On arithmetic functions orthogonal to deterministic sequences

We prove the equivalence of Sarnak's conjecture on Mobius orthogonality with a Kolmogorov type property conjectured by Veech for Furstenberg systems of the Mobius function. This yields a combinatorial condition on the Mobius 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 Mobius orthogonality is satisfied for all dynamical systems whose ergodic measures yield nilsystems.