The Discrete Value Distribution of the Modified Mellin Transform of the Fourth Power of the Riemann Zeta-Function

Let Z2(s) denote the modified Mellin transforms of the modulus of the fourth power of the Riemann zeta-function. This paper is devoted to the probabilistic properties of generalized discrete shifts Z2(s+iψ(k)), k∈N, with a certain differentiable function ψ(τ) satisfying some estimate connected to the mean square of the function Z2(s) and such that the sequence {κψ(k):k∈N} is uniformly distributed modulo 1 with every κ∈R∖{0}. We propose the condition that Z2(s+iψ(k)) in the space of analytic functions has a limit distribution concentrated at the point g0(s)≡0. Such a limit theorem is applied for the approximation of the function g0(s).