Mellin Convolution and its Extensions, Perron

(1)

Article

1

Mellin Convolution and its Extensions, Perron

2

Formula and Explicit Formulae

3

Jose Javier Garcia Moreta

4

Graduate student of Physics at the UPV/EHU (University of Basque country);In Solid State Physics;Practicantes Adan y Grijalba2 5 G;P.O 644 48920 Portugalete Vizcaya

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(Spain);[email protected]

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ABSTRACT: In this paper we use the Mellin convolution theorem, which is related to Perron's formula. Also

8

we introduce new explicit formulae for arithmetic function which generalize the explicit formulae of Weil for

9

other arithmetic functions different from the Von-Mangoldt function.

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Keywords: Riemann-weil formula; perron formula; explicit formulae; Riemann zeros

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MELLIN DISCRETE CONVOLUTION:

12

First of all we need to prove the following identity

13

1

1 ( )

( ) ( )

2

c i

s

n c i

n

a n f

F s G s x

x



i

  

  

 

_

 

 





(1)

Where

1

( )

s

n

a n

G s

n







is the Dirichlet generating functio of the coefficients a(n) and

14

1 0

( )

s

F s

_





dxf x x

 _{is the Mellin transform of the function}

15

16

The proof of (1) is easy and is obtained from the application of the Mellin convolution theorem

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[3] applied to the integral linear operator

18

19 

( ) )



₀

( ) ( )

L g x







dxf xt g t dt

1

( )

( ) (

)

n

g t



a n



x n







(2)

20

21

The Mellin transform of the integral operator defined in (2) is a product of 2 Mellin transforms.

22

From the property of the Dirac delta functions used to define the distribution g(t) we find from

23

the Mellin convolution theorem that

24

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1 (1 ) 1

0

1 ₀

( ) ( )

s

( )

s

( )

( ) ( )

n

dx

f xn a n x

F s dxx

g x

F s G s

 

 _ _{ }





















(3)

26

27

Now if we take the change of variable

x

1 x



and take the inverse Mellin transforms on both

28

sides of (3) we can prove (1) inmediatly

29

Now, if we set

1 (

1)

1 t>1

0

1 f

H t

t



 



 



 

_

 



inside the test function (1)then

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we recover Perron's formula [6] for the Coefficients of the Dirichlet series

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1

1 ( )

( )

2

c i s

n n x c i

x

a n H

a n

G s

n



i

s

  

   



_



_













with ₁

1

1 ( )

dx

_s

F s

s

x

 

 



(2)

But one of the best applications of our identity (1) is related to several Dirichlet

32

series (see [5]) in the form

1

( )

s

n

a n

G s

n







, Where G(s) includes powers or quotients of the

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Riemann zeta function for example

34

1

1 ( )

( )

s

n

s

n











1

'( )

( )

s

n

s

n

s

n















1

(2 )

( )

s

n

s

n

s

n













(3)

1

( )

| ( ) |

(2 )

s

n

s

n

s

n













1

(

1)

( )

s

n

s

n

s

n

















(4)

The definitions of the functions inside (5) and (6) are as follows

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1. The Möbius function,



( ) 1

n



if the number ‘n’ is square-free (not divisible by an square)

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with an even number of prime factors ,



( ) 0

n



if n is not squarefree and if the number

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‘n’ is square-free with an odd number of prime factors.

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2. The Von Mangoldt function



( ) log

n



p

, in case ‘n’ is a prime or a prime power and

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takes the value 0 otherwise

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3. The Liouville function



( ) ( 1)

n

 

( )n



( )

n

is the number of prime factors of the number

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‘n’

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4.

| ( ) |



n

is 1 if the number is square-free and 0 otherwise

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5.

|

1 ( )

1

p n

n

p







_





_







, the meaning of

p n

|

is that the product is taken only over the

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primes p that divide ‘n’.

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To obtain the coefficients of the Dirichlet series we can use the Perron formula

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1

1 1

( )

s s

n

a n

A x

G s

s

n

x

 

 





_

( )

1 ( )

2

c i s

n x c i

x

A x

a n

G s ds

i

s



 

  









(5)

If the function G(s) includes powers and quotients of the Riemann zeta function we can use

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Cauchy’s theorem to obtain the explicit formulae ,see Baillie [2] by using the step function

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1 (

1)

H x

f

x

 

 

_{ }

 

inside (1) and evaluate the inverse Mellin transform inside (1)

1 ( )

2

s

C

x

F s

i

s





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we find the well-known identities

50  

2

1 ( ) ( ) 2

' '( 2 )( 2 ) n

n x n

x x

M x n

n n



 

  

 

 

    

 



(6)

2

1

'(0)

( )

(0)

( 2 )

n

n x n

x

n

x

n







 

 







 









(7)





 

(2 )

( )

( ) 1

1/ 2

'

n x

x

L x

n



 





 



(3)

 

2

1

6

2 (

)

( )

| ( ) | 1

'

( 2 ) '( 2 )

n

n x n

x

n

Q x

n









 



 

 

 

 

_

 



 







(11)

 

2 2

2

1

1 3

(

1)

( 2

1)

( )

6 '

( 2 ) '( 2 )

n

n x n

x

n

x

n



 







 



 

 







 







(12)

Under the assumption that all the Riemann Non-trivial zeros are simple.Also we have for the

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Riemann zeta function and its derivatives

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2 1 2

( 1)

(2

1)(2 )!

'( 2 )

2

n

n n

n















 

1 '(0)

log 2

2 

 



1 (0)

2 

 

(13)

The reader will remember the relation between Perron's formula and our discrete convolution ,

53

using the work of Baillie [2] we will give different explicit formulae, to do so we use Cauchy's theorem

54

on complex integration and we also evaluate the closed mellin inverse transform inside (1)

55

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Using the residue theorem

1 ( ) ( )

2

s

C

F s G s x

i





where 'C' is a closed circuit including all the

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poles of the Dirichlet series G(s) , if we apply the Residue theorem inside (1) the contribution to the

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closed integral (Mellin transform) come from the Poles

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 The pole of the Riemann Zeta function at

s



1

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 The Riemann non-trivial zeros on the critical strip

0 Re( ) 1



s



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 The Trivial zeros of the Riemann zeta



( 2 ) 0



n



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 Poles of the Mellin transform (if any)

64

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This is what we have done insidethe expression (8-12) but we can generalize these results to

66

different test functions to compute more sums.

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Then we can obtain some interesting and useful new formulae , which we think they have never

69

been published (at least all of them)

70

2

1 1

1

( ) (1) ( ) ( 2 ) _n

n n

n

n f xF x F F n

x x

 



 

 

 

 _{ }   

 



(14)

2

1 1

( )

( 2 ) 1

( )

'( )

'( 2 )

n

n n

n

F

n

n f

x

n x

 





 



 

 



 





 

_

 



(15)

1

1 (2 ) ( )

( )

1

2 '( )

2

n

x

F

n f

F

x

 

 





 







 





 

_{ }

 

 

 

 



₍₁₆₎

2

2 2

1 1

6 (

1) ( )

( 2 ) ( 2

1)

( )

(2)

'( )

n

'( 2 )

n n

n

F

n

n f

F

x

n

 

 









 



 

 



 





 

_

 

(4)

2 2

1 1

6

2

2 (

)

(

)

| ( ) |

(1)

2 '( )

n

2 '( 2 )

n n

F

n

F n

n

n f

F

x

n









 



 

 

   

   

_

 

_

_

   

_

 

_

 



(18)

If the Mellin transform has poles inside the closed circuit 'C'

( ) ( )

s C

F s G s x



, then this poles

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will contribute with a remainder term due to the Residue theorem [1] in this case we have the extra

72

term

73





( )

Re

( ) ( )

s

k s k

r x

s F s G s x





with

1

0

( )

k

F k

dxf x x









 

(19)

This is what happens in Perron's formula , due to the step function

H x

(



1)

in this case there

74

is a pole at

s



0

since

F s

( )

1 s



this is why in formulae (8-12) there is a constant term.

75

76

These formulae (14-18) may be regarded as a generalization of the Riemann-Weil formula

77

78  

1 0

( )

1 ' 1

(0)

(log )

log

4

2

k

n k

n

ir

i

h

g

n

dr

h

n







 

  









 



_



_



_{ }









 







(20)

But we have used the Mellin integral transform , instead of the Fourier integral transform to

79

relate some new sums over primes and over Riemann Zeros

80

An easier derivation of our explicit formulae

81

There is an easier derivation for our explicit formulae, in general after Perron's formula is

82

applied we find the following identity

83

2 2 1

( )

d nr

n n

n x n

a

P Qx

h

x



c x



 

 

 





(21)

For some real constants

P Q d c

, , ,

₂_n

,

r

and a function

h

( )



,which includes the Riemann zeta

84

function and its first derivative.

85

Taking the distributional derivative for an step function of the form n n x

a





86

1 1 2 1

2

1 1

( ) d ( ) ( 2 ) nr

n n n

n x n n

d

a a x n dQx h x c nr x

dx

 



 

 

   

  

 

     

 









(22)

So if we apply a certain test function with a parameter 'x'

f

t

x

 

 

 

and its Mellin transform

87

defined by

(5)

 

1 1

0 0

( )

s s s s

t

dt f

t

x dt f t t

x F s

x

 

 

 

_

_

 

 



(23)

Then we find the desired explicit formula

89

2

1 1

( ) ( ) ( ) ( 2 ) ( 2 )

n n

n

a f dQF d h F c nr F nr

x 

 



 

 

 

    

   



(24)

to evaluate the derivative of the Step function n n x

d

a

dx

















we can use the identities (8-12) these

90

are defined and derived in Baillie [2] assuming that ALL the Non-trivial zeros of the Riemann Zeta

91

function are simple

92

93

A similar method can be applied to derive the Poisson summation formula, let be the Floor

94

function

 

x

, then we have a formula valid on the whole real line

95  





1

sin 2

1

2

n

nx

x

n







  



(25)

Taking the distributional derivative of (32) and using the Euler's formula for the cosine function

96  

1

(

) 1 2

cos(2

)

n n

d

x

x n

nx

dx





 

 







 



cos( )

2

ix ix

e

x





 (26)

Now if we use a test function inside (32) we have the Poisson summation formula

97

2

( )

inx

n n

f n

f x

e



 _ 



 









(27)

REFERENCES:

98

[1] Apostol Tom “Introduction to Analytic Number theory” ED: Springuer-Verlag,(1976)

99

[2] Baillie R. “Experiments with zeta zeros and Perron formula” Arxiv: 1103.6226

100

[3] Bertrand J, Bertrand P.Orvalez J. "The Mellin transform" The transforms and applications

101

handbook CRC Press LLC

102

[4]Garcia J.J "On the Evaluation of Certain Arithmetical Functions of Number TheoryandTheir Sums and a

103

Generalization of Riemann-Weil Formula http://vixra.org/abs/1310.0048

104

[5]H. W. Gould and Temba Shonhiwa "A catalog of interesting Dirichlet series" avaliable

105

athttps://projecteuclid.org/download/pdf_1/euclid.mjms/1316032830

106

[6] Perron O.," ZurTheorie der DirichletschenReihen", J. reineangew. Math. 134, 1908, pp. 95-143.

107

[7] Titchmarsh, E. C. “The Theory of Functions”, 2nd ed. Oxford, England: Oxford University Press,

108