Saturday, November 24, 2012

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)










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