Functional equations characterizing totients in regular convolution rings

Let A be a regular convolution. An arithmetical function f is A -multiplicative if $$f(1)=1$$ f ( 1 ) = 1 and $$f(mn) = f(m)f(n)$$ f ( m n ) = f ( m ) f ( n ) for all positive integers m , n with m, n ∈ A(mn) m , n ∈ A ( m n ) , and f is an A -totient if there exist A -multiplicative functions fT f T and fV f V such that f=fT *AfV⁻¹ f = f T ∗ A f V - 1 . In this paper, we present various functional equations characterizing A -totients, derived from fundamental properties of Euler’s totient function. Our exposition also provides a historical perspective.