Liouville’s function λ(n) : Number Theory #Liouville’s #function #λ(n) #: #Number #Theory

               

Liouville’s function λ(n) :

                   We define λ(n) =  1                           if n = 1

                                         = ( -1 ) ^a1+a2+…+ak  if n = p₁^a1 . p₂^a2 . … . p_k^ak

Note :

           Liouville's function Λ (n) is an important example of a completely multiplicative

           function.

Theorem : 

              For every n ≥ 1 , we have 𝜮 λ(d)  =   1    if n is a square

                                                                     = 0      otherwise.

             Also , Λ^-1 (n) = | 𝛍 (n) | for all n.

 Proof : 

            Let n ≥ 1.

            Suppose g (n) =  𝜮 λ(d) for d/n

            First we prove g is multiplicative.

            For  m,n 𝞊 Z⁺, we have 

               g(mn) =  𝜮 𝜮 λ(d) λ(d') for d/m , d'/n

                          = 𝜮  λ(d) 𝜮   λ(d') for d/m and d'/n

               g(mn) = g (m) g(n) ∀ m,n 𝞊 Z⁺

                 ∴ g is multiplicative .                 

            For some prime power p^a ,

            g(p^a) =   λ(1) + λ(p) + λ(p²) + … + λ(p^a)

                     = 1 -1 +1 – 1 +…+ ( -1 )^a

                    = 0        if a is odd

                   = 1        if a is even.

         Hence, if n = 𝜫 p_i^ai    we have g (n) = 𝜫 g ( p_i^ai) .

         Also g (n) =  0   if for odd a_i            | since if a_i is odd then g ( p_i^ai) = 0 |

                         = 1    if for even a_i        | since if a_i is even then g ( p_i^ai) = 1 |

        ∴ g ( n )  =  1   if n is square

                        =   0   Otherwise

     Also λ^-1 (n) = 𝛍 (n) λ (n)     

                        = 𝛍 (n) 𝛍 (n)

                        = 𝛍² (n)

            λ^-1 (n) = | 𝛍 (n) |.

                                   **      Hence The Proof   **

 * Dirichlet Multiplication

 * Euclidean Algorithm

 * Fundamental Theorem of Arithmetic

 *  Properties of Numbers

 * Historical Introduction to Number Theory 

 * GCD of morethan 2 numbers

 *   The Mobius Function 𝝻 ( n ) .

 *  The Euler Totient Function 

#Liouville’s #function #λ(n) #: #Number #Theory