Zeros of high derivatives of the Riemann Zeta function

TL;DR

This paper investigates the zeros of high derivatives of the Riemann zeta function, revealing regions free of zeros, describing their distribution, and showing a periodic convergence pattern towards the critical line, indicating increased regularity.

Contribution

It introduces a novel generalization of non-vanishing results for zeta derivatives and uncovers a periodic pattern in the zero distribution of these derivatives.

Findings

Existence of zero-free regions for zeta derivatives in the right half-plane

Sharp estimates for the number of zeros in critical strips

Zeros' distribution converges periodically to the critical line as derivative order increases

Abstract

The main aim of this paper is twofold. First we generalize, in a novel way, most of the known non-vanishing results for the derivatives of the Riemann zeta function 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 the k-th derivative of the zeta function, we describe a unexpected phenomenon, which 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 are not randomly scattered to the right of the line Re(s)=1, 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 the zeta function on the line Re(s)=1/2.