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

TL;DR

This paper establishes new lower bounds for the maximum of derivatives of the Dedekind zeta function of cyclotomic fields on the critical line, using advanced convolution formulas and resonance methods, refining previous results in the field.

Contribution

It introduces novel lower bound techniques for Dedekind zeta derivatives, generalizing prior work and improving understanding of their extremal behavior near the critical line.

Findings

Established lower bounds for derivatives on the critical line.

Generalized previous bounds using convolution formulas.

Refined bounds near the critical line using resonance methods.

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.