[Paper Review] New zero-free regions for the derivatives of the Riemann Zeta Function

Thomas Binder, Sebastian Pauli+1 more|arXiv (Cornell University)|Jan 1, 2011

Analytic Number Theory Research12 references3 citations

TL;DR

This paper establishes new zero-free regions for the k-th derivatives of the Riemann zeta function ζ(k)(s) in the right half-plane, proving the existence of infinite sequences of such regions where no zeros can exist. It further reveals a surprising periodic convergence of zeros toward central lines in 'critical strips' as k increases, showing that higher derivatives exhibit highly regular, predictable two-dimensional zero distributions—contrasting with the chaotic one-dimensional distribution of ζ(s) on the critical line.

ABSTRACT

Abstract. The main aim of this paper is twofold. First we generalize, in a novel way, most of the known non-vanishing results for ζ(k)(s) by establishing the existence of an infinite sequence of regions in the right half-plane where these derivatives cannot have any zeros; and then, in the rare regions of the complex plane that do contain zeros of ζ(k)(s) (named “critical strips ” in analogy with the classical case of ζ(s)), we describe a unexpected phenomenon, which – especially for the hitherto-neglected high derivatives ζ(k)(s) – implies great regularities in their zero distributions. In particular, we prove sharp estimates for the number of zeros in each of these new critical strips, and we explain how they converge, in a very precise, periodic fashion, to their central, “critical ” lines, as k increases. This not only shows that the zeros of ζ(k)(s) are not randomly scattered to the right of the line σ = 12, but that, in many respects, their two-dimensional distribution eventually becomes much simpler and more predictable than the one-dimensional behavior of the zeros of ζ(s) on the line σ = 12. 1.

Motivation & Objective

• To generalize existing non-vanishing results for ζ(k)(s) by identifying infinite sequences of zero-free regions in the right half-plane.
• To investigate the distribution of zeros in the rare regions—termed 'critical strips'—where ζ(k)(s) can vanish, especially for high-order derivatives.
• To uncover structural regularities in the two-dimensional distribution of zeros of ζ(k)(s) as k increases.
• To quantify the number of zeros in each critical strip and describe their convergence to central lines with increasing k.
• To compare the predictability of ζ(k)(s) zero distributions with the classical, one-dimensional zero distribution of ζ(s) on σ = 1/2.

Proposed method

• Extends known non-vanishing theorems for ζ(k)(s) via a novel analytical framework to construct infinite sequences of zero-free regions in the right half-plane.
• Introduces the concept of 'critical strips'—regions in the complex plane where ζ(k)(s) may have zeros, analogous to the classical critical strip for ζ(s).
• Applies asymptotic analysis and complex function theory to study the behavior of ζ(k)(s) in these strips, particularly as k → ∞.
• Derives sharp estimates for the number of zeros in each critical strip using contour integration and density arguments.
• Analyzes the convergence of zero locations toward central lines in the strips, showing periodic, predictable patterns as k increases.
• Uses comparison with the classical Riemann zeta function to highlight the increased regularity in higher derivatives.

Experimental results

Research questions

• RQ1 What infinite sequences of zero-free regions exist for ζ(k)(s) in the right half-plane, and how can they be systematically constructed?
• RQ2 How do the zeros of ζ(k)(s) distribute within the critical strips, and what structural patterns emerge as k increases?
• RQ3 To what extent do the zeros of ζ(k)(s) converge periodically toward central lines in the critical strips with growing k?
• RQ4 How does the two-dimensional distribution of zeros of ζ(k)(s) compare in complexity and predictability to the one-dimensional distribution of ζ(s) on σ = 1/2?
• RQ5 What explains the emergence of such high regularity in the zero distribution of high-order derivatives, despite the apparent randomness in lower-order cases?

Key findings

• The paper proves the existence of an infinite sequence of zero-free regions in the right half-plane for ζ(k)(s), generalizing known non-vanishing results.
• Zeros of ζ(k)(s) are confined to rare 'critical strips' in the complex plane, analogous to the classical critical strip for ζ(s).
• For high derivatives, zeros exhibit a periodic, convergent pattern toward central lines within each critical strip as k increases.
• Sharp estimates are derived for the number of zeros in each critical strip, providing quantitative control over their distribution.
• The two-dimensional distribution of zeros of ζ(k)(s) becomes significantly more regular and predictable than the one-dimensional distribution of ζ(s) on σ = 1/2.
• The results demonstrate that higher-order derivatives of the zeta function display a surprising level of structural order, challenging the assumption of increasing complexity with k.