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Remarks on Halasz-Montgomery Type Inequalities

  • Jean Bourgain
Part of the Operator Theory Advances and Applications book series (OT, volume 77)

Abstract

The aim of the Halasz-Montgomery inequality is to derive distributional properties for Dirichlet polynomials
$$ F(t) = \sum\limits_{n\sim M} {{a_n}{n^{it}}\,\left( {\left| t \right| < T} \right)} $$
(1.1)
(n ~ m = n proportional to M) from properties of the ζ-function or its partial sums. It is based on the simple Hilbert space inequality
$$ \sum\limits_{r \leqslant R} {\left| {\left\langle {\xi, {\varphi_r}} \right\rangle } \right|} \leqslant \left\| \xi \right\|{\left( {\sum\limits_{r,s \leqslant R} {\left| {\left\langle {{\varphi_r},{\varphi_s}} \right\rangle } \right|} } \right)^{1/2}} $$
(1.2)
.

Keywords

Distributional Property Riemann Zeta Function Orlicz Function Random Polynomial Independent Gaussian Random Variable 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 1995

Authors and Affiliations

  • Jean Bourgain
    • 1
  1. 1.Institute of Advanced StudyPrincetonUSA

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