On Value Distribution of One Epstein Zeta-Function

The paper is devoted to Epstein zeta-function ζ3(s), s=σ+it, of a quadratic form of three variables. We obtain a probabilistic characterization of the function ζ3(s) in terms of weak convergence of probability measures in the space of analytic functions defined by means of shifts ζ3(s+iψ(τ)), where ψ(τ) is a differentiable function increasing to infinity with a monotonic derivative satisfying certain growth conditions. The Gram function is an example of ψ(τ). We prove a limit theorem for ζ3(s) with an explicitly given limit measure—the distribution of an analytic random element. For the proof, the mean square estimate for ζ3(s) is applied. The obtained limit theorem characterizes the chaotic behavior of the function ζ3(s) in the space of analytic functions and is applied for approximation of a certain class of analytic functions by shifts ζ3(s+iψ(τ)). This shows the impact of ζ3(s) in approximation function theory.