  This is the first installment in a series of papers devoted to examining
certain aspects of the asymptotic value distribution and distribution of zeros
manifested by members of a broad class of linear combinations of L-functions in
the vicinity of the critical line Re(s) = 1/2. The results that we ultimately
plan to discuss will entail a certain amount of probabilistic underpinning.
$\hspace{.16em}$Entire functions of Beurling-Selberg type turn out to provide a
very convenient inroad into the type of technical estimate that one needs.
$\hspace{.16em}$In the present paper, after first reviewing the basic
properties of these functions, we'll proceed to develop several relatively
simple (but highly suggestive) probabilistic examples in which the functions'
efficacy can be rather plainly seen. The path we take has the added advantage
of enabling several key ideas for later developments to already become visible
in an embryonic form.
