Remarks on Halasz-Montgomery Type Inequalities

The aim of the Halasz-Montgomery inequality is to derive distributional properties for Dirichlet polynomials 1.1 $$ F(t) = \sum\limits_{n\sim M} {{a_n}{n^{it}}\,\left( {\left| t \right| < T} \right)} $$ (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 1.2 $$ \sum\limits_{r \leqslant R} {\left| {\left\langle {\xi, {\varphi_r}} \right\rangle } \right|} \leqslant \left\| \xi \right\|{\left( {\sum\limits_{r,s \leqslant R} {\left| {\left\langle {{\varphi_r},{\varphi_s}} \right\rangle } \right|} } \right)^{1/2}} $$ .