# Number of non-primes in the set of units modulo $n$

## TL;DR

This paper investigates the properties of an arithmetic function counting non-prime units modulo n, providing insights into its behavior and distribution within number theory.

## Contribution

It introduces and analyzes the function f, a novel arithmetic function counting non-prime units modulo n, expanding understanding of unit groups.

## Key findings

- Characterization of f for various n
- Distribution patterns of non-prime units
- Relationships between f and classical arithmetic functions

## Abstract

In this work, we studied various properties of arithmetic function $\tilde{\varphi}$, where $\tilde{\varphi}(n)=|\{m\in \mathbb{N} | 1\le m\le n, (m,n)=1, \mbox{$m$ is not a prime}\}|.$