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Extreme values of derivatives of the Dedekind zeta function of a cyclotomic field

This paper establishes new lower bounds for the maximum of derivatives of the Dedekind zeta function of a cyclotomic field on and near the critical line by employing a double convolution formula, special GCD sums, and the resonance method, thereby generalizing and refining previous results by Bondarenko et al. and Yang.

Original authors: Zhonghua Li, Yutong Song, Qiyu Yang, Shengbo Zhao

Published 2026-01-15

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CC BY 4.0

Original authors: Zhonghua Li, Yutong Song, Qiyu Yang, Shengbo Zhao

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 world of numbers as a vast, turbulent ocean. In this ocean, there are special waves called Zeta functions. These aren't just any waves; they are the "heartbeat" of the number system, revealing secrets about how prime numbers (the building blocks of all numbers) are distributed.

For a long time, mathematicians have been trying to find the highest peaks these waves can reach. Knowing how high a wave can get helps us understand the ocean's most extreme storms.

This paper, written by Zhonghua Li and his team, is about measuring the highest peaks of a specific, more complex type of wave: the Dedekind zeta function of a cyclotomic field.

Here is a simple breakdown of what they did and what they found:

1. The New Wave: Cyclotomic Fields

Think of the standard Zeta function as a single, famous lighthouse beam. The authors are studying a "family" of lighthouses (called cyclotomic fields) that are built using roots of unity (a fancy way of saying they are constructed from specific geometric patterns of numbers).

These family lighthouses are actually made up of the original lighthouse plus several other smaller, related beams. Because they are more complex, their waves can potentially get much higher than the original single beam. The authors wanted to prove exactly how high these new, complex waves can get.

2. The Challenge: Measuring the "Slopes" (Derivatives)

Usually, mathematicians measure the height of the wave itself. But this paper focuses on the derivatives.

Analogy: If the wave is a rollercoaster track, the "height" is how high the coaster goes. The "derivative" is how steep the track is at that moment.
The authors wanted to know: How steep can the track get at the very top of the wave?
Why does this matter? If you know how steep the wave can get, you get a sharper, more precise understanding of the wave's behavior.

3. The Tool: The "Resonance Method" (The Tuning Fork)

To find these extreme heights, the authors used a technique called the Resonance Method.

The Metaphor: Imagine you have a giant, complex musical instrument (the number system). You want to find the one specific note that makes the instrument vibrate the loudest.
The authors built a special "tuning fork" (called a resonator). They designed this tuning fork to match the specific frequencies of the number system. When they "struck" the system with their tuning fork, it amplified the signal, revealing the maximum possible height (or steepness) of the wave.

4. The Secret Ingredient: GCD Sums

To make their tuning fork work perfectly, they needed a special mathematical tool called GCD Sums (Greatest Common Divisor sums).

The Metaphor: Think of GCD sums as a way to count how many pairs of numbers share a common "ancestor."
The authors combined their tuning fork with these sums to create a "double-version convolution formula." This is like using a high-tech lens to focus the resonance even more sharply, allowing them to see the extreme values that were previously hidden.

5. The Results: Breaking the Record

The paper establishes two main new records:

Result A (On the Critical Line): They proved that on the "critical line" (the most important, central path of the ocean), the steepness of the wave can reach a specific, massive height.
- The Catch: They found that because these new waves are more complex, the mathematical "loss" in their calculation meant the peak wasn't quite as high as the absolute theoretical maximum (it was off by a factor of $\sqrt{2}$ ), but it was still a significant improvement over previous estimates.
- Significance: This proves that the derivatives of these complex waves can get just as "wild" as the waves themselves.
Result B (Near the Critical Line): They also looked at areas slightly away from the center. They found that even in these "nearby" zones, the waves still reach incredible heights, though the formula for the height changes slightly depending on how far you are from the center.

6. Why This Matters (According to the Paper)

The authors state that their work generalizes and refines previous work by other mathematicians (like Yang, Bondarenko, and Soundararajan).

They took a method that was used for simple waves and successfully applied it to these complex, multi-layered waves.
They provided a more precise "lower bound." In math, a "lower bound" is like saying, "We know for a fact this wave must get at least this high." They pushed that "must" line higher than anyone else had before for this specific type of wave.

Summary

In short, Li and his team built a better "tuning fork" to listen to the loudest, steepest parts of a complex mathematical ocean. They proved that these waves are capable of reaching extreme heights, refining our understanding of how these fundamental number patterns behave at their most chaotic moments. They didn't just guess the height; they mathematically proved a minimum height that these waves must exceed.

Technical Summary: Extreme Values of Derivatives of the Dedekind Zeta Function of a Cyclotomic Field

Problem Statement
This paper investigates the extreme values of the derivatives of the Dedekind zeta function, $\zeta_K^{(\ell)}(s)$ , associated with a cyclotomic field $K = \mathbb{Q}(\omega_d)$ , where $\omega_d$ is a $d$ -th root of unity ( $d \ge 3$ ). The study focuses on establishing lower bounds for the maximum modulus of these derivatives on the critical line ( $\Re(s) = 1/2$ ) and in the region immediately adjacent to it ( $1/2 \le \sigma \le 1/2 + 1/\log_2 T$ ).

While extreme values of the Riemann zeta function $\zeta(s)$ and Dirichlet $L$ -functions have been extensively studied using the resonance method and GCD sums, extending these results to the derivatives of Dedekind zeta functions presents specific challenges. Unlike $\zeta(s)$ , the coefficients $a_K(n)$ of the Dirichlet series for $\zeta_K(s)$ are not completely multiplicative. This non-completely multiplicative nature complicates the direct application of standard GCD sum techniques used in previous works by Bondarenko et al. and Yang.

Methodology
The authors employ the resonance method, a technique originally proposed by Voronin and refined by Soundararajan, Aistleitner, and Bondarenko/Seip. The core strategy involves constructing a "resonator" function $R(t)$ designed to amplify the values of $\zeta_K^{(\ell)}(s)$ at specific points.

Key methodological components include:

Auxiliary Function Construction: The authors define an auxiliary function $G(s) = 1 + a_K(2)2^{-s} + (-1)^\ell \zeta_K^{(\ell)}(s)$ to ensure the coefficients of its Dirichlet series satisfy $a(n) \ge a_K(n)$ . This allows for the derivation of lower bounds for the derivatives.
Double-Version Convolution Formula: To connect the integral of the resonator against the zeta function to arithmetic sums, the authors utilize a double-version convolution formula (Lemma 2.2). This formula relates the integral of $G(\sigma+it+iy)G(\sigma-it+iy)$ to a sum involving the coefficients $a(n)$ and a kernel function $K(y)$ .
Weighted GCD Sums: The proof relies on establishing lower bounds for weighted GCD sums, specifically $S_{1/2}(M, a_K)$ . The authors construct a specific set of integers $M$ based on primes in arithmetic progressions modulo $d$ . A significant technical hurdle addressed is the non-completely multiplicative nature of $a_K(n)$ . To overcome this, the authors introduce a modified multiplicative function $a'_K(n)$ to facilitate the estimation of the GCD sums, noting that this procedure incurs a loss of a factor of $\sqrt{2}$ compared to the case of completely multiplicative functions.
Kernel Functions: The proofs utilize specific smooth kernel functions $K_\eta(u)$ and their Fourier transforms to control the error terms in the integral estimates, following the approaches of Bondarenko, Seip, and de la Bretèche.

Key Contributions and Results
The paper establishes two primary theorems providing lower bounds for the maximum of the derivatives:

Theorem 1.1 (Critical Line): For sufficiently large $T$ and uniformly for $d \ll (\log_2 T)^A$ , the authors prove:
$\max_{t \in [0, T]} \left| \zeta_K^{(\ell)}\left(\frac{1}{2} + it\right) \right| \ge \exp\left( \left(\sqrt{\phi(d)} + o(1)\right) \sqrt{\frac{\log T \log_3 T}{\log_2 T}} \right)$
This result demonstrates that the lower bound for the derivatives on the critical line attains the same order of magnitude as the bound for the function $\zeta_K(s)$ itself (where $\ell=0$ ). The exponent includes the factor $\sqrt{\phi(d)}$ , reflecting the degree of the field extension.
Theorem 1.2 (Near the Critical Line): For $\sigma$ in the range $1/2 \le \sigma \le 1/2 + 1/\log_2 T$ , the authors establish:
$\max_{t \in [0, T]} \left| \zeta_K^{(\ell)}(\sigma + it) \right| \ge \exp\left( \left(\sqrt{\frac{\phi(d)}{e-1}} + o(1)\right) \frac{(\log T)^{1-\sigma} (\log_3 T)^\sigma}{(\log_2 T)^\sigma} \right)$
This extends the estimate to a neighborhood of the critical line. The authors note that the leading term in the exponent is smaller than in Theorem 1.1 due to limitations in bounding the series $\sum p^{-2\sigma}$ .
Supplementary Results (Theorems 5.1 and 5.2): The paper briefly presents further refinements for specific ranges of $\sigma$ (e.g., $\sigma_D = 1/2 + D/\log_2 T$ and $1/2 \le \sigma \le 2/3$ ), showing that derivatives attain extreme values of the same order of magnitude as those found for $\zeta(s)$ in previous literature.

Significance and Claims
The authors claim that their results generalize and refine existing work:

Generalization of Yang's Work: The results extend Yang's (2022) findings on the extreme values of derivatives of $\zeta(s)$ to the context of Dedekind zeta functions of cyclotomic fields.
Refinement of Bondarenko et al.: Theorem 1.1 refines the results of Bondarenko et al. (2023) by providing a lower bound specifically for the derivatives of $\zeta_K(s)$ on the critical line.
Handling Non-Multiplicativity: A significant contribution is the successful adaptation of the resonance method to handle the coefficients $a_K(n)$ , which are not completely multiplicative. The authors explicitly acknowledge that this adaptation results in a loss of a factor of $\sqrt{2}$ in the exponent compared to the ideal case of completely multiplicative functions, a detail they attribute to the necessity of using the auxiliary function $a'_K(n)$ .
Optimality: The paper suggests that while the exponent in Theorem 1.1 matches the order of the function itself, the specific constant factor might be improvable if methods for optimizing GCD sums (like those of de la Bretèche and Tenenbaum) could be fully adapted to the non-completely multiplicative setting, though the current method yields the $\sqrt{\phi(d)}$ factor.

The paper concludes that in the critical strip, the derivatives of $\zeta_K(s)$ exhibit extreme values comparable in magnitude to those of the Riemann zeta function derivatives, with the growth rate modulated by the Euler totient function $\phi(d)$ of the cyclotomic field's conductor.