Application of Stochastic Elements in the Universality of the Periodic Zeta-Function: The Case of Short Intervals

Let a={am:m∈N} be a multiplicative periodic sequence of complex numbers. In this paper, we consider the approximation of analytic functions defined in the strip {s=σ+it:1/2<σ<1} by shifts ζ(s+iτ;a) of the zeta-function defined, for σ>1, by ζ(s;a)=∑m=1∞amm−s and by analytic continuation elsewhere. Using stochastic techniques, we obtain that the set of the above shifts approximating a given analytic function has a positive lower density (or density with at most countably many exceptions) in the interval [T,T+V] with T23/70⩽V⩽T1/2 as T→∞. The proofs are based on a limit theorem with an explicitly given limit probability measure in the space of analytic functions.