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

Published online by Cambridge University Press:  20 November 2018

R. Sivaramakrishnan
R. Sivaramakrishnan*
Affiliation:
Department of Mathematics, University of Calicut, Kerala 673635, India
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Abstract

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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

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 22 , Issue 2 , 01 June 1979 , pp. 207 - 220
Copyright
Copyright © Canadian Mathematical Society 1979

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