characters.
$F(\chi) = \prod_{j=1}^4 L(1/2, \chi\chi_j) M(1/2, \chi\chi_j)$
The mollifier is a short Dirichlet polynomial designed to approximate the inverse of the $L$ -function.
Random Model Reduction: The core technical step involves reducing the computation of moments of the weighted $L$ -values to a random model. Let $X(p)$ be independent random variables uniformly distributed on the unit circle. The authors define a random variable $L(s, X)$ mimicking the $L$ -function and a corresponding random mollifier $M(X)$ . They prove that the characteristic function of the weighted sum of the $L$ -values over the family of characters is asymptotically equivalent to the expectation of the corresponding random model (Corollary 3.4).
Twisted First Moments: A crucial input is the asymptotic evaluation of the twisted first moment of the fourfold product of $L$ -functions, established in [4]. This allows the authors to compute the moments of the mollified $L$ -values. The proof relies on a "swap" structure in the random model to handle cross-correlations between the four twists. Unlike previous two-dimensional results where decorrelation was governed by fixed Rankin–Selberg $L$ -functions, the four-dimensional case requires controlling cross-correlations of the form $\sum \chi_j(p)\chi_k(p)/p$ where the modulus grows with $q$ .
Analytic Continuation and Contour Shifting: The authors analyze the generating function $E[L(1/2+s, X)M(X) \exp(i \sum u_j P_{\chi_j}(X))]$ . By shifting the contour of integration and utilizing bounds on high mollified moments (derived from ideas of Soundararajan, Harper, and Lester–Radziwiłł), they isolate the contribution of the pole at $s=0$ and bound the error terms.

Key Contributions and Results

Theorem 1.1 (Weighted Central Limit Theorem): Assuming GRH and under the condition $D^{272} \ll q^{11/16 - 1/2000}$ , the joint distribution of the normalized logarithms of the four central values converges to a product of four independent standard Gaussian distributions. Specifically, for intervals $U_j \subset \mathbb{R}$ :
$\mu_F \left\{ \chi : \frac{\log |L(1/2, \chi\chi_j)|}{\sqrt{\frac{1}{2}\log \log q}} \in U_j, \forall j \right\} = \frac{1}{4\pi^2} \int_{U_1 \times \dots \times U_4} e^{-\sum x_j^2/2} d\mathbf{x} + O_\epsilon((\log \log q)^{-1/2+\epsilon})$
This establishes that the four normalized values are asymptotically independent standard Gaussians under the weighted measure.
Corollary 1.2 (Simultaneous Growth and Non-Vanishing): As a direct application of Theorem 1.1, the authors prove that for any fixed $c > 0$ :
1. A positive proportion of characters $\chi \pmod q$ satisfy $|L(1/2, \chi\chi_j)| > \exp(c\sqrt{\log \log q})$ simultaneously for $j=1, \dots, 4$ .
2. A positive proportion of characters satisfy $0 < |L(1/2, \chi\chi_j)| < \exp(-c\sqrt{\log \log q})$ simultaneously for $j=1, \dots, 4$ .

Significance and Claims
The paper claims to go significantly beyond previous lower bounds (such as the $1/\log q$ bound in [20]) by establishing simultaneous growth of order $\exp(c\sqrt{\log \log q})$ . The authors emphasize that this result is only possible due to the two-sided control provided by the weighted central limit theorem. Previous one-sided central limit theorems (e.g., Hough [8]) provided only upper or lower bounds on the distribution, which were insufficient to guarantee simultaneous large values for multiple twists.

The work highlights two specific technical advancements:

Decorrelation Mechanism: The decorrelation of the four twists is governed by Dirichlet characters with growing conductors ( $D_j D_k$ ), requiring uniform control of cross-correlations that decay as $\log \log \log q$ , rather than the $O(1)$ decay seen in fixed-form settings.
Dimensionality: The proof establishes genuine four-dimensional joint Gaussianity. This is a stronger structural fact than pairwise independence, requiring the full six-term "swap" structure of the random model to cancel correctly, rather than merely vanishing pairwise correlations.

The authors note that the result relies on GRH to ensure the necessary bounds on the $L$ -functions and to facilitate the analytic continuation arguments. The paper does not claim to remove the GRH assumption but utilizes it to establish the asymptotic distribution and the resulting non-vanishing proportions.

The Big Discovery: A Weighted Forecast

Why This is a Big Deal

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