Zeta-Gamma Function

GZ.ID.1:  Fundamental Zeta-Gamma  function
the relation between Zeta function and Gamma function can be represented in the following form:

Proof

Given

(1)

(2)

 (3)

(4)

substituting (4) into (3)

(5)

(6)

(7)

(8)

Based on Geometric Series of Exponential (GS.ID.2),

(9)

(10)

(11) 

based on (10) and (11) , 

 

 (12) 

 

(13) 

(14) 

 

 (15) 

 (16) 

GZ.ID.2:   Zeta-Gamma  function (Trigonometric)

Proof

Given

(1)

 (2)

(3)

(4)

substituting (4) into (1) ,

(5)

(6)

(7)

substituting (2) into (7) :

(8)

from (16),  at (GZ.ID.1)  ,

(9)

(10)

based on (9) and (10) , then

 multiplying and dividing the denominator by   ,   and based on (3) therefore 

GZ.ID.3:   Zeta-Gamma  function

 

Given

(1)

 (2)

 (3)

substitute (3) into (2) , 

(4)

 (5)

(6)

differentiate (6) ,

(7)

substitute (7) into (5) ,

(8)

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