Monday, April 11, 2011

Riemann zeta function


firstly , I recommend you  to read these  Riemann zeta function  , Euler product  to have fundamental understanding of the Riemann zeta function.

 RZ.ID.1:  

 
p is a prime number

 
Proof
  given

(1)


(2)

(3)
cancelling out the similarities, therefore 
(4)

 RZ.ID.2: 


Proof
derived by multiplying (1) and (4)


 RZ.ID.3:  


proof:



 by differentiating the above equation
 



 RZ.ID.4: 


Proof

(1)
replacing

therefore,
(2)


from (1) and (2)
 


multiplying and dividing  by -1


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