On the frequency of Titchmarsh's phenomenon for \(\zeta{}(s)\). VIII
type: Item
instance of: scholarly article
title: On the frequency of Titchmarsh's phenomenon for \(\zeta{}(s)\). VIII
zbMATH Open document ID: 0761.11033
DOI: 10.1007/BF02837174
author: R. Balasubramanian
author: K. Ramachandra
author: Ayyadurai Sankaranarayanan
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1993-01-16T00:00:00Z
review text: Let \(z=e^{i\theta}\) where \(\theta\) is a constant in \([0,2\pi)\) and let \(G(\sigma,t)=|(\zeta(\sigma+it))^ z|\) where \(\sigma\) is a constant satisfying \({1\over 2}\leq\sigma\leq 1\). Also \(t\geq 1\). Then the authors prove the following three theorems. Theorem 1. We have \[ \max_{T\leq t\leq T+H} G(1,t)\geq e^ \gamma \lambda(\theta)(\log\log H-\log\log\log H)+O(1) \] where \(\gamma\) is the Euler's constant and \(\lambda(\theta)=\prod_ p \lambda_ p(\theta)\) where \(\lambda_ p(\theta)=(1-1/p)((1-s^ 2/p^ 2)^{1/2}(-{c/p})^{- c}\exp(s\sin^{-1} (s/p))\), where \(c+is=\exp(i\theta)\). The conditions on \(H\) and \(T\) are \(T\geq H\geq C\log\log\log T\), \(T\geq T_ 0\) where \(C\) and \(T_ 0\) are certain positive constants. Theorem 2. Let \(\alpha\) be a constant in \((1/2,1)\). Then \[ \max_{T\leq t\leq T+H}G(\alpha,t)>\exp(C_ 1(log H)^{1-\alpha}/\log\log H) \] where \(C_ 1\) is a certain positive constant depending on \(\alpha\) and \(T^{1/3}\leq H\leq T\). If \(\theta=0\) then the conditions on \(H\) and \(T\) are \(T\geq H\geq C'\log\log T\); where \(C'\) is a certain positive constant. Theorem 3. (On Riemann hypothesis) In Theorem 2 the condition on \(H\) can be relaxed to \(T\geq H\geq C'\log\log T\). Also under this condition on \(H\), there holds \(\max_{T\leq t\leq T+H} G(1/2,t)>\exp(C_ 2(\log H/\log\log H)^{1/2})\) where \(C_ 2\) is a positive numerical constant. Remark 1. Where \(\theta=0\), Theorem 3 is true even if we do not assume Riemann hypothesis. Remark 2. The results of this paper were inspired by the result \[ \max_{0\leq t\leq T} G(\alpha,t)>\exp(C_ 3(\log H)^{1- \alpha}/(\log\log H)^ \alpha) \] where \(C_ 3>0\) depends on \(\alpha\) and \(1/2<\alpha<1\) due to H. L. Montgomery. It may be noted that K. Ramachandra and A. Sankaranayanan have obtained recently \(C_ 3=C_ 4(\alpha-1/2)^{1/2}(1-\alpha)^{-1}\) by the method of Montgomery. Here \(C_ 4>0\) is a numerical constant. Also the quantity \((\alpha- 1/2)^{1/2}\) can be replaced by 1 if we…
reviewed by: K. Ramachandra
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11M26
zbMATH DE Number: 92800
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: Titchmarsh series
zbMATH Keywords: Omega theorems
MaRDI profile type: Publication
cites work: An Improvement on a Theorem of Titchmarsh on the Mean Square of |ζ(12+it)|
cites work: Progress towards a conjecture on the mean value of Titchmarsh series, III
cites work: Extreme values of the Riemann zeta function
cites work: Q3916646
cites work: On the Frequency of Titchmarsh's Phenomenon for Ζ(S)
cites work: On the frequency of Titchmarsh's phenomenon for ζ(s)-VII
cites work: Q3988154
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full work available at URL: https://doi.org/10.1007/bf02837174
OpenAlex ID: W4232191945
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description: scientific article; zbMATH DE number 92800