On Self-Approximation of the Riemann Zeta Function in Short Intervals

The Riemann hypothesis (RH) says that all zeros of the Riemann zeta function ζ(s), s=σ+it, in the strip {s∈C:0<σ<1} lie on the line σ=1/2. There are many equivalents of RH in various terms. In this paper, we propose equivalents of RH in terms of self-approximation, i.e., of the approximation of ζ(s) by ζ(s+iτ), τ∈R, in the interval τ∈[T,T+U] with Tη⩽U⩽T, η=1273/4033. We show that the RH is equivalent to the positivity of lower density and (with some exception for the accuracy of approximation) the density of the set of approximating shifts ζ(s+iτ). For the proof, a probabilistic approach and mean square estimates for ζ(s) in short intervals are applied.