Editing Multiplicative function (section)

== Rational arithmetical functions ==

An arithmetical function ''f'' is said to be a rational arithmetical function of order <math>(r, s)</math> if there exists completely multiplicative functions ''g''<sub>''1''</sub>,...,''g''<sub>''r''</sub>, 
''h''<sub>''1''</sub>,...,''h''<sub>''s''</sub> such that 
<math display="block"> 
f=g_1\ast\cdots\ast g_r\ast h_1^{-1}\ast\cdots\ast h_s^{-1}, 
</math>
where the inverses are with respect to the Dirichlet convolution.  Rational arithmetical functions of order <math>(1, 1)</math> are known as totient functions, and rational arithmetical functions of order <math>(2,0)</math> are known as quadratic functions or specially multiplicative functions. Euler's function <math>\varphi(n)</math> is a totient function, and the divisor function <math>\sigma_k(n)</math> is a quadratic function. 
Completely multiplicative functions are rational arithmetical functions of order <math>(1,0)</math>. Liouville's function <math>\lambda(n)</math> is completely multiplicative. The Möbius function <math>\mu(n)</math> is a rational arithmetical function of order <math>(0, 1)</math>.
By convention, the identity element <math>\varepsilon</math> under the Dirichlet convolution is a rational arithmetical function of order <math>(0, 0)</math>.

All rational arithmetical functions are multiplicative.  A multiplicative function ''f'' is a rational arithmetical function of order <math>(r, s)</math> if and only if its Bell series is of the form 
<math display="block"> 
{\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)}}
</math>
for all prime numbers <math>p</math>.

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