No one cares

]]>*[Corrected, thanks – T.]*

Well, you should indeed use \dots and not \ldots or \cdots. (TeX takes care of the correct placement of the ellipses.)

]]>Yes; one replaces the minus sign here by the value of the multiplicative function at the given prime.

]]>Thanks for the correction! Yes, there are additional dynamical systems obeying (2), (3), (4). Here is one. Consider the space formed by quotienting by the diagonal copy of the integers; this is also the inverse limit of for natural numbers . This is a compact abelian group that is also a vector space over (multiplication by comes from taking inverse limits of the scaling maps from to ). One can take to be the compact abelian group with Haar measure, with an affine map acting on this space by where is the number of prime factors of counting multiplicity), with and . (This is basically the system that is blocking us from establishing local Fourier uniformity of the Liouville function.)

]]>*[Corrected, thanks – T.]*