Summary of Beta - Dr. Huang

Summary

1 Dirichlet Beta Generating Functions

sech x , sec x and csc x can be expanded to Fourier series and Taylor series. And if the termwise higher order integration of these is carried out, Dirichlet Beta at a natural number are obtained.

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

The work that obtain the non-automorphism formulas by removing lower betas from these is done in the next chapter

" 2 Formulas for Dirichlet Beta " .

In this chapter, we obtain the following polynomials from the beta generating functions of each family of sech, sec and csc . Where, Dirichlet Beta and Dirichlet Lambda are as follows.

 ^{( )} x = Σ

r=0



( 2 r +1 )

^x

( -1 )

^r

,  ^{( )} 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 = Σ

r=0



( 2 r +1 )

ⁿ

( -1 )

^r

e

^-(2r+1)x

- 2 ( -1 )

ⁿ

Σ

r=0



( 2 r + n ) ! E

_2r

x

^2r+n

- Σ

s=1 n-1

s !

( -1 )

^s

x

^s

 ⁽ n - s ⁾

Σ

r=0



( 2 r +1 )

²ⁿ⁺¹

( -1 )

^r

sin {( 2 r +1 ) x }

- 2 ( -1 )

ⁿ

Σ

r=0



( 2 n +1+2 r ) !

  E

_2r

x

^2n+1+2r

= Σ

s=0 n-1

( 2 s +1 ! )

( -1 )

^s

x

^2s+1

 ^{( 2} n -2 s ⁾

Σ

r=0



( 2 r +1 )

²ⁿ

( -1 )

^r

cos {( 2 r +1 ) x }

- 2 ( -1 )

ⁿ

Σ

r=0



( 2 n +2 r ) !

  E

_2r

x

^2n+2r

= Σ

s=0 n-1

( 2 s ) !

( -1 )

^s

x

^2s

 ^{( 2} n -2 s ⁾

Σ

r=0



( 2 r +1 )

²ⁿ⁺¹

( -1 )

^r

cos {( 2 r +1 ) x }

= Σ

s=0 n

( 2 s ) !

( -1 )

^s

x

^2s

 ^{( 2} n +1-2 s ⁾

Σ

r=0



( 2 r +1 )

²ⁿ

( -1 )

^r

sin {( 2 r +1 ) x }

= Σ

s=0 n-1

( 2 s +1 ! )

( -1 )

^s

x

^2s+1

 ^{( 2} n -1-2 s ⁾

 ^{( 2} n ⁾ =

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

ⁿ

  ^{2 }

^2n-1

 ^{log 4  - H}

^2n-1



+ 2 ( -1 )

ⁿ

Σ

r=1



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

 2

^2r

-2  _{ } B

_2r

  ^{2 }

^2n-1+2r

- Σ

s=1 n-1

( 2 s -1 ! ) ( -1 )

^s

  ^{2 }

^2s-1

^{ ( 2 n +1-2 s )}

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

 ^{( -} n ⁾ = 2

ⁿ⁺¹

1 Σ

r=0 n

( -1 )

^r_n

K

_r

n =1 , 2 , 3 , 

 ^{( -2} n ⁾ = 2

²ⁿ⁺¹

1 Σ

r=0 2n

( -1 )

^r_2n

K

_r

n =1 , 2 , 3 , 

= 2 n E

_2n

n =1 , 2 , 3 , 

Where, n

K

_r is a kind of Eulderian Number and is defined as follows.

n

K

_r

= Σ

k=0 r

( -1 )

^k

  ^{n +1} _k ^{( 2 r +1-2 k )}

ⁿ

^{n =1 , 2 , 3 , }

2 Formulas for Dirichlet Beta

Here, removing the lower betas from the the automorphism formulas in the previous chapter, we obtain the following non-automorphism formulas. 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, gamma function and incomplete gamma function were as follows.

 ^{( )} p = 

₀^

^t

^p-1

^e

^-t

^{dt ,  ( p,x ) =} 

_x^

^t

^p-1

^e

^-t

^dt

2.1 Formulas for Beta at natural number For 0 < x   /2

,

 ^{( )} n = Σ

r=0



Σ

s=0 n-1

s ! ( 2 r +1 )

^s

x

^s

( 2 r +1 )

ⁿ

( -1 )

^r

e

^-(2r+1)x

+ 2 x

ⁿ

Σ

r=0



  ^- 2 ⁿ r ( n +2 r ) ! E

_2r

x

^2r

Especially,

 ^{( )} n = Σ

r=0



Σ

s=0 n-1

s !2

^s

( 2 r +1 )

^s

( 2 r +1 )

ⁿ

( -1 )

^r

e

^-(r+1/2)

+ 2

ⁿ⁺¹

1 Σ

r=0



  ^- 2 ⁿ r ( n +2 r ) !2

^2r

E

_2r

 ^{( )} n = Σ

r=0



Σ

s=0 n-1

s ! ( 2 r +1 )

^s

( 2 r +1 )

ⁿ

( -1 )

^r

e

^-(2r+1)

+ 2 1 Σ

r=0



  ^- 2 ⁿ r ( n +2 r ) ! E

_2r

Example

 ( ) 4 = Σ

r=0



 ¹⁺ 1!2

¹



2r+1 +

2!2

²

( 2r+1 )

²

+

3!2

³

( 2r+1 )

³

( 2r +1 )

⁴

( -1 )

^r

e

-r-2 1

+ 2

⁵

1 Σ

r=0



  ^-4 2r ( 4+2r ) !2

^2r

E

_2r

 ( ) 4 = Σ

r=0



 ¹⁺ 1! 

2r+1 + 2!

(2r+1)

²

+ 3!

(2r+1)

³

( 2r +1 )

⁴

( -1 )

^r

e

^-2r-1

+ 2 1 Σ

r=0



  ^-4 2r ( 4+2r ) ! E

_2r

2.2 Formulas for Beta at even number For 0 < x   /2

,

 ^{( 2} n ⁾ = Σ

r=0



Σ

s=0 n-1

( 2 s ) !

  E

_2s

{( 2 r +1 ) x }

^2s

( 2 r +1 )

²ⁿ

( -1 )

^r

cos {( 2 r +1 ) x }

- 2 ( -1 )

ⁿ

Σ

r=0



 ^Σ

s=0



n-1

 ^{2 n} 2 ⁺² s ^r  E

_2s

( 2 n +2 r ) !

  E

_2r

x

^2n+2r

 ^{( 2} n ⁾ = - x 1 Σ

r=1



Σ

s=0 n-1

( 2 s ) !

( -1 )

^s

B

_2s

 2

^2s

-2 {(  2 r +1 ) x }

^2s

( 2 r +1 )

²ⁿ⁺¹

( -1 )

^r

sin {( 2 r +1 ) x }

+ ( -1 )

ⁿ

2 x

²ⁿ

Σ

r=1



 ^Σ

_s=0



n-1

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

 2

^2s

-2  B

_2s

2 r

  E

_2r

x

^2r

Especially,

 ^{( 2} n ⁾ = 2 ( -1 )

^n-1

Σ

r=0



 ^Σ

s=0



n-1

 ^{2 n} 2 ⁺² s ^r  E

_2s

( 2 n +2 r ) !

  E

_2r

  ^{2 }

^2n+2r

Example

 ( ) 4 = - 2 1 Σ

r=0



   ⁴⁺² 0 ^r E

₀

+   ⁴⁺² 2 ^r E

2

 ( 4+2 r ) !

  E

_2r

  ^{2 }

^4+2r

 ( ) 6 = 2 1 Σ

r=0



   ^6+2r 0 E

₀

+   ^6+2r 2 E

₂

+   ^6+2r 4 E

4



( 6+2 r ) !

  E

_2r

  ^{2 }

^6+2r

2.3 Formulas for Beta at odd number For 0 < x   /2

,

 ^{( 2} n -1 ⁾ = Σ

r=0



Σ

s=0 n-1

( 2 s ) !

  E

2s

{( 2 r +1 ) x }

^2s

( 2 r +1 )

^2n-1

( -1 )

^r

cos {( 2 r +1 ) x }

 ^{( 2} n -1 ⁾ = - x 1 Σ

r=0



Σ

s=0 n-1

( 2 s ) !

( -1 )

^s

B

_2s

 2

^2s

-2 {(  2 r +1 ) x }

^2s

( 2 r +1 )

²ⁿ

( -1 )

^r

sin {( 2 r +1 ) x }

Especially,

 ^{( 2} n -1 ⁾ = 4



( 2 n -2 ! )

 E

_2n-2



  ^{2 }

^2n-2

2.4 Formulas for Beta at complex number

When

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 = Σ

r=0



 ^{( )} p





p , ( 2 r +1 ) x



( 2 r +1 )

^p

( -1 )

^r

+ 2 x

^p

Σ

r=0



  ^- 2 ^p r  ⁽ p +1+2 r ⁾ E

_2r

x

^2r

Especially,

 ^{( )} p = Σ

r=0



 ^{( )} p

 ⁽ p , 2 r +1 ⁾

( 2 r +1 )

^p

( -1 )

^r

+ 2 1 Σ

r=0



  ^- 2 ^p r  ⁽ p +1+2 r ⁾ E

_2r

3 Global definition of Dirichlet Beta and Generalized Euler Number

Diriclet beta function is defined on the whole complex plane with patches as follows.

 ^{( )} p =  ^{  Σ}

^r=1

^{ 2 ( ( 2 -1}

^1-p

^{r -1 ) cos}

^r-1

⁾

^p

^{p 2   ( 1- p ) Σ}

^r=1

( 2 r -1 )

^1-p

^{Re ( ) p 0} ( -1 )

^r-1

Re ( ) p < 0

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

n

B

_r

= Σ

s=0 r

( -1 )

^r-sr

Ｃ

s

  ^{s - 2 1}

ⁿ

^{r =0 , 1 , 2 ,  ,n}

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

Definition 3.2.1

 ^{( )} p = Σ

r=1



2

^r+1

1 Σ

s=1 r

( -1 )

^s-1

  ^r _s ^{( 2 s -1 )}

^-p

Furthermore, by using this sequence, Euler Number can be defined on the whole complex plane.

Definition 3.3.1

E

_p

= Σ

r=1



2

^r

1 Σ

s=1 r

( -1 )

^s-1

  ^r _s ^{( 2 s -1 )}

^p

4 Completed Dirichlet Beta

In 4.1, symmetric functional equations are derived from functional equations.

Formula 4.1.1

  ² _

^1+z

^{  2 1 + 2 z   ( ) z =}   ² _

^2-z

^{  2 1 + 1- 2 z   ( 1- z )}

  ² _

²

3+z

  ^{4 3 + 2 z}  ^  ^{2 1 + z}  ⁼   ² _

²

3-z

  ^{4 3 - 2 z}  ^   ^{2 1 - z}

In 4.2, we define the completed Dirichlet beta functions

 ( ) z ,  ( ) z

as follows, respectively.

 ^{( )} z =   ² _

^1+z

^{   1+ 2 z  ( ) z}

 ^{( )} z =   ² _

²

3+z

  2 

1  2 

3 +z   2 

1 + z

Then, Formula 4.1.1 is expressed as follows.

 ^{( )} z =  ^{( 1-} z ⁾

 ^{( )} z =  ^{( -} z ⁾

From the latter, we can see that

 ^{( )} z

is an even function. Therefore,

 ( ) z

has the same properties as completed Riemann zeta function

 ( ) z

. ( See " 07 Completed Riemann Zeta ". ) And, as in

 ( ) z

, the following theorem holds.

Theorem 4.2.1

If Dirichlet beta 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 )}

05 Factorization of Completed Dirichlet Beta

In 5.1, the following Hadamard product is derived.

Formula 5.1.1

Let completed beta function be as follows.

 ^{( )} z =   ² _

^1+z

^{   1+ 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 

2



3log -2

 - log2 - 4log

 

4 3 z

Π

k=1



 ^{1- z}

k



z e

^zk

z

 ^{( )} z = e 

2



3log -2

 - log2 - 4log

 

4 3 z

Π

n=1



 ^{1- x}

n



2

+ y

_n²

2x

_n

z

+

x

_n²

+ y

_n²

z

²

e

^xn

2+y_n² 2x_n z

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

4log

 

4 3 +

2

 + log2 - 2 3log

= 1.08088915

Π

n=1



 ^1-  x

_n

 iy

_n



²



1  ^1-  x

_n

 i y

_n



²



1 =  ^{( -1 ) =} 1.16624361

In 5.2, 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 5.2.2

Let



be Euler-Mascheroni constant, non-trivial zeros of Dirichlet beta 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

_n²

+ 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

= 4log    4 3 +

2

 + log 2 - 2 3 log 

= 0.07778398

Formula 5.2.3 ( Special values )

When non-trivial zeros of Dirichlet beta 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 =  ^{( -1 ) = 1.1662436}

Theorem 5.2.4

Let non-trivial zeros of Dirichlet beta function are

x

_n

+ i y

_n

n =1, 2, 3, 

and



be Euler-Mascheroni constant.

If the following expression holds, non-trivial zeros whose real parts is not

1/2

do not exist.

Σ

n=1



1/4 + y

_n²

1 = 4 log    ^{4 3 + 2  + log 2 - 3 log 2  =} 0.07778398

Incidentally, when this was calculated using 10000

y

_r, both sides coincided with the decimal point 3 digits.

In 5.3, we show that

 ^{( )} z

is factored completely.

Theorem 5.3.1 ( Factorization of 

^{( )}z )

Let Dirichlet beta function be

 ^{( )} z

, the non-trivial zeros are

z

_n

= x

_n

 i y

_n

n =1, 2,3, 

and completed beta function be as follows.

 ^{( )} z =   ² _

^1+z

^{   1+ 2 z  ( ) z}

Then,

 ( ) z

is factorized as follows.

 ^{( )} z = Π

n=1



 ^1- x

_n²

+ y

_n²



2 x

_n

z +

x

_n²

+ y

_n²

z

²

In 5.4, we first derive the factorization of

 ^{( )} z

.

Theorem 5.4.1 ( Factorization of



^{( )}z )

Let Dirichlet beta function be

 ^{( )} z

, the non-trivial zeros are

z

_n

= x

_n

 i y

_n

n =1, 2,3, 

and completed beta function be as follows.

 ^{( )} z =   ² _

²

3+z

  2 

1  2 

3 +z   2 

1 + z

Then,

 ( ) z

is factorized as follows.

 ^{( )} z =  ( ) 0 Π

n=1



 ^1-



x

_n

-1/2

²

+ y

_n²



2

_

x

_n

-1/2

_

z

+



x

_n

-1/2

²

+ y

_n²

z

²

Where,

 ( ) 0 = Π

n=1



x

_n²

+ y

_n²



x

_n

-1/2

²

+ y

_n²

=   

2

^3/2

   4

3    2

1 = 0.98071361

And, using this theorem and Theorem 4.2.1 in the previous section, we obtaine the following theorem.

Theorem 5.4.4

When Dirichlet beta function is

 ( ) z

and the non-trivial zeros are

z

_n

= x

_n

 iy

_n

n =1 , 2 , 3 , 

, If the following expression holds, non-trivial zeros whose real parts is not

1/2

do not exist.

Π

r=1



1/4 + y

_r²

y

_r²

=   ² _

^3/2

^{   4 3    2 1 =} 0.98071361

(4.4₀) Incidentally, when this was calculated using 10000

y

_r, both sides coincided with the decimal point 4 digits.

06 Zeros on the Critical Line of Dirichlet Beta

In 6.1, substituting

z = 0+iy

for the completed Dirichlet beta

 ^{( )} z

,



h

( ) z =   ² _

²

3+iy

  ^{2 1}  ^{2 3 + iy}   ^  ^{2 1 + iy} 

We use this to calculate the zeros on the critical line. However, this function is too small in absolute value and can only find the zeros up to

y =917

.

So we normalize



_h

( ) y

and define the following sign function.

sgn ( ) y = -

 

_h

( ) y 



h

( ) y

= -   ² _

^{i y}

_{    }

2

1  2 

3 + _{iy }  2 

1 + _iy

  2 

1  2 

3 + iy   2 

1 + iy

Using this sign function

sgn ( ) y

, we can find the zeros at large

y

^.

In 6.2, multiplying this sign function

sgn ( ) y

by the absolute value of the Dirichlet beta

 ( 1/2+i y )

, we obtain a smooth function

B ( ) y

^.

B ( ) y = sgn ( ) y  ^  ^{2 1 + iy}   ^{= -}   ² _

^{i y}

_{    }

2

1  2 

3 + i y

  2 

1  2 

3 + _{i y}

  2 

1 + _{i y}

Using this

B ( ) y

function, we can find the zeros on the critical line of

 ( ) z

by the intersection of the curve and the

y

-axis In 6.3, first, a lemma is prepared.

Lemma

When

f ( ) z

is a complex function defined on the domain

D

, the following expression holds.

e

i Im log f ( )z

=



f ( ) z



f ( ) z

Applying this lemma to the gamma function in the 6.2 ,

B ( ) y = -   ² _

^{i y}

_{    }

2

1  2 

3 + _{i y}

  2 

1  2 

3 + _{i y}

  2 

1 + _{i y}

= -    2

^{i y}

e

i Im log 



2



1



2



3+

i y

  2 

1 + _{i y}

= - e

i



^{Im log }



2

 

1



2



3+

i y

- 2 ylog

4



  ^{2 1 + i y} 

From this, we obtain

B ( ) y = - e

^{i ( )y}

  ^{2 1 + i y} 

^where,

^{ ( ) y = Im log }  ^{2 1}  ^{2 3 + i y}   ^{- 2 y log 4 }

This is Riemann-Siegel style

B

function .

2012.04.13



2019.03.27 Updated

Kano Kono Alien's Mathematics