The Möbius function

The Möbius function

Dilafruz Sh.Usmonova

May 29, 2020

Abstract

The German mathematician August Ferdinand Möbius introduced in 1832 the classical Möbius function μ(n) , which an important multiplicative function in number theory and combinatorics. In this paper we study Möbius function, Möbius theorem and some properties of this function.

Keywords and Phrases : Multiplicative function ,arithmetic function, distinct primes
1 Introduction

We define an arithmetical function µ

Let us recall a definition of arithmetic function;

Definition 1. An arithmetic function is a function defined on the positive integers which takes values in the real or complex numbers.

Definition 2. An arithmetic function is multiplicative if for any relatively prime :


Problem
If length is equal to , how many words from X with different cycles from x?
2.The Möbius function

The Möbius function is an arithmetic function of a natural number argument n with µ(1)=1, 

if  is divisible by the square of a prime number, otherwise   ,where k  is the number of prime factors of n.
Definition. The Möbius function is defined by



This definition and the following expression are equally valid:



Examples.