Representation of Arithmetic Functions and Free Probability

In this monograph we study a class of representations of arithmetic functions and corresponding operator-theoretic and free-probabilistic properties. We associate given arithmetic functions to certain matrices and study free-probabilistic structures on such matrices determined by prime-powers. By understanding such matrices as operators acting on an “indefinite” inner product space we derive operator-theoretic properties of them. This study is one of the frontier works providing connections between modern number theory and operator theory via free probability with help of representation theory.