  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.
