Generalized Universality for Compositions of the Riemann Zeta-Function in Short Intervals

In the paper, the approximation of analytic functions on compact sets of the strip { s = σ + i t ∈ C ∣ 1 / 2 < σ < 1 } by shifts F ( ζ ( s + i u 1 ( τ ) ) , … , ζ ( s + i u r ( τ ) ) ) , where ζ ( s ) is the Riemann zeta-function, u 1 , … , u r are certain differentiable increasing functions, and F is a certain continuous operator in the space of analytic functions, is considered. It is obtained that the set of the above shifts in the interval [ T , T + H ] with H = o ( T ) , T → ∞ , has a positive lower density. Additionally, the positivity of a density with a certain exceptional condition is discussed. Examples of considered operators F are given.