Abstract For any positive integer q, let A(q) denote the set of all regular numbers of the modulo q in the interval 1≤m≤q. A new arithmetic function is introduced on the basis of A(q). The arithmetic properties of the function are studied by elementary methods and by applying the properties of trigonometric sums. Using this arithmetic property, we study the computational problem of an infinite series containing the function, and obtain the concrete form of the arithmetic function when it is equal to 1, then derive an interesting identity containing the function.
