Low moments of Dirichlet series
type: Item
instance of: scholarly article
title: Low moments of Dirichlet series
zbMATH Open document ID: 1340.30003
published in: Acta Mathematica Hungarica
publication date: 2015-01-07T00:00:00Z
review text: In order to bound the mean value of multiplicative functions precise bounds for integrals like \[ I(q)=\int_{-1}^1\left| \sum_p\frac{a_p\log p}{p^{\sigma+it}}\right| ^qdt,\quad\text{and}\quad J(q)=\int_{-1}^1\left| \sum_{n\geq1}\frac{b_n}{n^{\sigma+it}}\right| ^qdt \] are needed. It can easily be deduced that \[ I(q)\ll(\sigma-1)^{1-q},\quad J(q)\ll(\sigma-1)^{1-q} \] for any \(2\leq q\leq4\) in the case when \(| a_p| \leq1\) for all prime \(p\) and \(| b_n| \leq1\) for all positive integer \(n\). The goal of the authors is to extend these estimations for \(1\leq q\leq 2\). However, they point out that not exactly these relations extend to this wider range of parameters but the following is true: \[ I(q)\ll(\sigma-1)^{-q/2},\quad J(q)\ll(\sigma-1)^{-q/2} \] for any \(1\leq q\leq2\).
Mathematics Subject Classification ID: 30B50
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 6384134
zbMATH Keywords: Dirichlet series
zbMATH Keywords: multiplicative function
author: Hugh L Montgomery
author: Gérald Tenenbaum
reviewed by: István Mező
MaRDI profile type: Publication
full work available at URL: s10474-014-0457-6
OpenAlex ID: W1989777613
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DOI: 10.1007/S10474-014-0457-6
description: scientific article; zbMATH DE number 6384134