Joint Universality in Short Intervals with Generalized Shifts for the Riemann Zeta-Function

In the paper, the simultaneous approximation of a tuple of analytic functions in the strip { s = σ + i t ∈ C : 1 / 2 < σ < 1 } by shifts ( ζ ( s + i φ 1 ( τ ) ) , … , ζ ( s + i φ r ( τ ) ) ) of the Riemann zeta-function ζ ( s ) with a certain class of continuously differentiable increasing functions φ 1 , … , φ r is considered. This class of functions φ 1 , … , φ r is characterized by the growth of their derivatives. It is proved that the set of mentioned shifts in the interval [ T , T + H ] with H = o ( T ) has a positive lower density. The precise expression for H is described by the functions ( φ j ( τ ) ) 1 / 3 ( log φ j ( τ ) ) 26 / 15 and derivatives φ j ′ ( τ ) . The density problem is also discussed. An example of the approximation by a composition F ( ζ ( s + i φ 1 ( τ ) ) , … , ζ ( s + i φ r ( τ ) ) ) with a certain continuous operator F in the space of analytic functions is given.