Summary of Zeta

(1)

Summary

1 Zeta Generating Functions

Both of hyperbolic functions and trigonometric functions can be expanded to Fourier series and Taylor series. And if the termwise higher order integration of these is carried out, Riemann Zeta Functions are obtained.

Where, these are automorphisms which are expressed by lower zetas. However, in this chapter, we stop those so far.

The work that obtain the non-automorphism formulas by removing lower zetas from these are performed subsequent to Chapter2 .

In this chapter, we obtain the following polynomials from the zeta generating functions Where, Riemann Zeta, Dirichlet Eta and Dirichlet Lambda are as follows.

 ^{( )} x = Σ

r=1



r

^x

1 ,  ^{( )} x = Σ

r=1



r

^x

( -1 )

^r-1

,  ^{( )} x = Σ

r=1



( 2 r -1 )

^x

1

Bernoulli numbers and Euler numbers are as follows.

B

₀

=1 , B

₂

=1/6 , B

₄

=-1/30 , B

₆

=1/42 , B

₈

=-1/30 , ^ E

0

=1 , E

2

=-1 , E

4

=5 , E

6

=-61 , E

8

=1385 , ^

Harmonic number is

H

_s

= Σ

t=1 s

1/t =  ( 1+s ) + 

 ( ) n =

( n -1 ! ) ( - x )

^n-1

 log x - H

_n-1

 + 2 ( -1 )

ⁿ

n ! x

ⁿ

+ Σ

r=1



r

ⁿ

e

^-rx

- ( -1 )

ⁿ

Σ

r=1



2 r ( 2 r + n -1 ! ) B

_2r

x

^2r+n-1

- Σ

s=1 n-2

s !

( -1 )

^s

x

^s

 ⁽ n - s ⁾

 ^{( )} n = - 2 ( -1 )

ⁿ

n ! x

ⁿ

- Σ

r=1



( -1 )

^r

r

ⁿ

e

^-rx

+ ( -1 )

ⁿ

Σ

r=1



2 r ( 2 r + n -1 ! )

 2

^2r

-1  B

_2r

x

^2r+n-1

- Σ

s=1 n-1

s !

( -1 )

^s

x

^s

 ⁽ n - s ⁾

 ^{( )} n = 2 ( -1 )

^n-1

( n -1 ! ) x

^n-1

 ^log ² ^x ^- ^H

^n-1

 ⁺ ^Σ

r=1



( 2r -1 )

ⁿ

e

^-(^2r-1⁾^x

+ 2 ( -1 )

ⁿ

Σ

r=1



2 r ( 2 r + n -1 ! )

 2

^2r

-2  B

_2r

x

^2r+n-1

- Σ

s=1 n-2

s !

( -1 )

^s

x

^s

 ⁽ n - s ⁾

Σ

r=1



r

^2n-1

cosrx

- ( 2 n -2 ! ) ( -1 )

ⁿ

x

^2n-2

 log x - H

2n-2

 +( -1 )

ⁿ

Σ

r=1



2 r ( 2 r +2 n -2 ! )

  B

_2r

x

^2r+2n-2

= Σ

s=0 n-2

( 2 s ) !

( -1 )

^s

x

^2s

 ⁽ ² n -1-2 s ⁾

Σ

r=1



r

²ⁿ

sin rx

- ( 2 n -1 ! ) ( -1 )

ⁿ

x

^2n-1

 log x - H

_2n-1

 +( -1 )

ⁿ

Σ

r=1



2 r ( 2 r +2 n -1 ! )

  B

_2r

x

^2r+2n-1

= - Σ

s=1 n-1

( 2 s -1 ! )

( -1 )

^s

x

^2s-1

 ⁽ ² n +1-2 s ⁾

Σ

r=1



( -1 )

^r-1

r

^2n-1

cosrx

-( -1 )

ⁿ

Σ

r=1



2 r ( 2 r +2 n -2 ! )

 2

^2r

-1  _{ } B

_2r

x

^2r+2n-2

= Σ

s=0 n-1

( 2 s ) !

( -1 )

^s

x

^2s

 ⁽ ² n -1-2 s ⁾

Σ

r=1



( -1 )

^r-1

r

²ⁿ

sin rx

-( -1 )

ⁿ

Σ

r=1



2 r ( 2 r +2 n -1 ! )

 2

^2r

-1  _{ } B

_2r

x

^2r+2n-1

= - Σ

s=1 n

( 2 s -1 ! )

( -1 )

^s

x

^2s-1

 ⁽ ² n +1-2 s ⁾

(2)

Σ

r=1



( 2r -1 )

^2n-1

cos {( 2r-1 ) x }

- 2( 2 n -2 ! ) ( -1 )

ⁿ

x

^2n-2

 ^log 2 

x -H

_2n-2

- 2 ( -1 )

ⁿ

Σ

r=1



2r ( 2 r +2 n -2 ! )

 2

^2r

-2 

_

B

_2r_

x

^2r+2n-2

= Σ

s=0 n-2

( 2s ) !

( -1 )

^s

x

^2s

 ( 2n -1-2s )

Σ

r=1



( 2r -1 )

²ⁿ

sin {( 2r-1 ) x }

- 2( 2n -1 ! ) ( -1 )

ⁿ

x

^2n-1

 ^log 2 

x -H

_2n-1

- 2 ( -1 )

ⁿ

Σ

r=1



2r( 2r +2n -1 ! )

 2

^2r

-2 

_

B

_2r_

x

^2r+2n-1

= - Σ

s=1 n-1

( 2 s -1 ! ) ( -1 )

^s

x

^2s-1

 ( 2n +1-2s )

Furthermore, if the termwise higher order differentiation of the Fourier series of each family of tanh, cot and tan are carried out, the following expressions are obtained.

 ⁽ ^- n ⁾ =

2

ⁿ⁺¹

 1-2

ⁿ⁺¹



1 Σ

r=1 n

( -1 )

^r-1_n

D

_r

n =1 , 2 , 3 , 

 ⁽ ^1-2 n ⁾ =

2

²ⁿ

 1-2

²ⁿ

 ( -1 )

^n-1

T

_2n-1

n =1 , 2 , 3 , 

= ( -1 )

^2n-1

2 n B

2n

n =1 , 2 , 3 , 

Where, n

D

_r are the Eulderian Numbers and

T

_n-1 are the tangent numbers. These are defined as follows respectively.

n

D

_r

= Σ

k=0 r-1

( -1 )

^k

  ⁿ ⁺¹ _k ⁽ ^r ^- ^k ⁾

ⁿ

^, ^T

^n-1

⁼ ²

ⁿ

^ ²

ⁿ

^-1 ^ n

  B

_n

By-products

- log 0 =  ^{( )} ¹ ⁼  ^{( )} ^{1 -} 2

 i

2 Formulas for Riemann Zeta at natural number

In this chapter, removing the lower zetas from automorphism formulas in the previous chapter, we obtain non-automorphism formulas for Riemann Zeta at natural number.

Where, Bernoulli numbers and Euler numbers are as follows.

B

0

=1 , B

2

=1/6 , B

4

=-1/30 , B

6

=1/42 , B

8

=-1/30 , ^ E

0

=1 , E

2

=-1 , E

4

=5 , E

6

=-61 , E

8

=1385 , ^

And Harmonic number is

H

_s

= Σ

t=1 s

1/t =  ( 1+s ) + 

For

0 < x  2 

,

 ^{( )} n =

( n -1 ! ) x

^n-1

 n -1 

1 - 2 n

x + Σ

r=1



Σ

s=0 n-1

s ! ( ) xr

^s

r

ⁿ

e

^xr

1 - Σ

r=1



  2 ^- r -1 ⁿ 2 r ( n -1+2 r ) ! B

_2r

x

^n-1+2r

 ^{( )} n =

( n -1 ! ) x

^n-1

 n -1 

1 + 2 n ( n -1 ) x

- log x

+ Σ

r=1



Σ

s=0 n-2

s ! ( ) xr

^s

r

ⁿ

e

^xr

1 - Σ

r=1



  ^1- 2 r ⁿ 2 r ( n -1+2 r ) ! B

_2r

x

^n-1+2r

For

0 < x  

,

 ^{( )} n =

2

^n-1

- 1 2

^n-1

 ² ^x ⁿ

ⁿ

^! ^- ^Σ

^r=1^

^Σ

^n-1^s=0

^{( )} ^xr ^s ^!

^s

r

ⁿ

e

^xr



( -1 )

^r

+ Σ

r=1



  2 ^- r -1 ⁿ 2 r ( n -1+2 r ) !

 2

^2r

-1  B

_2r

x

^n-1+2r

(3)

 ^{( )} n =

2

ⁿ

-1 2

ⁿ

 2( n -1 !( ) n -1 ) x

^n-1

+ Σ

r=1



Σ

s=0 n-1

s !



x ( 2 r -1 )

^s

( 2 r -1 )

ⁿ

e

^-x(^2r-1⁾

+ 2 1 Σ

r=1



  2 ^- r -1 ⁿ 2 r ( n -1+2 r ) ! 

 2

^2r

-2  B

_2r

x

^n-1+2r

 ^{( )} n = 2

ⁿ

-1

2

ⁿ

 2( n -1 ! ) x

^n-1

 n -1 

1 - log 2

x + Σ

r=1



Σ

s=0 n-2

s !



x ( 2 r -1 )

^s

( 2 r -1 )

ⁿ

e

^-x(^2r-1⁾

+ 2 1 Σ

r=1



  ^1- ² r ⁿ ² r ( n -1+2 r ) ! 

 2

^2r

-2  B

_2r

x

^n-1+2r

Especially,

 ^{( )} n =

2 n !( n -1 ) n +1

+ Σ

r=1



Σ

s=0 n-1

s ! r

^s

r

ⁿ

e

^r

1 - Σ

r=1



  2 ^- r -1 ⁿ 2 r ( n -1+2 r ) ! B

_2r

 ^{( )} n =

n !( n -1 ) 2

^n-1

+ Σ

r=1



Σ

s=0 n-1

s ! ( 2 r )

^s

r

ⁿ

e

^2r

1 - Σ

r=1



  2 ^- r -1 ⁿ 2 r ( n -1+2 r ) ! B

_2r

2

^n-1+2r

 ( ) n =

2 n !( n -1 ) n

²

+1

+ Σ

r=1



Σ

s=0 n-2

s ! r

^s

r

ⁿ

e

^r

1 - Σ

r=1



  ^1- 2 r ⁿ 2 r ( n -1+2 r ) ! B

_2r

 ^{( )} n =

2

^n-1

- 1 2

^n-1

 ² n ! 

1 - Σ

r=1



Σ

s=0 n-1

s!

r

^s

r

ⁿ

e

^r

( -1 )

^r

+ Σ

r=1



  2 ^- r -1 ⁿ 2 r ( n -1+2 r ) !

 2

^2r

-1  B

_2r

 ( ) n =

2

ⁿ

-1 2

ⁿ

 2( n -1 !( ) n -1 )

1 + Σ

r=1



Σ

s=0 n-1

s ! ( 2 r -1 )

^s

( 2 r -1 )

ⁿ

e

^-(^2r-1⁾

+ 2 1 Σ

r=1



  2 ^- r -1 ⁿ 2 r ( n -1+2 r ) ! 

 2

^2r

-2  B

_2r

 ^{( )} n = 2

ⁿ

-1

2

ⁿ

 2( n -1 ! )

1  n -1 

1 + log 2 + Σ

r=1



Σ

s=0 n-2

s ! ( 2 r -1 )

^s

( 2 r -1 )

ⁿ

e

^-(^2r-1⁾

+ 2 1 Σ

r=1



  ^1- 2 r ⁿ 2 r ( n -1+2 r ) ! 

 2

^2r

-2  B

_2r

 ^{( )} n = 2

ⁿ

-1

2

ⁿ

 ( n -1 !( ) n -1 ) 2

^n-2

+ Σ

r=1



Σ

s=0 n-2

s ! ( 4 r -2 )

^s

( 2 r -1 )

ⁿ

e

^-(^4r-2⁾

+ 2 1 Σ

r=1



  ^1- ² r ⁿ ² r ( n -1+2 r ) ! 

 2

^2r

-2  B

_2r

2

^n-1+2r

Example

 ( ) 5 = 2  5!4

6 + Σ

r=1



 ¹⁺ ^1! ^r

¹

⁺ ^2! ^r

²

⁺ ^3! ^r

³

⁺ ^4! ^r

⁴

 _r

⁵

¹ _e

^r

^- ^Σ

_r=1



  2r ^-5 -1 2r( 4+2r ) ! B

_2r

 ( ) 5 = 5!4

2

⁴

+ Σ

r=1



 ¹⁺ ^1! ² ^r ⁺ ⁽ ² ^2! ^r ⁾

²

⁺ ⁽ ² ^3! ^r ⁾

³

⁺ ⁽ ² ^4! ^r ⁾

⁴

 _r

⁵

¹ _e

^2r

^- ^Σ

_r=1^

  2r ^-5 -1 2r( 4+2r ) ! B

_2r

2

^4+2r

 ( ) 5 = 2  5!4

5

²

+1 + Σ

r=1



 ¹⁺ 1! 

r

¹

+ 2!

r

²

+ 3!

r

³

r

⁵

e

^r

1 - Σ

r=1



  ^-4 2r 2r( 4+2r ) ! B

_2r

 ( ) 5 = 2

⁴

- 1

2

⁴

 ^2 ¹ ^5! ^- ^Σ

r=1





 ¹⁺ 1! 

r

¹

+ 2!

r

²

+ 3!

r

³

+ 4!

r

⁴

r

⁵

e

^r

( -1 )

^r

+ Σ

r=1



  2 ^-5 r -1 2r( 4+2r ) !

 2

^2r

-1  B

_2r

(4)

 ( ) 3 = 7 8  8

1 + Σ

r=1



 ¹⁺ 1! 

2r -1

+ 2!

( 2r -1 )

²

( 2r -1 )

³

e

^-(^2r-1⁾

+ 2 1 Σ

r=1



  2 ^-3 r -1 2r( 2+2r ) ! 

 2

^2r

-2  B

_2r

 ( ) 3 = 7 8  4

1  2 

1 +log 2 + Σ

r=1



 ¹⁺ 1! 

2r -1

( 2r -1 )

³

e

^-(^2r-1⁾

+ 2 1 Σ

r=1



  ^-2 2r 2r( 2+2r ) ! 

 2

^2r

-2  B

_2r

 ( ) 3 = 7 8  2

1 + Σ

r=1



 ¹⁺ 1! 

4r -2

( 2r -1 )

³

e

^-(^4r-2⁾

+ 2 1 Σ

r=1



  ^-2 2r 2r( 2+2r ) ! 

 2

^2r

-2  B

_2r

2

^2+2r

3 Formulas for Riemann Zeta at odd number

In this chapter, we obtain non-automorphism formulas for Riemann Zeta at odd number.

Where, Bernoulli numbers, Euler numbers and tangent numbers are as follows.

B

0

=1 , B

2

=1/6 , B

4

=-1/30 , B

6

=1/42 , B

8

=-1/30 , ^ E

0

=1 , E

2

=-1 , E

4

=5 , E

6

=-61 , E

8

=1385 , ^ T

₁

=1 , T

₃

=2 , T

₅

=16 , T

₇

=272 , T

₉

=7936 , 

And Harmonic number is

H

_s

= Σ

t=1 s

1/t =  ( 1+s ) + 

For

0 < x < 2 

,

 ( 2n +1 ) = - x 1 Σ

r=1



Σ

s=0 n

( -1 )

^s

( 2 s ) !

 2

^2s

-2  B

_2s

( rx )

^2s

r

²ⁿ⁺²

sin rx

+( -1 )

ⁿ

x

²ⁿ

 ^Σ

_s=0



n

( 2s ) !( 2n +1-2s ) !

 2

^2s

-2  B

_2s

H

_2n+1-2s

+ Σ

r=1



Σ

s=0 n

( 2s ) !( 2n +1+2r -2s ) !

 2

^2s

-2  B

_2s

2r

 

B

_2r

x

^2r

 ( 2n +1 ) = Σ

r=1



Σ

s=0 n

( 2s ) !



E

_2s

( rx )

^2s

r

²ⁿ⁺¹

cos rx

-( -1 )

ⁿ

x

²ⁿ

 ^Σ

s=0



n

( 2s ) !( 2n -2s ) ! E

_2s

H

_2n-2s

+ Σ

r=1



Σ

s=0 n

( 2s ) !( 2n +2r -2s ) ! E

_2s

2r

 

B

_2r

x

^2r

For

0 < x  

,

 ( 2n +1 ) = 2

²ⁿ

-1

2

²ⁿ

 x 1 Σ

r=1



Σ

s=0 n

( 2s ) !

( -1 )

^s

B

_2s

 2

^2s

-2 (  rx )

^2s

r

²ⁿ⁺²

( -1 )

^r

sin rx

- ( -1 )

ⁿ

x

²ⁿ

Σ

r=1



Σ

s=0 n

( 2s ) !( 2n +1+2r -2s ) !

 2

^2s

-2  B

_2s

2r 

 2

^2r

-1 

_{ }

B

_2r

x

^2r

 ( 2n +1 ) = - 2

²ⁿ

-1

2

²ⁿ

 ^Σ

r=1



Σ

s=0 n

( 2s ) !



E

_2s

( rx )

^2s

r

²ⁿ⁺¹

( -1 )

^r

cos rx

- ( -1 )

ⁿ

x

²ⁿ

Σ

r=1



Σ

s=0 n

( 2s ) !( 2n +2r -2s ) ! E

_2s

2r 

 2

^2r

-1 

_{ }

B

_2r

x

^2r

 ( 2n +1 ) =

2

²ⁿ⁺¹

-1 2

²ⁿ⁺¹

Σ

r=1



Σ

s=0 n

( 2s ) !



E

_2s

( 2r -1 ) x

^2s

( 2r -1 )

²ⁿ⁺¹

cos



( 2r -1 ) x



-

2

²ⁿ⁺¹

-1 ( -1 )

ⁿ

( 2 x )

²ⁿ

 ^Σ

s=0



n

( 2 s ) !( 2 n -2 s ) ! E

2s

H

2n-2s

- Σ

r=1



Σ

s=0 n

( 2 s ) !( 2 n +2 r -2 s ) ! E

2s 

2

²^r

-2

_

B

2r_

2 r x

²^r

Especially,

 ( 2n +1 ) = ( -1 )

ⁿ



²ⁿ

 ^Σ

_s=0



n

( 2s ) !( 2n +1-2s ) !

 2

^2s

-2  B

_2s

H

_2n+1-2s

+ Σ

r=1



Σ

s=0 n

( 2s ) !( 2n +1+2r -2s ) !

 2

^2s

-2  B

_2s

2r

 

B

_2r



^2r

 ( 2n +1 ) =

2

²ⁿ

-1 ( -1 )

^n-1

( 2  )

²ⁿ

Σ

r=1



 ^Σ

_s=0



n

( 2s ) !( 2n +1+2r -2s ) !

 2

^2s

-2  B

_2s

T

_2r-1

  2



^2r

(5)

 ( 2n +1 ) =

2

²ⁿ⁺¹

-1

( -1 )

ⁿ^-1



²ⁿ

 ^Σ

s=0



n

( 2 s ) !( 2 n -2 s ) ! E

2s

H

2n-2s

- Σ

r=1



Σ

s=0 n

( 2 s ) !( 2 n +2 r -2 s ) ! E

2s 

2

²^r

-2

_

B

2r_

2 r 1   2



^2r

Example

 ( ) 5 = 

⁴

 ²¹⁶⁰⁰ ²⁶⁹ ⁺ ^Σ

_r=1





 ^- ( 2r +5 ! ) 

1 +

6( 2r +3 ! )

1 -

360( 2r +1 ! ) 7

2r

 

B

_2r



^2r

 ( ) 5 = - 15 16 

⁴

Σ

r=1



 ^- ⁽ 2r +5 ! ⁾ 

1 +

6( 2r +3 ! )

1 -

360( 2r +1 ! ) 7

2r

 2

^2r

-1 

_{ }

B

_2r



^2r

 ( ) 5 = 31



⁴

 ²⁸⁸ ⁸³ ⁺ ^Σ

r=1





 ( 4+2r ) ! 

1 -

2( 2+2r ) !

1 +

24( 2 r ) ! 5

2 r



2

²^r

-2

_

B

_2r_

  2



^2r

4 Formulas for Riemann Zeta at even number

In this chapter, we obtain non-automorphism formulas for Riemann Zeta at even number.

Where, Bernoulli numbers and Euler numbers are as follows.

B

0

=1 , B

2

=1/6 , B

4

=-1/30 , B

6

=1/42 , B

8

=-1/30 , ^ E

0

=1 , E

2

=-1 , E

4

=5 , E

6

=-61 , E

8

=1385 , ^

For

0 < x < 2 

,

 ( 2n ) = - x 1 Σ

r=1



Σ

s=0 n-1

( 2s ) !

( -1 )

^s

B

_2s

 2

^2s

-2 ( )  rx

^2s

r

²ⁿ⁺¹

sin rx

- 2( 2n ) !



B

_2n

x

²ⁿ

 ^ ²

²ⁿ

^{-2 -} ^ x 

  2

²ⁿ⁺¹

-2 

 ( 2n ) = Σ

r=1



Σ

s=0 n-1

( 2s ) !

 

E

_2s

( ) rx

^2s

r

²ⁿ

cos rx

- 2( 2n ) !



E

_2n

x

²ⁿ

+ 2



( 2n ) !

2

²ⁿ

 2

²ⁿ

-1 

_

B

_2n_

x

^2n-1

For

0 < x  

,

 ⁽ ² n ⁾ =

2

²ⁿ

-2 2

²ⁿ

Σ

r=1



Σ

s=0 n-1

( 2 s ) !

( -1 )

^s

B

_2s

 2

^2s

-2 ( )  rx

^2s-1

r

²ⁿ

( -1 )

^r

sin rx

+ 2( 2 n ) !

 B

_2n

 ( 2 x )

²ⁿ

 ⁽ ² n ⁾ = -

2

²ⁿ

-2 2

²ⁿ

 ^Σ

^r⁼¹^

^Σ

ⁿ^s⁼⁰^-1

^{ } ^E

^2s

⁽ ² ^s ^{( )} ⁾ ^rx ^!

^2s

⁽ ^-1 ⁾ _r

^r²ⁿ

^cosrx ^- ^ ²⁽ ^E

²ⁿ

² ⁿ ^ ⁾ ^x ^!

²ⁿ



 ⁽ ² n ⁾ = -

2

²ⁿ

-1 2

²ⁿ

Σ

r=1



Σ

s=0 n-1

( 2 s ) !

( -1 )

^s

B

_2s

 2

^2s

-2 {(  2 r -1 ) x }

^2s-1

( 2 r -1 )

²ⁿ

sin

^

( 2 r -1 ) x

^

+ 2



( 2 n ) ! 2

²ⁿ

_ B

_2n

_ x

^2n-1

 ⁽ ² n ⁾ =

2

²ⁿ

-1 2

²ⁿ

Σ

r=1



Σ

s=0 n-1

( 2 s ) !

  E

_2s

{( 2 r -1 ) x }

^2s

( 2 r -1 )

²ⁿ

cos

^

( 2 r -1 ) x

^

+ 4



( 2 n ) ! 2

⁴ⁿ

_ B

_2n

_ x

^2n-1

Especially,

 ⁽ ² n ⁾ =

2( 2 n ) !

 B

_2n

 ( 2  )

²ⁿ

 ⁽ ² n ⁾ = -

2

²ⁿ

-2

1  ^Σ

^r⁼¹^

^Σ

ⁿ^s⁼⁰^-1

^{ } ^E

^2s

⁽ ² ^s ⁽ ⁾ ^ ^! ^r ⁾

^2s

r

²ⁿ



( -1 )

^r

- 2( 2 n ) !

 E

_2n

 

²ⁿ

By-products

Σ

s=0 n-1

( 2 s ) !( 2 n -2 s ) !

 2

^2s

-2  B

_2s

= -

( 2 n ) !0!

 2

²ⁿ⁺¹

-2  B

_2n

(6)

Σ

s=0 n-1

( 2 s ) !( 2 n -1-2 s ) ! E

2s

=

( 2 n ) ! 2

²ⁿ

 2

²ⁿ

-1  B

2n

5 Formulas for Riemann Zeta at complex number

In this chapter, we obtain the formulas for Riemann Zeta at a complex number by processing " 2 Formulas for Riemann Zeta at natural number "

Where, Bernoulli numbers are as follows.

B

₀

=1 , B

₂

=1/6 , B

₄

=-1/30 , B

₆

=1/42 , B

₈

=-1/30 , ^

And gamma function and incomplete gamma function are

 ^{( )} p = 

₀^

^t

^p-1

^e

^-t

^dt ^, ^ ⁽ ^p,x ⁾ ⁼ 

_x^

^t

^p-1

^e

^-t

^dt

And

p

is a complex number such that

p 1 , 0 , -1 , -2 , 

. For

x = u + v i s . t . 0 < | | x  2  , u 0

,

 ^{( )} p =

 ^{( )} p x

^p^-1

 p -1 

1 - 2 p

x + Σ

r=1



 ^{( )} p r

^p

 ⁽ p , xr ⁾ - Σ

r=1



  2 ^- r -1 ^p 2 r  ⁽ p +2 r ⁾ B

_2r

x

^p^-1+2r

 ^{( )} p = _ ( ) p x

^p-1

 p -1 

1 + 2p ( p -1 ) x

- log x + Σ

r=1



 ( p -1 ) r

^p

 ( p -1, xr ) - Σ

r=1



  ^1-p 2r 2r  ( p +2r ) B

_2r

x

^p-1+2r

For

x = u + v i s . t . 0 < | | x   , u 0

,

 ^{( )} p =

2

^p-1

- 1 2

^p-1

 ² ^ ⁽ p +1 ⁾ 

x

^p

- Σ

r=1



 ^{( )} p

 ⁽ p , xr ⁾ r

^p

( -1 )

^r

+ Σ

r=1



  2 ^- r -1 ^p 2 r  ⁽ p +2 r ⁾

 2

^2r

-1  B

_2r

x

^p-1+2r

 ^{( )} p =

2

^p

-1 2

^p

 2( p -1 )  ^{( )} p x

^p-1

+ Σ

r=1



 ^{( )} p ( 2 r -1 )

^p





p , x ( 2 r -1 )



+ 2 1 Σ

r=1



  2 ^- r -1 ^p ² r  ( p +2 r ) 

 2

^2r

-2  B

_2r

x

^p-1+2r

 ^{( )} p = 2

^p

-1

2

^p

 ²  ^{( )} p x

^p-1

 p -1 

1 - log 2

x + Σ

r=1



 ⁽ p -1 ( ⁾ 2 r -1 )

^p





p -1 , x ( 2 r -1 )



+ 2 1 Σ

r=1



  ^1- ² r ^p 2 r  ⁽ p +2 r ⁾ 

 2

^2r

-2  B

_2r

x

^p-1+2r

Especially,

 ^{( )} p =

2 ( p -1 )  ⁽ p +1 ⁾ p +1

+ Σ

r=1



 ^{( )} p r

^p

 ⁽ p , r ⁾ - Σ

r=1



  2 ^- r -1 ^p 2 r  ⁽ p +2 r ⁾ B

2r

 ^{( )} p =

2( p -1 )  ⁽ p +1 ⁾ p

²

+1

+ Σ

r=1



 ⁽ p -1 ⁾ r

^p

 ⁽ p -1 , r ⁾ - Σ

r=1



  ^1- 2 r ^p 2 r  ⁽ p +2 r ⁾ B

_2r

 ^{( )} p =

2

^p-1

- 1 2

^p-1

 ² ^ ⁽ p +1 ⁾ 

1 - Σ

r=1



 ^{( )} p

 ⁽ p , r ⁾ r

^p

( -1 )

^r

+ Σ

r=1



  2 ^- r -1 ^p 2 r  ⁽ p +2 r ⁾

 2

^2r

-1  B

_2r

 ^{( )} p = 2

^p

-1

2

^p

 ²⁽ ^p ^-1 ¹ ⁾ ^ ^{( )} ^p ⁺ ^Σ

r=1





 ( ) p ( 2r -1 )

^p

 ( p ,2r -1 ) + 2

1 Σ

r=1



  ² ^-p r -1 2r  ( p +2r )

 2

^2r

-2  B

_2r

 ^{( )} p = 2

^p

-1

2

^p

 ² ^ ¹ ^{( )} ^p ^ ^p ¹ ^-1 ^+log ² ^ ⁺ ^Σ

r=1





 ( p -1 ( ) 2r -1 )

^p

 ( p -1, 2r -1 ) + 2

1 Σ

r=1



  ^1- 2r ^p 2r  ( p +2r )

 2

^2r

-2  B

_2r

(7)

 ^{( )} p = 2

^p

-1

2

^p

 ⁽ ^p ^-1 ² ⁾

^p-2

^ ^{( )} ^p ⁺ ^Σ

r=1





 ( p -1 ( ) 2r -1 )

^p

 ( p -1 , 4 r -2 ) + 2

1 Σ

r=1



  ^1-p 2r 2r  ( p +2r )

 2

^2r

-2  B

_2r

2

^p-1+2r

6 Global definition of Riemann Zeta, and generalization of related coefficients

From Euler to Riemann, the zeta function was defined with patches as the domain was expanded.

 ^{( )} p =  ^Σ

^r

^1-2

⁼¹^

( 2 ¹  ^r ¹

^1-p^p

)

^1-p

^Σ

^r⁼¹^

⁽ ^-1 ^r ⁾

^p^r-1

^Re ⁰ ^p ^{( )} ^ ^ ^p ^Re ¹ ^{> 1} ^{( )} ^p ^ ¹ 2  ⁽ ^1- p ⁾

sin   ^p ² ^ _1-2 ¹

^p

^Σ

r=1



r

^1-p

( -1 )

^r-1

Re ( ) p  0 p  0

This is inconvenient. so, we focus on the following sequence.

n

B

_r

= Σ

s=1 r

( -1 )

^r-s

  ^r _s ^s

ⁿ

^r ⁼⁰ ^, ¹ ^, ² ^, ^ ^,n

Using this sequence, we can define the Zeta function on the whole complex plane as follows.

Definition 6.2.1

We difine the Riemann Zeta Function on the complex plane as follows.

 ^{( )} p =

1- 2

^1-p

1 Σ

r=1



2

^r+1

1 Σ

s=1 r

( -1 )

^s-1

  ^r _s ^s

^-p

^p ^1

Furthermore, by using this sequence, the following various coefficients can be generalized.

Generalized Stirling Number of the 2nd kind

S

2

( p , r ) = r !

1 Σ

s=1



( -1 )

^s-1

  ^r _s ^s

^p

^r ⁼¹ ^, ² ^, ³ ^, ^

Generalized Tangent Number

T

p

=  ^Σ

r=1

⁰ ^p ^{= 0}



2

^p-r

Σ

s=1 r

( -1 )

^s-1

  ^r _s ^s

^p

^p ^ ⁰

Generalized Bernoulli Number

B

_p

=  ^- ² ¹ p = 1 2

^p

-1

p Σ

r=1



2

^r+1

1 Σ

s=1 r

( -1 )

^s-1

  ^r _s ^s

^p-1

^p ^ ¹

7 Completed Riemann Zeta

In 7.1, we consider even function and odd function for complex function. Generally, we obtain the same results as for real-valued functions, but the results unique to complex-valued functions are as follows.

Theorem 7.1.3

Let

f ( ) z

be a complex function in the domain

D

.

(1)

If

f ( ) z

is an even function , both the real part and the imaginary part are even functions.

(2)

If

f ( ) z

is an odd function , both the real part and the imaginary part are odd functions.

(8)

Theorem 7.1.4

Let

f ( ) z

be a complex function in the domain

D

. Then,

if

f ( ) z

is an even function or an odd function , 

f ( ) z

² is an even function.

In 7.2, we study complex conjugate properties. Especially when the function

f ( ) z

is an even function or an odd function with complex conjugate properties, two important theorems are obtained.

Theorem 7.2.3

When

f ( x,y ) = u ( x,y ) + iv ( x,y )

is a function with the complex conjugate property in the domain

D

^,

(1)

if

f ( x,y )

is an even function,

u ( x,y ) = u ( x, - y ) = u ( - x,y ) = u ( - x, - y ) v ( x,y ) = - v ( x, - y ) = - v ( - x,y ) = v ( - x, - y )

(2)

if

f ( x,y )

is an odd function,

u ( x,y ) = u ( x, - y ) = - u ( - x,y ) = - u ( - x, - y ) v ( x,y ) = - v ( x, - y ) = v ( - x,y ) = - v ( - x, - y )

Corollary 7.2.3

Let

f ( x,y ) = u ( x,y ) + iv ( x,y )

be a function with the complex conjugate property in the domain

D

^.

Then, the followings hold for any real number

x , y  D

^.

(1)

When

f ( x,y )

is an even function ,

v ( x, 0 = 0 ) , v ( 0 ,y ) = 0

.

(2)

When

f ( x,y )

is an odd function ,

u ( 0 ,y ) = 0 , v ( x, 0 = 0 )

.

Theorem 7.2.4

When

f ( ) z

is a function with the complex conjugate property in the domain

D

and has a zero

z

1

= x

1

+ iy

1

 x

₁

 0 

,

(1)

if

f ( ) z

is an even function,

- x

1

- iy

1

, x

1

- iy

1

, - x

1

+ iy

1 are also zeros of

f ( ) z

^.

(2)

if

f ( ) z

is an odd function,

- x

1

- i y

1

, x

1

- iy

1

, - x

1

+ iy

1 are also zeros of

f ( ) z

^.

In 7.3, symmetric functional equations are derived from functional equations.

Formula 7.3.1 (

Riemann

)



^-²

z

   2

z  ( ) z = 

^- ²

1-z

  2 

1- z  ( 1-z ) z  0, 1



^-²

1



2



1+z

  2 

1  2 

1 + z   2 

1 + z = 

^-²

1



2



1-z

  2 

1  2 

1 -z   2 

1 - z

Where,

z  1/2

In 7.4, we define the completed Riemann zeta functions

 ^{( )} z ,  ^{( )} z

as follows, respectively. These are a little different from Landau's definition.

 ^{( )} z = - z ( 1- z ) 

^-²

z

   ² ^z ^ ^{( )} ^z

 ^{( )} z = -  ² ¹ ⁺ ^z   ² ¹ ^- ^z  ^

^-²¹

^

²¹⁺^z

^ ^  ² ¹  ² ¹ ⁺ ^z   ^  ² ¹ ⁺ ^z 

Then, the following equations hold from Formula 7.3.1 .

 ^{( )} z =  ⁽ ^1- z ⁾

 ^{( )} z =  ⁽ ^- z ⁾

From the latter, we can see that

 ( ) z

is an even function. Therefore, Thoerem 7.2.3 (1) and Corollary 7.2.3 hold for the real part

u ( x,y )

and the imaginary part

v ( x,y )

^of

 ^{( )} z

as they are. And from Theorem 7.2.4 , the following very important theorem is obtained.

(9)

Theorem 7.4.1

If Riemann zeta function

 ( ) z

has a non-trivial zero whose real part is not

1/2

, the one set consists of the following four.

1/2+ 

₁

 i ^

1

, 1/2- 

₁

 i ^

1

( 0 < 

₁

^< ^1/2 ⁾

08 Factorization of Completed Riemann Zeta

In 8.1, the following Hadamard product is shown.

Formula 8.1.1 ( Hadamard product of  ^{( )} z ⁾

Let completed zeta function be as follows.

 ^{( )} z = - z ( 1- z ) 

^-²

z

   ² ^z ^ ^{( )} ^z

When non-trivial zeros of

 ^{( )} z

are

z

_k

= x

_k

 i y

_k

k =1, 2,3, 

and



is Euler-Mascheroni constant,

 ( ) z

is expressed by the Hadamard product as follows.

 ^{( )} z = e 

^{log2 +} 2



log -1-2

 z

Π

k=1



 ^1- ^z

k



z e

^z^k

z

 ^{( )} z = e 

^{log2 +} 2



log -1-2

 z

Π

n=1



 ^1- x

n



2

+ y

n 2

2 x

n

z +

x

n 2

+ y

n

2

z

²

e

^xⁿ

2+y_n² 2x_nz

And, the following special values are obtained.

Π

n=1



 ^1- x

_n²

+ y

_n²



2 x

_n

-1 e

^xⁿ

2+y_n² 2x_n

= e

1+2

-log2 - 2 log

= 1.02336448

Π

n=1



 ^1-  x

_n

 iy

_n



²



1  ^1-  x

_n

 i y

_n



²



1 =

3 

In 8.2, we consider how the formulas in the previous section are expressed when non-trivial zeros whose real part is 1/2 and non-trivial zeros whose real part is not 1/2 are mixed. Then, we obtain the following theorems.

Theorem 8.2.2

Let



be Euler-Mascheroni constant, non-trivial zeros of Riemann zeta function are

x

_n

+ i y

_n

n =1, 2, 3, 

. Among them, zeros whose real part is 1/2 are

1/2  i y

_r

r =1, 2,3, 

and zeros whose real parts is not 1/2 are

1/2 

_s

i 

_s



0 < 

_s

< 1/2



s =1, 2,3, 

. Then the following expressions hold.

Π

n=1



 ^1- ^x

n



2

+ y

_n²

2x

_n

-1

= 1

Σ

n=1



x

_n²

+ y

_n²

2x

_n

= Σ

r=1



1/4 + y

_r²

1 + Σ

s=1





1/2 

_s²

 

_s²



1 2 

_s

+



1/2 

_s²

 

_s²

1 2 

_s

Σ

n=1



x

_n²

+ y

_n²

2x

_n

= 1 + 2

 - log 2 - 2 log 

= 0.0230957

Formula 8.2.3 ( Special values )

When non-trivial zeros of Riemann zeta function are

x

_k

 i y

_k

k =1, 2, 3, 

, the following expressions hold.

Π

n=1



 ^1- x

_n

 i y

_n



1  ^1- x

_n

 iy

_n



1 = 1

Π

n=1



 ¹⁺ x

_n

 i y

_n



1  ¹⁺ x

_n

 iy

_n



1 =

3 

Summary of Zeta

Summary

1 Zeta Generating Functions

 ( ) x = Σ

r

1 ,  ( ) x = Σ

r

( -1 )

,  ( ) x = Σ

( 2 r -1 )

1

B

=1 , B

=1/6 , B

=-1/30 , B

=1/42 , B

=-1/30 ,  E

=1 , E

=-1 , E

=5 , E

=-61 , E

=1385 , 

H

= Σ

1/t =  ( 1+s ) + 

 ( ) n =

( n -1 ! ) ( - x )

 log x - H

 + 2 ( -1 )

n ! x

+ Σ

r

e

- ( -1 )

Σ

2 r ( 2 r + n -1 ! ) B

x

- Σ

s !

( -1 )

x

 ( n - s )

 ( ) n = - 2 ( -1 )

n ! x

- Σ

( -1 )

r

e

+ ( -1 )

Σ

2 r ( 2 r + n -1 ! )

 2

-1  B

x

- Σ

s !

( -1 )

x

 ( n - s )

 ( ) n = 2 ( -1 )

( n -1 ! ) x

 log 2 x - H

 + Σ

( 2r -1 )

e

+ 2 ( -1 )

Σ

2 r ( 2 r + n -1 ! )

 2

-2  B

x

- Σ

s !

( -1 )

x

 ( n - s )

Σ

r

cosrx

- ( 2 n -2 ! ) ( -1 )

 ^{( )} x = Σ

1 ,  ^{( )} x = Σ

,  ^{( )} x = Σ

=-1/30 , ^ E

=1385 , ^

 ⁽ n - s ⁾

 ^{( )} n = - 2 ( -1 )

 ⁽ n - s ⁾

 ^{( )} n = 2 ( -1 )

 ^log ² ^x ^- ^H

 ⁺ ^Σ

 ⁽ n - s ⁾

 ⁽ ² n -1-2 s ⁾

 ⁽ ² n +1-2 s ⁾

-1  _{ } B

 ⁽ ² n -1-2 s ⁾

-1  _{ } B

 ⁽ ² n +1-2 s ⁾

 ^log 2 