Fractional moments of the \(\zeta\)-function
type: Item
instance of: scholarly article
title: Fractional moments of the \(\zeta\)-function
zbMATH Open document ID: 0592.10034
DOI: 10.1007/BF01158400
author: I. Sh. Dzhabbarov
published in: Mathematical Notes
publication date: 1985-01-01T00:00:00Z
review text: The author considers \(2\lambda\)-th (\(\leq 4)\) fractional moments of \(\zeta\) (s) in the right half of the critical strip. He proves the following theorem. Let \(0<\lambda \leq 2\), \(T\geq 10^ 3\), \(\sigma \geq 1/2+(\log \log \log T)^ 2/\log \log T\). Then we have the formula \[ \int^{T}_{1}| \zeta (\sigma +it)|^{2\lambda} dt=C(\sigma,\lambda)T+O(\Delta (T^{1-(2\sigma -1)/\quad 2(3- 2\sigma)}+T^{1-(2\sigma -1)/(2-\sigma)}(1-\lambda (1-\sigma)))), \] \(C(\sigma,\lambda)=\sum^{\infty}_{n=1}\tau^ 2_{\lambda}(n) n^{- 2\sigma}\), \(\tau_{\lambda}(n)\) are the coefficients of the Dirichlet series for \(\zeta^{\lambda}(s)\) for Re s\(>1\), and \(\Delta =Exp\{C_ 0 (\log \log \log T)^ 2/\log \log T\}\), \(C_ 0>0\) being an absolute constant.
Mathematics Subject Classification ID: 11M06
zbMATH DE Number: 3950634
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: fractional moments
MaRDI profile type: Publication
cites work: Q5803461
cites work: Q3936818
cites work: AN EXAMPLE IN THE THEORY OF THE SPECTRUM OF A FUNCTION
cites work: Q3864597
cites work: Topics in multiplicative number theory
full work available at URL: https://doi.org/10.1007/bf01158400
OpenAlex ID: W2067981679
description: scientific article; zbMATH DE number 3950634