The Mangoldt Function ⴷ ( n ) : Number Theory #The #Mangoldt #Function #ⴷ ( n ) : #Number #Theory

 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 = 2¹ = P^m , here m = 1

                ∴   ⴷ ( 2 ) = log 2.

    For n = 3 , we have 3 = 3¹ = p^m , here m = 1

                  ∴  ⴷ ( 3 ) = log 3

    For n = 4, we have 4 = 2² = p^m , here m = 2

                   ∴ ⴷ ( 4 ) = log 2

    For n = 5 , we have 5 = 5¹ = p^m , here m = 1

                    ∴ ⴷ ( 5 ) = log 5

     For n = 7 , we have 7 = 7¹ = p^m , here m = 1

                   ∴ ⴷ( 7 ) = log 7

    For n = 8 , we have 8 = 2³ = p^m , here m = 3

                    ∴ ⴷ ( 8 ) = log 2

    For n = 9, we have 9 = 3² = p^m , here m = 2

                     ∴ ⴷ ( 9 ) = log 3 .

Here is a short table of values of ⴷ ( n ) :

      n :    1          2            3          4          5          6         7          8          9          10   

ⴷ(n) :   0       log2    log3    log2    log5     0      log7   log2   log3       0

 

Note :

   This Mangoldt’s function ⴷ plays a central role in the distribution of primes.

Result :  

                   If n  ≥ 1, we have log n  = Σ ⴷ (d) for d/n

Proof :

                   Suppose n is a positive integer .

                     For n = 1 , log 1 = 0 = ⴷ (1)

                         i.e. the result is true for n = 1.

                    Assume n > 1.

                             Write           n =  𝚷 p_k ^ak,         1 ≤ k ≤ p.

                                             Log n = Σ a_k log p_k .                          -----à ①

                      Now consider Σ ⴷ (d) for d/n.

                        In the above sum , the only non-zero terms come from those divisors d                              of  the form p_k^m  for

                             m =  1 , 2 , 3 ,… , a_k and k = 1 , 2 , 3 , … , r.

                        ence Σ ⴷ (d) = Σ Σ  ⴷ (p_k^m) 

                                           = Σ Σ  log p_k

                                           = Σ a_k log p_k

                               Σ ⴷ (d) = log n.     | since from ① |

                               *** Hence The Proof  ***

* 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 

 #The #Mangoldt #Function #ⴷ ( n ) :  #Number #Theory