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