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In the first part of this expository paper, we present and discuss theinterplay of Dirichlet polynomials in some classical problems of number theory,notably the Lindel\"of Hypothesis. We review some typical properties of theirmeans and continue with some investigations concerning their supremumproperties. Their random counterpart is considered in the last part of thepaper, where a analysis of their supremum properties, based on methods ofstochastic processes, is developed.

Dirichlet polynomials: some old and recent results, and their interplay in number theory