The mean value of Hardy's function in short intervals
type: Item
instance of: scholarly article
title: The mean value of Hardy's function in short intervals
zbMATH Open document ID: 1343.11074
author: Matti Jutila
published in: Indagationes Mathematicae. New Series
publication date: 2015-12-08T00:00:00Z
review text: For real \(t\), Hardy's function is defined by \[ Z(t)=\chi\left(\tfrac{1}{2}+it\right)^{-1/2} \zeta\left(\tfrac{1}{2}+it\right), \] where \(\zeta(s)\) is the Riemann zeta-function and \(\chi(s)\) is the gamma factor determined by \(\zeta(s)=\chi(s)\zeta(1-s)\). We note that \(Z(t)\) is a real-valued function and that \(|Z(t)|=|\zeta(\frac12+it)|\) holds for any real \(t\). In this paper, the author studies \[ I(T,H,U)=\int_T^{T+H}\left(\int_0^U Z(t+u)\,du\right)^2 dt, \] where \(0<U\leq T^{1/2}\leq H\leq T\). By three different approaches the author gives several formulas of the type \(I(T,H,U)=M(T,H,U)+E(T,H,U)\). Here \(M(T,H,U)\) is written explicitly and \(E(T,H,U)\) is regarded as an error term under certain condition among \(T\), \(H\) and \(U\).
reviewed by: Hirotaka Akatsuka
Mathematics Subject Classification ID: 11M06
zbMATH DE Number: 6517638
zbMATH Keywords: Riemann's zeta-function
zbMATH Keywords: Hardy's function
zbMATH Keywords: mean value
MaRDI profile type: Publication
full work available at URL: j.indag.2015.08.006
OpenAlex ID: W1244323882
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DOI: 10.1016/J.INDAG.2015.08.006
description: scientific article; zbMATH DE number 6517638