The Mobius function mu(n) is minus one to the number of distinct
prime factors of n if n has no square factors and zero otherwise.
Understanding the randomness (often referred to as the `Mobius
randomness principle' in this function is a fundamental and very
difficult problem. We will explain a precise dynamical formulation of
the randomness and report on recent advances establishing it. In
particular the disjointness of the resulting Mobius Flow from
horocycle flows and related horocycle dynamics at ``prime times''.
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