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

On Gaussians, Zeros, and Linear Combinations of L-functions. Part A: Entire Functions of Beurling-Selberg Type and Probability Distributions