Square-Reduced Residue Systems (Mod r) and Related Arithmetical Functions

We define a square-reduced residue system (mod r) as the set of integers a (mod r) such that the greatest common divisor of a and r, denoted by (a, r), is a perfect square ≥ 1 and contained in a residue system (mod r). This leads to a Class-division of integers (mod r) based on the 'square-free' divisors of r. The number of elements in a square-reduced residue system (mod r) is denoted by b(r). It is shown that(1)(2)In view of (2), b(r) is said to be 'specially multiplicative'. The exponential sum associated with a square-reduced residue system (mod r) is defined bywhere the summation is over a square-reduced residue system (mod r).B(n, r) belongs to a new class of multiplicative functions known as 'Quasi-symmetric functions' and(3)As an application, the sum is considered in terms of the Cauchy-composition of even functions (mod r). It is found to be multiplicative in r. The evaluation of the above sum gives an identity involving Pillai's arithmetic function