Some problems on mean values of the Riemann zeta-function
type: Item
instance of: scholarly article
title: Some problems on mean values of the Riemann zeta-function
zbMATH Open document ID: 0858.11045
author: Aleksandar Ivić
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1997-03-25T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93643
full work available at URL: jtnb8-1.html
review text: This paper begins by asking for which \(H,\alpha,\beta\) (depending on \(T)\) one has \[ \int_T^{T+H} \bigl|\zeta (\textstyle {1\over 2} + it) \bigr|^\alpha dt\leq \int_T^{T+H} \bigl|\zeta (\textstyle {1\over 2} + it) \bigr|^\beta dt. \] The author gives a partial answer and states various conjectures which would yield better results. Results and conjectures on the error term in the asymptotic formula for \(\int^T_0 |\zeta (\sigma + it) |^4dt\) for \(0< \sigma<1\) are then discussed. In particular it is shown that the integral is \[ {\zeta^4 (2\sigma) \over \zeta (4\sigma)} T+O(T^{2-2 \sigma} \log^3T). \] It is also asked whether one can find a \(\sigma<1\) for which \[ \int^T_0 \bigl|\zeta (\textstyle {1\over 2} + it) \bigr|^4 \bigl|\zeta (\sigma+it) \bigr|^2dt \ll_\varepsilon T^{1+ \varepsilon}. \] It seems to the reviewer that \(\sigma = 61/62\) is acceptable, since \(\zeta(\sigma+it)\) may then be approximated by a Dirichlet polynomial of length \(T^{1/5}\).
reviewed by: D. R. Heath-Brown
Mathematics Subject Classification ID: 11M06
zbMATH DE Number: 912501
zbMATH Keywords: mean values of the Riemann zeta-function
zbMATH Keywords: fractional moments
zbMATH Keywords: fourth power moment
zbMATH Keywords: error term
zbMATH Keywords: asymptotic formula
MaRDI profile type: Publication
OpenAlex ID: W2316758950
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DOI: 10.5802/JTNB.159
description: scientific article; zbMATH DE number 912501