A harmonic analysis of directed graphs from arithmetic functions and primes

Abstract

In this paper, we study groupoid actions acting on arithmetic functions. In particular, we are interested in the cases where groupoids are generated by directed graphs. By fixing a prime p and a graph G, we establish a noncommutative free probabilistic structure embedded in the algebra of all arithmetic functions. We act the additive group (ℝ, +), the flow, on the free probability space dependent both on p and on G, and construct uncountable families of free probability spaces determined by the flow. We study fundamental properties of such a family.