Representation of Arithmetic Functions and Free Probability
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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...
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.
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