  To study arithmetic structures of natural numbers, we introduce a notion of
entropy of arithmetic functions, called anqie entropy. This entropy possesses
some crucial properties common to both Shannon's and Kolmogorov's entropies. We
show that all arithmetic functions with zero anqie entropy form a C*-algebra.
Its maximal ideal space defines our arithmetic compactification of natural
numbers, which is totally disconnected but not extremely disconnected. We also
compute the $K$-groups of the space of all continuous functions on the
arithmetic compactification. As an application, we show that any topological
dynamical system with topological entropy $\lambda$, can be approximated by
symbolic dynamical systems with entropy less than or equal to $\lambda$.
