On mean values of Dirichlet polynomials
type: Item
instance of: scholarly article
title: On mean values of Dirichlet polynomials
zbMATH Open document ID: 0756.11023
DOI: 10.1007/BF01199063
author: Antal Balog
author: Glyn Harman
published in: Archiv der Mathematik
publication date: 1992-09-27T00:00:00Z
review text: For fixed non-zero real numbers \(\alpha\) and \(\beta\) it is shown that \[ \int_ 0^ T|\sum_{N<n\leq 2N} a_ n n^{i\alpha t} |^ 2\;|\sum_{M<m\leq 2M} b_ m m^{i\beta t} |^ 2\;dt\ll(T+MN)MN\log 2MNT, \] for \(T,M,N\geq 1\) and arbitrary coefficients satisfying \(| a_ n|,| b_ m|\leq 1\). When \(\alpha=\beta\) this is an easy consequence of the usual mean-value estimate for a single Dirichlet polynomial, but the general case appears to require treatment from first principles. The proof depends on an estimate for the number of solutions of the inequality \[ \left|\left({n'\over n}\right)^ \alpha-\left({m'\over m}\right)^ \beta \right|\leq\Delta. \] The authors also discuss the possible extensions of their result to products of 3 or more Dirichlet polynomials, and give a partial result using mean-values of the Riemann zeta-function.
Mathematics Subject Classification ID: 11M99
Mathematics Subject Classification ID: 11M06
zbMATH DE Number: 62248
zbMATH Keywords: Dirichlet polynomials
zbMATH Keywords: mean-values
zbMATH Keywords: Riemann zeta-function
reviewed by: D. R. Heath-Brown
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01199063
OpenAlex ID: W2072555252
description: scientific article; zbMATH DE number 62248