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

Sivaramakrishnan, R.

Canadian mathematical bulletin, Tome 22 (1979) no. 2, pp. 207-220

Résumé

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 by where 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

Détail

DOI : 10.4153/CMB-1979-028-1