  We prove Veech's conjecture on the equivalence of Sarnak's conjecture on
M\"obius orthogonality with a Kolmogorov type property of Furstenberg systems
of the M\''obius function. This yields a combinatorial condition on the
M\"obius function itself which is equivalent to Sarnak's conjecture. As a
matter of fact, our arguments remain valid in a larger context: we characterize
all bounded arithmetic functions orthogonal to all topological systems whose
all ergodic measures yield systems from a fixed characteristic class (zero
entropy class is an example of such a characteristic class) with the
characterization persisting in the logarithmic setup. As a corollary, we obtain
that the logarithmic Sarnak's conjecture holds if and only if the logarithmic
M\''obius orthogonality is satisfied for all dynamical systems whose ergodic
measures yield nilsystems.
