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

TL;DR

The paper investigates extreme values of derivatives $\zeta_{\mathbb{K}}^{(\ell)}(s)$ of the Dedekind zeta function for cyclotomic fields $\mathbb{K}=\mathbb{Q}(\omega_d)$ on the critical line and in a near-critical region. It blends a double convolution framework with weighted GCD sums and employs refined resonators to transfer lower bounds from GCD-sum problems to the magnitude of $\zeta_{\mathbb{K}}^{(\ell)}(s)$, achieving uniform results in $d$ up to $(\log_2 T)^A$. The main contributions are Theorem 1.1, giving a critical-line lower bound of order $\exp((\sqrt{\phi(d)}+o(1))\sqrt{(\log T\log_3 T)/\log_2 T})$, and Theorem 1.2, providing a near-critical-line bound of order $\exp((\sqrt{\phi(d)/(e-1)}+o(1))(\log T)^{1-\sigma}(\log_3 T)^{\sigma}/(\log_2 T)^{\sigma})$ for $\tfrac12 \le \sigma \le \tfrac12+\tfrac{1}{\log_2 T}$. These results extend the landscape of omega-type bounds from $\zeta(s)$ and Dirichlet $L$-functions to Dedekind zeta functions of cyclotomic fields, offering sharper exponents and uniform conductor-parameter control, with potential implications for zero-distribution and value-distribution problems in families of $L$-functions.

Abstract

In this paper, we establish a lower bound for the maximum of derivatives of the Dedekind zeta function of a cyclotomic field on the critical line. Employing a double version convolution formula and combing special GCD sums, our result generalizes the work of Bondarenko et al. in 2023. We also set a lower bound by the resonance method when the real part is near the critical line, both of the above results refine part of Yang's work in 2022.

Theorems & Definitions (21)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Remark
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4: bondarenko2023dichotomy, Lemma 5