Arithmetic Functions in Harmonic Analysis and Operator Theory

The main purpose of this chapter is to introduce some new tools from harmonic analysis and the theory of operator algebras into the study of arithmetic functions, i.e., functions defined from the natural numbers ℕ $$\mathbb{N}$$ to the complex numbers ℂ $$\mathbb{C}$$ . The cases are from number theory (for example, Dirichlet L-functions, etc.), from the theory of moments, and from probability theory (e.g., generating functions). Algebras of arithmetic functions and their representations are considered. In particular, direct decompositions and tensor-factorizations of arithmetic functions are studied. One can do this with a reduction over the primes; and with the use of free probability spaces, one for every prime. The algebras are represented in Kreĭn spaces. The notion of freeness here is analogous to independence in classical statistics. As an application, the study of certain representations of countable discrete groups is considered.