Classification on arithmetic functions and corresponding free-moment $L$-functions

摘要

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 ($ullet $ ); $(t_{1},t_{2})ullet (s_{1},s_{2})=(t_{1}s_{1},t_{1}s_{2}+t_{2}s_{1}).$