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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> is a rational arithmetical function of order <math>(0, 0)</math>.
All rational arithmetical functions are multiplicative.
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