Remarks on Halasz-Montgomery Type Inequalities

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


The aim of the Halasz-Montgomery inequality is to derive distributional properties for Dirichlet polynomials
$$ F(t) = \sum\limits_{n\sim M} {{a_n}{n^{it}}\,\left( {\left| t \right| < T} \right)} $$
(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
$$ \sum\limits_{r \leqslant R} {\left| {\left\langle {\xi, {\varphi_r}} \right\rangle } \right|} \leqslant \left\| \xi \right\|{\left( {\sum\limits_{r,s \leqslant R} {\left| {\left\langle {{\varphi_r},{\varphi_s}} \right\rangle } \right|} } \right)^{1/2}} $$


Distributional Property Riemann Zeta Function Orlicz Function Random Polynomial Independent Gaussian Random Variable 
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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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