Bulletin of the
Korean Mathematical Society BKMS

In this paper, we provide a classification of arithmetic functions in terms of identically-free-distributedness, determined by a fixed prime. We show then such classifications are free from the choice of primes. In particular, we obtain that the algebra $\mathfrak{A}_{p}$ of equivalence classes under the quotient on $\mathcal{A}$ by the identically-free-distributedness is isomorphic to an algebra $\mathbb{C}^{2},$ having its multiplication ($\bullet $ ); $(t_{1},t_{2})\bullet (s_{1},s_{2})=(t_{1}s_{1},t_{1}s_{2}+t_{2}s_{1}).$

Keywords: arithmetic functions, arithmetic algebra, linear functionals, arithmetic probability spaces, free-moment $L$-functions

MSC numbers: 05E15, 11G15, 11R04, 11R09, 11R47, 11R56, 46L10, 46L40, 46L53, 46L54, 47L15, 47L30, 47L55