一个算术函数及其有关的恒等式

对任意的正整数q，设A(q)表示模q在区间1≤m≤q中所有正则数的集合。在A(q)基础上引入一个新的算术函数，借助初等方法以及三角和性质研究了该函数的算术性质；利用此算术性质研究了包含该函数的一个无穷级数的计算问题，给出了此算术函数等于1时的具体形式，进而给出了一个包含该函数的一个有趣的恒等式。

An arithmetical function and related identity

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.