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

Abstract

In the paper, the approximation of analytic functions on compact sets of the strip {s=σ+it∈C∣1/2<σ<1} by shifts F(ζ(s+iu1(τ)),…,ζ(s+iur(τ))), where ζ(s) is the Riemann zeta-function, u1,…,ur 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.