Topology of natural numbers and entropy of arithmetic functions

Compactification of natural numbers is studied in association with their arithmetics. Stone-Gelfand-Naimark's theory on function algebras and abelian C*-algebras is applied to characterize the associated compact Hausdorff spaces. Entropy for arithmetic functions is introduced. It is shown that zero entropy functions form a C*-algebra. Sarnak's Mobius Disjointness Conjecture is studied in association with this algebra and its maximal ideal space. Connections between geometry and number theory are further discussed.