Selberg Identity : Number Theory #Selberg #Identity #: #Number #Theory

Selberg Identity :

    For n ≥ 1 , we have   ⴷ(n) logn + 𝜮 ⴷ(d) ⴷ(n/d) = 𝜮 μ(d) log² (n/d).

Proof :

     Suppose n ≥ 1 .

  We know that log n = ( ⴷ * u ) (n)

  ⇒ u(n)logn = ( ⴷ * u ) (n) | since u(n) = 1)

 ⇒ u’(n) = ( ⴷ * u ) (n)    ……………①

By differentiating on both sides , we get

  u’’(n) = ( ⴷ * u )’ (n)

             = ( ⴷ’ * u ) (n) +( ⴷ * u’ ) (n)

            =  ( ⴷ’ * u ) (n) + ( ⴷ *  ( ⴷ * u ) (n)    | Since from   ① |

 u’’(n)  =   ( ⴷ’ * u ) (n) + (( ⴷ *   ⴷ)) * u ) (n)                                                                            

By multiplying on both sides with μ = u’ we get

         (u’’ * μ)(n)  = ⴷ’(n) + (ⴷ *   ⴷ)(n)

⇒     𝜮 μ(d) u’’(n/d) = ⴷ(n) logn + 𝜮 ⴷ(d) ⴷ(n/d) …………. ㉣

 Now

 u’’(n/d) = ( u’(n/d) )’

             = ( u(n/d)log (n/d) )’

             = ( u(n/d) log (n/d) ) log(n/d)

             = u(n/d) log²(n/d)

             = log²(n/d)         | since u(n) = 1 for all n|

∴ from ㉣ , we have

     𝜮 μ(d) u’’(n/d) = ⴷ(n) logn + 𝜮 ⴷ(d) ⴷ(n/d)

⇒ 𝜮 μ(d) log²(n/d) = ⴷ(n) logn + 𝜮 ⴷ(d) ⴷ(n/d)  

                      *** Hence The Proof ***      

  * Bracket Function 

* Perfect Number

 *Historical background of Number Theory

 * 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) 

  * Congruences

 * Fermat's Theorem

 * Wilson's Theorem  

#Selberg #Identity #: #Number #Theory