Large values of derivatives of the Riemann zeta function on vertical homogeneous progressions

TL;DR

The authors establish lower bounds for the maxima of derivatives $\zeta^{(j)}(s)$ on vertical homogeneous progressions, showing that the discrete (homogeneous) setting attains the same order of magnitude as the continuous case. They extend Soundararajan’s resonance method to arithmetic progressions by constructing tailored resonators and analyzing associated quadratic forms, obtaining explicit leading constants: $\lambda(A)=\frac{1}{\sqrt{2}\,(e-1)\,e^A}$ for $\sigma=\tfrac12+\tfrac{A}{\log_2 N}$ and a Dickman-function-based constant $D_j(A)$ for $\sigma=1-\tfrac{A}{\log_2 N}$. Specifically, they prove $\max_{\sqrt{N}\le \ell \le N}|\zeta^{(j)}(\sigma+i\alpha\ell)| \ge \exp((\lambda(A)+o(1))\sqrt{\frac{\log N\,\log_3 N}{\log_2 N}})$ and $\max_{\sqrt{N}\le \ell \le N}|\zeta^{(j)}(\sigma'_A+i\alpha\ell)| \ge (D_j(A)+o(1))(\log_2 N)^{j+1}$ under natural ranges for $j$. The results demonstrate that large zeta-derivative values on vertical progressions mirror the continuous-case behavior, validating resonance-method techniques in discrete settings.

Abstract

In this paper, we establish lower bounds for the maximum of derivatives of the Riemann zeta function on vertical homogeneous progressions. When the real part $σ$ lies within a suitable range, we show that the discrete case has a similar order of magnitude to the continuous case, using the resonance method.

Theorems & Definitions (19)

• Theorem 1.1
• Remark
• Theorem 1.2
• Remark
• Lemma 2.1: yang2024extreme, Lemma 1
• Lemma 2.2: montgomery1971topics, Lemma 1.4
• Lemma 2.3
• proof
• Proposition 3.1
• proof