devotion & mathematics

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 commuta...

 Mangoldt Function :  Definition             For every n ≥ 1 we define                       ⴷ ( n )  =  { log p    if n = p m for some  prime p and some m   ≥ 1              =  0    Otherwise    Since 1, 6, 10 can not be expressed as prime power , ⴷ( 0) = ⴷ(6) = ⴷ(10 ) = 0.     For n = 2, we have 2 = 2 1 = P m , here m = 1                 ∴    ⴷ ( 2 ) = log 2.     For n = 3 , we have 3 = 3 1 = p m , here m = 1                   ∴   ⴷ ( 3 ) = log 3     For n = 4, we have 4 = 2 2 = p m , here m = 2        ...

The Euler Totient Function Ჶ(n) :                                                      If n ≥ 1 , the Euler Totient Function Ჶ(n) is defined to              be the number of positive integers not exceeding n which are relatively prime to             n ;  thus ,                         Ჶ ( n ) = 𝚺 1 for 1 ≤ k ≤ n, here 𝚺 indicates that the sum is extended over                                                                       those k relatively prime to n.  For Example :      Suppose n = 1.       ...

Number theory , like many other branches of mathematics , is often concerned  with sequences of real or complex numbers. In number theory such sequences are called arithmetic functions. Definition :                    A real or complex - valued functions defined on  the positive integers is called an arithmetical function or a number- theoretic function. The Mobius Function μ (n) :                    The Mobius function μ is defined as follows :                         μ(1) = 1 ;                     If n > 1 , write n = p 1 a1 + p 2 a2 + … + p k ak . Then                                           μ( n) = ( -1 ) k if a 1 = a 2 = … = a k =1     ...