Dirichlet Series of Divisor Function

 DF.ID.1: Dirichlet series of first and second order  divisor function:

 

proof
Euler product representation of Dirichlet series of divisor function:

 (1.a)

 (1.b)

 (2.a)

 (2.b)

according to 

from (1.a) and (2.a):

 

 (3.a)

from (1.b) and (2.b):  

 

  (3.b)

The derivative of geometric series (GS.ID.1):

 

 (4)

 

 (5.a)

 

 

 (5.b)

from ( 3.a) and (4) , the Dirichlet series of first order divisor function:

(6)

from (3.b) and (5.b) , the Dirichlet series of second order divisor function:

 

 (7)

DF.ID.2: Dirichlet series of first  order  divisor function: of squared number

 

proof

 (8)

 

  (8.b)

 (8.c)

given that :

 (9)

The derivative of previous series:

 (10)

 (11)

from (8.c)(10) and (11), 

 (12)

 

 (13)

 DF.ID.3: Dirichlet series of  sum of higher order  divisor function:

Proof

 (1)

 (2)

 (3)

 (4)

 (5)

 (6)

 (7)

replacing (7) in (6) gives:

 (8)

 (9)

 (10)

 (11)

 (12)

 (13)