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

Abstract

In the paper, the simultaneous approximation of a tuple of analytic functions in the strip {s=σ+it∈C:1/2<σ<1} by shifts (ζ(s+iφ1(τ)),…,ζ(s+iφr(τ))) of the Riemann zeta-function ζ(s) with a certain class of continuously differentiable increasing functions φ1,…,φr is considered. This class of functions φ1,…,φr 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 (φj(τ))1/3(logφj(τ))26/15 and derivatives φj′(τ). The density problem is also discussed. An example of the approximation by a composition F(ζ(s+iφ1(τ)),…,ζ(s+iφr(τ))) with a certain continuous operator F in the space of analytic functions is given.