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Zeros of higher derivatives of Riemann zeta function

Mithun Kumar Das, Sudhir Pujahari

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 $ζ^{(k)}$. Further, we showed that almost all non-trivial zeros of Matsumoto-Tanigawa's $η_k$-function cluster near the critical line.

Zeros of higher derivatives of Riemann zeta function

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 . Further, we showed that almost all non-trivial zeros of Matsumoto-Tanigawa's -function cluster near the critical line.

Paper Structure

This paper contains 7 sections, 10 theorems, 101 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Zeros of derivatives of zeta functions
  3. Mean square of the product of $\zeta^{(k)}$ with mollifier
  4. Zeros of Matsumoto-Tanigawa's $\eta_k$-function
  5. Preliminaries
  6. Proof of theorems
  7. Acknowledgement

Key Result

Theorem \oldthetheorem

Let $0 < \theta < \frac{2k+1}{4(k+1)}$ and $\rho_k = \beta_k + i \gamma_k$ be zero of $\zeta^{(k)}$ such that $T \leq \gamma_k \leq T+H$ for an integer $k\geq 1$. Then, for any $H=T^a, \frac{1}{2} +\theta < a \leq 1$ we have

Figures (2)

  • Figure 1: Plot for $f(k)=\frac{1}{4\pi}\operatorname{log}{(1+\frac{4k}{\sqrt{12k^2-3}})}$ and $g(k)=\frac{1}{4\pi}\operatorname{log}{2}$, $4\leq k\leq 1000$.
  • Figure 2: Comparison between the plots of $(a,\theta)$ for Theorem \ref{['Thm1']} and Corollary \ref{['coro5']}

Theorems & Definitions (22)

  • Theorem \oldthetheorem
  • Corollary 1
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Remark 1
  • Corollary 2
  • Remark 2
  • Remark 3
  • Theorem \oldthetheorem
  • Remark 4
  • ...and 12 more