In the first part of the 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 next considered in the second part of the paper. An
analysis of their supremum properties, which is entirely based on methods of
stochastic processes, is presented. Some complementary results and related
questions are included in the last section of the paper.