  The paper presents two arithmetical versions of the Nyman-Beurling
equivalence with the Riemann hypothesis, proved by classical, quasi elementary,
number-theoretic methods, based on an integrated version of the classical
combinatorial identity for Moebius numbers. These proofs also give insight into
the troublesome phenomenon that many natural sequences of Beurling functions
tending to the indicator function of (0,1) both pointwise and in L1 norm do not
converge in L2 norm.
