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

Abstract

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.