Universality of the zeta function in short intervals

We improve the universality theorem of the Riemann zeta-function in short intervals by establishing universality for significantly shorter intervals $$[T,T+H]$$ [ T , T + H ] . Assuming the Riemann Hypothesis, we prove that universality in such short intervals holds for H=(log T)B H = ( log T ) B with an explicitly given $$B>0$$ B > 0 . Unconditionally, we show that for the same H the set of real numbers τ ∈ [T,T+H] τ ∈ [ T , T + H ] such that ζ (s+iτ ) ζ ( s + i τ ) approximates an arbitrary given analytic function has a positive upper density.