Dirichlet Multiplication : Number Theory #Dirichlet #Multiplication #: #Number #Theory

The Dirichlet product of arithmetical functions : 

Definition : 

                  If f and g are two arithmetical functions then we define their Dirichlet product ( or Dirichlet convolution )  to be the arithmetical function f *g  defined by the equation

    (f * g)(n) = 𝜮 f(d) g(n/d) for d/n

Notation :

              The symbol N will be used for the arithmetical function for which N(n) = n ∀ n.

    From this notation , we can relate the Mobius function ( 𝝁 ) and Euler totient function ( Ჶ ) by

       Ჶ = 𝝁 * N.

Note : 

            The Concept Dirichlet  multiplication helps clarify interrelations between various arithmetical                    functions.

Result : 

            Dirichlet multiplication is commutative and associative. i.e. for any arithmetical functions f , g, h , we have 

                      f * g = g * f                  ( commutative )

           f * ( g * h ) = (f * g ) * h         ( Associative )

Proof : 

            Let Ⱥ be the set of all arithmetical functions defined on set of all positive integers.

       Suppose f , g , h ϵ Ⱥ .

         i.e. f , g, h are three arithmetical functions. 

      For f, g , the Dirichlet multiplication can also be written as 

                       (f * g ) ( n ) = 𝜮 f (a) g( b )  for a.b = n.

      Commutative Law : 

         We have  (f * g ) ( n ) = 𝜮 f (a) g ( b ) for a.b = n

                                            = 𝜮 g (b ) f ( a ) for b.a = n

                                            = ( g * f ) ( n )  

                   ∴ ( f * g ) ( n ) = ( g * f ) ( n ) ∀ n 

                         Hence f * g = g * f ∀ f , g 𝛜 Ⱥ 

             ∴ Dirichlet Multiplication is commutative.

        Associative Law : 

             Let   k = g * h. 

         ∴    k(m) = 𝜮 g(b) h(c) for b . c= m

               Now [ f * ( g * h ) ] (n) = ( f * k ) ( n )

                                                     = 𝜮 f(a) k(m) for a.m = n

                                                     = 𝜮 f(a) . 𝜮 g(b)h(c)   for a.m = n and b.c=m

                        [ f * ( g * h ) ] (n) = 𝜮 f(a) g(b) h(c) for a.b.c = n               ..........> I

             Let l = f * g.            

         ∴   l(p) = 𝜮 f(a) g(b) for a.b = p.              

               Also   [ ( f * g ) * h ] (n) =   ( l * h ) (p) 

                                                      = 𝜮 l(p) h(c) for p.c = n

                                                      = 𝜮 f(a) g(b) 𝜮 h(c) for a.b = p and p.c = n

                         [ ( f * g ) * c ] (n) = 𝜮 f(a) g(b) h(c) for a.b.c = n                  ..........> II  

         ∴ From I and II we have   [ f * ( g * h ) ] (n) =   [ ( f * g ) * h ] (n) ∀ n

                                ∴    [ f * ( g * h ) ] =   [ ( f * g ) * h ]   ∀ f,g,h 𝛜 Ⱥ.

                                       Hence Dirichlet product is Associative.      

* 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 

   #Dirichlet #Multiplication #: #Number #Theory