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