RATIONAL ARITHMETICAL FUNCTIONS RELATED TO CERTAIN UNITARY ANALOGS OF GCD TYPE MATRICES

An arithmetical function f is a rational arithmetical function of order (r,s) if it can be written as the Dirichlet convolution of r completely multiplicative functions and s inverses of completely multiplicative functions. In this paper we show that pseudo-unitarily semimultiplicative functions and a related generalization of the unitary analog of Euler’s totient function are rational arithmetical functions of orders (1,2) and (2,3). These functions arise from the theory the so-called pseudo-LCUM and GCUD reciprocal pseudo-LCUM matrices, where GCUD and pseudoLCUM stand for the greatest common unitary divisor and an extension of the least common unitary multiple.