Zeros of higher derivatives of Riemann zeta function

TL;DR

This paper extends results on the zeros of higher derivatives of the Riemann zeta function, improving mollifier techniques and zero density estimates, and shows zeros of a related function cluster near the critical line.

Contribution

It introduces refined mollifier methods for higher derivatives of zeta, leading to sharper zero density bounds and insights into zeros of Matsumoto-Tanigawa's eta_k function.

Findings

Improved mollifier length in short intervals

Refined error terms in zero density estimates

Zeros of eta_k-function cluster near the critical line

Abstract

In this article, we extend the result of Conrey [5, Theorem 2] to shorter intervals for higher-order derivatives of the zeta function. That is we study the mean value of the product of two finite order derivatives of the zeta function multiplied by a mollifier in short intervals. In this process, we obtain better mollifier length in some short intervals compared to the length of mollifier implied by Conrey's result. These finer studies allow us to refine the error term of some classical results of Levinson and Montgomery [13], Ki and Lee [11] on zero density estimates of $\zeta^{(k)}$. Further, we showed that almost all non-trivial zeros of Matsumoto-Tanigawa's $\eta_k$-function cluster near the critical line.