Title: An arithmetical equation with respect to regular convolutions
Abstract: It is well known that Euler's totient function $$\phi $$ satisfies the arithmetical equation $$ \phi (mn)\phi ((m, n))=\phi (m)\phi (n)(m, n) $$ for all positive integers m and n, where (m, n) denotes the greatest common divisor of m and n. In this paper we consider this equation in a more general setting by characterizing the arithmetical functions f with $$f(1)\ne 0$$ which satisfy the arithmetical equation $$ f(mn)f((m,n)) = f(m)f(n)g((m, n)) $$ for all positive integers m, n with $$m,n \in A(mn)$$ , where A is a regular convolution and g is an A-multiplicative function. Euler's totient function $$\phi _A$$ with respect to A is an example satisfying this equation.