On a problem connected with zeros of \(\zeta\) (s) on the critical line
type: Item
instance of: scholarly article
title: On a problem connected with zeros of \(\zeta\) (s) on the critical line
zbMATH Open document ID: 0624.10033
DOI: 10.1007/BF01540522
published in: Monatshefte für Mathematik
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178336
review text: The paper deals with estimates of \(\zeta (+it)\) and zeros of \(Z^{(k)}(t)\), where \(Z(t)=\chi^{-}(+it)\zeta (+it),\chi (s)=\zeta (s)/\zeta (1-s).\) The main result, based on the technique of exponential averaging and the theory of exponent pairs, is the following: Let \(k\geq 1\) be a fixed integer. Then every interval \([T, T+H]\) contains a zero of odd order of \(Z^{(k)}(t)\) if \(T\geq T_ 0\) and \(H\gg T^{a(k)} \log T,\) where \[ a(k)=\min (3/37,\quad \mu ()+\epsilon)\;if\;k=1,\quad and \] \[ a(k)=\min (27/(164k+168),\quad (1/k)\mu ()+\epsilon)\;if\;k\geq 2, \] and \(\mu (\sigma)=\limsup_{t\to \infty}\log | \zeta (\sigma +it)| / \log t\). The bounds for a(k) are the sharpest ones hitherto. Bounds for \(\zeta (+iT)\) are also derived, provided that \(\zeta (+it)\) or Z'(t) does not vanish in certain short intervals containing T.
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11L40
zbMATH DE Number: 4012735
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: zeros on critical line
zbMATH Keywords: Riemann-Siegel formula
zbMATH Keywords: exponential sums
zbMATH Keywords: exponential averaging
zbMATH Keywords: exponent pairs
author: Aleksandar Ivić
MaRDI profile type: Publication
cites work: An Improvement on a Theorem of Titchmarsh on the Mean Square of |ζ(12+it)|
cites work: Q4723831
cites work: Q3220660
cites work: On the order of zeta(1/2+it) and Delta(R)
OpenAlex ID: W2079376057
description: scientific article; zbMATH DE number 4012735