Interpolation and Extrapolation Statements equivalent to the Riemann Hypothesis

The goal of this paper is twofold; on one hand we wish to present some statements that can be formulated in terms of Interpolation theory which are equivalent to the truth or the falseness of the Riemann Hypothesis, on the other hand we will use a key result of the Jawerth-Milman extrapolation to improve on a well-known criterion by Beurling and Nyman for the Riemann Hypothesis giving sharper sufficient conditions to the celebrated hypothesis.