Bell Series of an Arithmetical Function : Number Theory #Bell #Series #of #an #Arithmetical #Function #: #Number #Theory

           

               E. T. Bell used formal power series to study properties of multiplicative arithmetical functions .

Definition : [ Bell Series ] 

  Given an arithmetical function f and a prime p , we denote by f_p(x) the formal power series

        f_p(x) = 𝜮 f(pⁿ) xⁿ and call this the Bell series of f modulo p.

Note :

          Bell series are especially useful when f is multiplicative.

Examples :

·         The Bell series for the Mobius function 𝛍 is given by 𝛍_p(x) = 1-x.

·         The Bell series for the Euler ‘s totient function ⲫ is given by

                ⲫ_p(x)  =   (1-x) / (1-px)

·         The Bell series for Completely multiplicative functions f is given by  

               f_p(x) = 1 / (1-f(p) x)

·         The Bell series for identity function I is given by  I_p(x) = I

·         The Bell series for unit function u is given by u_p(x) = 1/(1-x)

·         The Bell series for Power function N^a is given by N_p^a (x) = 1/(1-pⁿx)

·         The Bell series for Liouville’s function ƛ is given by ƛ_p(x) = 1/(1+x)

         

Uniqueness :

         Let f and g be multiplicative functions then f = g if and only if f_p(x) = g_p(x) for all           primes p.

Proof :

          Suppose f and g are multiplicative .

   Necessary Condition :

          Suppose f = g

                i.e. f(x) = g(x) for all x

         and hence f (pⁿ ) = g( pⁿ ) for all prime p and all n > 0

ð  f_p(x) = g_p(x) for all prime p 

   Sufficient Condition :

       Suppose  f_p(x) = g_p(x) for all primes p.

                 ∴    f (pⁿ) = g (pⁿ ) for all n > 0.

         Since f and g are multiplicative and f (pⁿ) = g (pⁿ ) for all prime p and all n > 0 ,

            We have f (x) = g(x ) for all  x

            ∴                   f = g .

                                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 

 * Formal Power Series 

 * Liouville’s function λ(n) 

  #Bell #Series #of #an #Arithmetical #Function #: #Number #Theory