On the zeros of higher derivatives of Hardy's \(Z\)-function
type: Item
instance of: scholarly article
title: On the zeros of higher derivatives of Hardy's \(Z\)-function
zbMATH Open document ID: 0980.11038
author: Kohji Matsumoto
author: Yoshio Tanigawa
published in: Journal of Number Theory
publication date: 2002-02-27T00:00:00Z
review text: The function in question is \[ Z(t) := \chi^{-1/2}({\textstyle{1\over 2}} + it)\zeta({\textstyle{1\over 2}} + it),\quad \chi(s) = \zeta(s)/\zeta(1-s) = 2^s\pi^{s-1}\sin\left({\pi s\over 2}\right) \Gamma(s), \] so that \(Z(t)\) is real for real \(t\) and \(|Z(t)|= |\zeta({1\over 2}+it)|\). Hardy's function \(Z(t)\) is especially useful in investigations of zeros of the Riemann zeta-function on the critical line \(\Re s = {1\over 2}\). \textit{R. J. Anderson} [J. Math. Anal. Appl. 118, 323-340 (1986; Zbl 0592.10035)] proved that the number of zeros of \(Z'(t)\) in \((0, T] \) is \[ \leq {T\over 2\pi}\log{T\over 2\pi} - {T\over 2\pi} + O(\log T), \] with equality holding if the Riemann Hypothesis (that all complex zeros of \(\zeta(s)\) have real parts equal to \({1\over 2}\)) is true. The present authors successfully generalize Anderson's method and prove that the number of zeros of \(Z^{(k)}(t)\) in \((0, T] \) is \[ \leq {T\over 2\pi}\log{T\over 2\pi} - {T\over 2\pi} + O_k(\log T),\leqno(1) \] with equality holding if the Riemann Hypothesis is true. To this end they introduce the meromorphic function \(\eta_k(s)\) for each \(k\), precisely defined in the text, whose zeros on \(\Re s = {1\over 2}\) essentially coincide with the zeros of \(Z^{(k)}(t)\). They prove that the number of zeros of \(\eta_k(s)\) in a suitable rectangle is given by the expression in (1).
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 11M26
zbMATH DE Number: 1275046
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: zeros
zbMATH Keywords: higher derivatives of Hardy's \(Z\)-function
zbMATH Keywords: Riemann hypothesis
reviewed by: Aleksandar Ivić
MaRDI profile type: Publication
full work available at URL: jnth.1998.2341
OpenAlex ID: W2016164022
cites work: On the function \(Z(t)\) associated with the Riemann zeta-function
cites work: A Mean Value Theorem for the Riemann Zeta-Function at its Relative Extrema on the Critical Line
cites work: Q4074972
cites work: On a problem connected with zeros of \(\zeta\) (s) on the critical line
cites work: Q3926684
cites work: Uniform approximations and zeros of the derivatives of the Hardy \(Z\)- function in short intervals
cites work: Q4324004
cites work: ON THE RIEMANN ZETA FUNCTION
cites work: Q5803461
cites work: Q3202187
DOI: 10.1006/JNTH.1998.2341
Recommended article: Q3470546
Recommended article: Behaviour of the argument of the Riemann zeta function on the critical line
Recommended article: A note on the zeros of the derivatives of Hardy's function Z(t)$Z(t)$
Recommended article: A note on Hardy's theorem
Recommended article: Q4029822
Recommended article: A note on the zeros of the zeta-function of Riemann
Recommended article: On the primitive of the Hardy function \(Z(t)\)
Recommended article: Q4232570
Recommended article: On large distances between neighbouring zeros of the Riemann zeta-function
Recommended article: On a theorem of Levinson
description: scientific article; zbMATH DE number 1275046