On Equivalents of the Riemann Hypothesis Connected to the Approximation Properties of the Zeta Function

The famous Riemann hypothesis (RH) asserts that all non-trivial zeros of the Riemann zeta function ζ(s) (zeros different from s=−2m, m∈N) lie on the critical line σ=1/2. In this paper, combining the universality property of ζ(s) with probabilistic limit theorems, we prove that the RH is equivalent to the positivity of the density of the set of shifts ζ(s+itτ) approximating the function ζ(s). Here, tτ denotes the Gram function, which is a continuous extension of the Gram points.