Multiplicative function
Rational arithmetical functions

An arithmetical function f is said to be a rational arithmetical function of order (r, s) {\displaystyle (r,s)} if there exists completely multiplicative functions g,..., g, h,..., h such that f = g 1 ∗ ⋯ ∗ g r ∗ h 1 − 1 ∗ ⋯ ∗ h s − 1, {\displaystyle f=g_{1}\ast \cdots \ast g_{r}\ast h_{1}^{-1}\ast \cdots \ast h_{s}^{-1},} where the inverses are with respect to the Dirichlet convolution. Rational arithmetical functions of order (1, 1) {\displaystyle (1,1)} are known as totient functions, and rational arithmetical functions of order (2, 0) {\displaystyle (2,0)} are known as quadratic functions or specially multiplicative functions. Euler's function φ (n) {\displaystyle \varphi (n)} is a totient function, and the divisor function σ k (n) {\displaystyle \sigma _{k}(n)} is a quadratic function. Completely multiplicative functions are rational arithmetical functions of order (1, 0) {\displaystyle (1,0)}. Liouville's function λ (n) {\displaystyle \lambda (n)} is completely multiplicative. The Möbius function μ (n) {\displaystyle \mu (n)} is a rational arithmetical function of order (0, 1) {\displaystyle (0,1)}.By convention, the identity element ε {\displaystyle \varepsilon } under the Dirichlet convolution is a rational arithmetical function of order (0, 0) {\displaystyle (0,0)}.

All rational arithmetical functions are multiplicative. A multiplicative function f is a rational arithmetical function of order (r, s) {\displaystyle (r,s)} if and only if its Bell series is of the form f p (x) = ∑ n = 0 ∞ f (p n) x n = (1 − h 1 (p) x) (1 − h 2 (p) x) ⋯ (1 − h s (p) x) (1 − g 1 (p) x) (1 − g 2 (p) x) ⋯ (1 − g r (p) x) {\displaystyle {\displaystyle f_{p}(x)=\sum _{n=0}^{\infty }f(p^{n})x^{n}={\frac {(1-h_{1}(p)x)(1-h_{2}(p)x)\cdots (1-h_{s}(p)x)}{(1-g_{1}(p)x)(1-g_{2}(p)x)\cdots (1-g_{r}(p)x)}}}} for all prime numbers p {\displaystyle p}.

The concept of a rational arithmetical function originates from R. Vaidyanathaswamy (1931).
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