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)

Gamma-Polylogarithm

GP.ID.1:

 

 

Proof

Given

 

 (1)

 therefore ,

 (2)

 (3)

 (4)

 (5)

 (6)

Based on the identity: 

 (7)

From (5) and (6) ,

Reciprocals of Binomial Coefficients

 

(1)

RBC.ID.1:   Trigonometric Identity

(2)

proof

(3)

(4)

by geometric series

 

(5)

replacing (5) by,

(6)

results

(7)

guided by the following identities:

(8)

(9)

(10)

from (7) , (8) , (9) , and (10) ,  The first part of (7):

(11)

The first part of (11) equals zero

(12)

(13)

(14)

from (7) , (8) , (9) , and (10) ,  The second  part of (7):

(15)

(16)

(17)

(18)

from (14 and (18) ,

(19)

if x=1/2 , then

(20)

RBC.ID.2:    Derivative of Trigonometric Identity-1

 (21)

Proof  

differntiate (19)  in respect to x

 

(22)

(23)

(24)

multiply and divide  the first part of right-hand side by

(25)

(26)

from (25) and (26) 

(27)

(28)

if x=1/2 

(29)

divide both sides by 4

(30)

RBC.ID.3:    Derivative of Trigonometric Identity-2

Proof

(31)

(32)

 

(33)

Using geometric mean

(34)

(35)

Replacing

(36)

(37)

(38)

From  (31) and (38) , Therefore ,

(39)