On The Generalized Multiplicative Euler Phi Function

TL;DR

This paper introduces a new generalized Euler phi function based on the group of units modulo n, explores its properties, and solves related equations, extending classical number theory concepts.

Contribution

It defines and analyzes the properties of the generalized Euler phi function ^k(n), providing explicit formulas and solutions to related equations.

Findings

Explicit form of ^k(n) derived

Properties of ^k(n) similar to classical (n) established

Complete solutions for certain equations involving ^k(n)

Abstract

The generalized group of units of the ring modulo $n$ was first introduced by El-Kassar and Chehade, written as $U^k(Z_n)$. This allows us to formulate a new generalization to the Euler phi function $\varphi(n)$, that represents the order of $% U^k(Z_n)$ and it is denoted by $\varphi ^{k}(n).$ In this paper, we introduce this newly defined function, where we compute its explicit form and examine some of its properties similar to that of $\varphi (n)$. In addition, we study some generalized equations involving $\varphi ^{k}(n)$ where complete solution is given for some equations by considering the general case and others for some particular cases.