The Fibonacci’s zeta function. Mathematical connections with some sectors of String Theory

The Fibonacci’s zeta function. Mathematical connections with some sectors of String Theory

Michele Nardelli1,2_{and Rosario Turco}

1_{Dipartimento di Scienze della Terra}

Università degli Studi di Napoli Federico II, Largo S. Marcellino, 10 80138 Napoli, Italy

2_{Dipartimento di Matematica ed Applicazioni “R. Caccioppoli”}

Università degli Studi di Napoli “Federico II” – Polo delle Scienze e delle Tecnologie Monte S. Angelo, Via Cintia (Fuorigrotta), 80126 Napoli, Italy

Abstract

In this paper we have described, in the Section 1, the Fibonacci’s zeta function and the Euler-Mascheroni constant and in the Section 2, we have described some sectors of the string theory: zeta strings, zeta nonlocal scalar fields and some Lagrangians with zeta Function nonlocality. In conclusion, in the Section 3, we have described some possible mathematical connections.

1. The Fibonacci’s zeta function [1], [2], [3], [4], [5], [6] The Fibonacci’s zeta function is defined as:

1

1

( )

( )

s k

F s

Fib k

∞ =

=

∑

(1) where s=a+jb.

The logarithm of the Fibonacci’s zeta and the Fiborial

( )

ln ( )

ln(

,

dove Fib Fib k k

F s

Fib

s

∞ ∏ # = = 1

−

=

# )

(2)

We have that Fib# is the Fiborial , i.e. the product to the infinity of the Fibonacci’s numbers. We can write the eq. (1) also as follows:

( )

1

1

2

3

5

8

...

ln1

ln1

ln 2

ln3

ln 5

ln8

( )

...

ln ( )

(ln1 ln1 ln 2 ln 3 ln 5 ln8 ...)

ln ( )

ln

( ) ln(

)

1

s

s

s

s

s

s

F s

s

s

s

s

s

s

F s e

e

e

e

e

e

F s

s

F s

Fib k

Fib

s

_k

−

−

−

−

−

−

=

+

+

+

+

+

+

−

−

−

−

−

−

=

+

+

+

+

+

+

= −

+

+

+

+

+

+

∞

∏

−

=

=

#

=

where we have applied a note property of the logarithms.

Generali zation of the Gamma function for various series

If we want to reach a possible functional equation for the Fibonacci’s zeta, then we need to ask first how to express the "Fibonacci’s Gamma”.

We known that the Gamma function Γ (x) is an extension of factorial, by an integer k to a real number x. The Gamma function was studied by Euler (see [1]) and has a significant importance in Number Theory and in the study of the Riemann zeta.

In general we know that Γ (x) enjoys some of the following properties: 1. Γ(k + 1) = k!, k ≥ 0;

2. Γ(1) = 1, Γ(x + 1) = x Γ(x) e Γ(x) is the unique solution of this functional equation for x > 0;

3. , x ≠ 0, -1, -2 …

(Euler form);

4. (Weierstrass form) where γ is the Euler-Mascheroni constant defined by

( )

1

1

lim

k

log

k _i

i

k

γ

_{→ ∞} =

=

∑

−

;

5. (property of reflection), (property of

extension) and Γ(1/2)2 = π.

From above we can tell which should be considered in the follow, both the Euler form and the Weierstrass form.

Generalized Euler form

We consider a series gk > 0, k > 0 and an extension g(x) > 0, x > 0 with the property that: g(k) = gk and there exists and isn’t zero. From the above property 3, we have:

,

1

( ) lim ( )

( )

( )

( )

lim ( )

, g(x+i) 0,i=0,1,2,...

( )

(

)

g _k g k x k k i

x

g x

x

g k

g i

x

g x

g x i

λ

→ ∞ → ∞ ₌

Γ

=

Γ

=

≠

+

∏

(3)

where λ(x) is to establish. Thence we obtain

If we choose λ(x) such that

for example: we obtain

(

1)

( )

( )

g

x

g x

g

x

Γ

+ =

Γ

(4)

If we take Γg(1) = 1 , we obtain . Furthermore, from eq. (3) we have:

such that

With this restriction, λ(k) can be extended to λ(x) or to have Γg(x) with additional properties. Using the eq. (4), Γg(x), x > 0 satisfy the relation

Example A

A simple generalization of gk = k is gk = ak + b with a, b choose such that gk > 0, k > 0. Thence, we obtain:

If we take λ(x) = 1, we obtain:

with

.

The Figure 1 concerning the product of the odd numbers.

Figure 1

(a) show Γg(x), an extension of the product to the real numbers, for a = 2, b =

-1. (b) show the comparison with the normal Gamma function Γ(x). Example B

For the Fibonacci’s numbers is:

where , i.e. the aurea ratio, and c = 5, and thence:

. The most simple and effective extension is:

and we obtain ( 1) / 2 1

( )

lim

x k x x k i F _k i x x i

F

F

x

F

F

φ

+ → ∞ ₌ +

Γ

=

∏

(5) where ΓF (k + 1) = (k!)F.

The Figure 2 show ΓF(x).

Figure 2

ΓF(x), an extension of the Fiborial to the real numbers with the property that ΓF(k + 1) = (k!)F. (a) show the

values for -2 ≤ x ≤ 1 (b) show the values for 0 ≤ x ≤ 5.

Weierstrass generalization form

Using the eq. (3) and assuming that λ(x) and g(x) are differentiable, we obtain:

1

' ( )

'( )

'( )

'(

)

lim

ln ( )

( )

( )

( )

(

)

k g x _i g

x

x

g x

g x i

g k

x

x

g x

g x i

λ

λ

→ ∞ ₌

Γ

₌

₋

₋



+

₋







Γ

_

∑

+

_

(6)

which suggests an Euler-Mascheroni generalized constant:

1

'( )

lim

ln ( )

( )

k g _x i

g i

g k

g i

γ

_{→ ∞} =





=

_

−

_



∑



(7)

(we remember that the Euler-Mascheroni constant can be written also as follow

( )

 

∫

∑

∞ = ∞ →









₋

=













₋

=

1 1

1

1

ln

1

lim

dx

x

x

n

k

n k n

γ

, (7b))

thence γ becomes the associated constant to the series gk = k.

Thence the Weierstrass generalized form (see property 4) becomes: '( ) ( ) 1

1

( )

(

)

exp

( )

( )

( )

g g i x x g i i g

g x

g x i

e

x

x

g i

γ

λ

  ∞ −    =

+

=

Γ

∏

(8)

While the generalized reflection formula becomes:

2 1

( ) (

)

( )

( )

(

)

( ) (

)

(

) (

)

g g i

x

x

g i

x

x

g x g x

g x i g x i

λ

λ

∞ =

−

Γ

Γ −

=

−

∏

+

− +

(9) and since Γg(1 - x) = g(-x)Γg(-x), we obtain 2 1

( ) (

)

( )

( )

(1

)

( )

(

) (

)

g g i

x

x

g i

x

x

g x

g x i g x i

λ

λ

∞ =

−

Γ

Γ

−

=

+

− +

∏

(10) 2 ₂ 1

1

1

(

) (

)

1

₂

₂

( )

1

1

2

(1 2)

(

₂

) (

₂

)

g i

g i

g

g

i g

i

λ

λ

∞ =

−

 

Γ

_{ }

=

+

−

+

 

∏

(11) Example A

For gk = ak + b , as before, the Weierstrass form is:

where

The functional equation Γg(x + 1) = (ax + b)Γg(x) , thence, becomes

Example B

For the Fibonacci’s numbers the Weierstrass form is:

(12) where γF = 0.6676539532 … , and the eq. (11) gives

since for i integer. We have that:

Alternatively, can be considered a closed form for the constant of the Fiborial (Sloane's A062073 [4])

(If we replace ΓF(1/2) with Γg(1/2) where and φ is the aurea ratio ,

the precedent result can give a closed form for the infinite product where φ > 1. This product is linked to the partitions functions and the q-series)

Other series and generalizations

Generalized Gamma functions can be obtained also for series as: , and etc. The Euler-Mascheroni generalized constant can also be used to introduce sequences that are well known. If then (A constant) such that:

For example, , z > 0 from with , the Riemann zeta.

can be extended to real numbers using:

The Beta function can be also generalized to:

The logarithmic derivative, analyzed previously, generalizes the digamma function

with . Thence:

This satisfies also other relations:

Considering

,

where is the k-mo harmonic number, the 'g-harmonic' number can be defined as .

Analogously, if we assume that g(x) is differentiable, also the properties of the poligamma function can be considered.

For example:

while

Thence a 'g-zeta' can be defined as

with k>1 when g(i) = ai + b , we obtain

that can be extended to the real numbers and becomes the Hurwitz zeta function:

For the Fibonacci’s zeta, thence, we can take in consideration the eq. (5) or the eq. (12).

1.1 The Euler-Mascheroni constant

The Euler-Mascheroni constant (see eq. (7b)) is a mathematical constant, used principally in number theory and in the mathematical analysis. It is defined as the limit of the difference between the truncated harmonic series and the natural logarithm:

( )













₊

₊

₊

₊

₊

₋

=

∞ →

_n

n

n

ln

1

...

4

1

3

1

2

1

1

lim

γ

,

( )

 

∫

∑

∞ = ∞ →









₋

=













₋

=

1 1

1

1

ln

1

lim

dx

x

x

n

k

n k n

γ

, (7b)

where

 

x

is the part integer function.

We have that

γ

has the following approximated value

γ

≅

0

,

5772156649

....

The constant can be defined in various modes by the integrals:

∫

∞

( )

∫

∫

∞ ₋ −

∫

∞ −













₋

+

=













₋

−

=













−

=

−

=

0 1 0 0 0

1

1

1

1

1

1

1

ln

ln

ln

dx

e

x

x

dx

e

x

e

dx

x

dx

e

x

x x x x

γ

. (7c)

The Euler-Mascheroni constant is related to the zeta function from the following expressions:

( )

(

)

(

)

∑

∞ = −

+

+

−

+













=

1 1

1

2

1

1

4

ln

m m m

m

m

ζ

π

γ

;

∑

∞

( )

= → →













−

−

=













₋

=

₊ 1 1 1

1

1

lim

1

1

lim

n s n s s

n

s

ζ

s

s

γ

. (7d)

The constant is also linked to the gamma function:

∑

= ∞ → ∞ →









₋









=













Γ

−

=

n k n x

k

n

k

n

n

x

x

1

1

lim

1

lim

γ

, (7e)

and to the beta function:

(

)

1

1

2

1

1

lim

2 / 1 1

+

−













₊

₊

Γ

+

Γ













Γ

=

+ ∞ →

_n

n

n

n

n

n

n

n n

γ

. (7f)

Now we describe two theorems related to the Euler-Mascheroni constant

γ

. Theorem (1).

For positive integers

a

we have

∫

{ }

=

−

∞ → a a

a

1

a

/

x

dx

1

1

lim

γ

where

γ

is the Euler-Mascheroni constant.

{ }

(

)





(

)

( ) ₍( ₎) _{( )}

∫

=

∫

−

∫

=

∫

−



_

∫

−



_



_

+

∫

−

_





_

+

+

∫



_



_



_

− a a a a a a a a a a a a

_x

dx

a

dx

x

a

dx

x

a

dx

x

a

dx

x

a

dx

x

a

dx

x

a

1 1 1 1 1 / 1 2 / 1 /

..

/2

/

/

/

/

=

∫

∑

−

₍

_{) (}

_{) (}

)

( )

∑

= =

+

−

=

−









+

−

−

−

−













=

a a i a i

a

i

a

a

a

i

a

i

a

a

i

a

a

dx

x

a

1 1 1 1

ln

1

. Therefore,

∫

{ }

( )

∑

( )

∑



=











₋

₊

=













₋

₊

=

= ∞ → = ∞ → ∞ → a a i a a i a a

i

a

a

i

a

a

a

a

dx

x

a

a

1 1 1

1

1

ln

lim

ln

1

lim

/

1

lim

=

1

−

_a

lim

_→∞

(

H

a

−

ln

( )

a

)

=

1

−

γ

. (7g) Theorem (2).

If

p

n denotes the nth prime number, then we get as

n

→

∞

∑

( )

( )

− = ∞ →

−

−

−

=

1 1

0

1

ln

1

1

lim

n i n i n

n

p

p

i

d

n

where

d

( )

i

=

p

i+1

−

p

i.

We start by using the result of theorem (1), as

n

→

∞

∫

∫

∫

∫

−

∫

− −





=









+













+













+













=













=

−

n n n n n p p p p p p p p n n n n n n n n n n

dx

x

p

p

dx

x

p

p

dx

x

p

p

dx

x

p

p

dx

x

p

p

1 1 1 2 1 1 2 1

1

1

...

1

1

1

1

γ

∑ ∫

∑ ∫

∑ ∫

− = − = − = + + +

=









−

=













=

1 0 1 0 1 0 1

1

1

1

1

1

n i p p n i p p n i p p n n n n n n i i i i i i

dx

x

p

p

dx

x

p

p

dx

x

p

p

( )

∑ ∫

− = +









−

=

1 0 1

1

ln

n i p p n n n

dx

x

p

p

p

i i . (7h)

Now we proceed in two directions: 1)

( )

∑

−

(

)

( )

∑

( )

∑

(

)

= − = − = + +



=

−

+

−

=







−

−

−

<

−

1 0 1 0 1 0 1 1

1

ln

1

1

ln

1

n i n i n i i i n i n i n i i n n

p

p

p

p

i

d

p

p

p

p

p

p

p

γ

( )

∑

( )

( )

∑

( )

( )

∑

( )

− = − = − =

−

=

+

−

<

−

+

−

=

1 0 1 0 1 1

ln

1

ln

1

ln

n i n i n i i n i n n n i n

p

i

d

p

p

i

d

p

p

p

p

i

d

p

. Hence we get

∑

( )

( )

− =

+

−

<

1 1

1

ln

n i n i

p

p

i

d

_γ

. 2)

( )

∑ ∫

( )

∑

(

)

( )

∑

( )

− = − = − = + + + +

=

−

=

−

−

>









−

=

−

1 0 1 0 1 0 1 1 1 1

ln

1

ln

1

ln

1

n i p p n i n i i n i n i i n n n n n i i

p

i

d

p

p

p

p

p

p

p

dx

x

p

p

p

γ

( )

∑

( )

− = +

−

−

=

1 1 1

1

ln

n i i n

p

i

d

p

. (7i) Hence we get

∑

( )

( )

− = +

+

−

>

1 1 1

2

ln

n i n i

p

p

i

d

_γ

.

Combining, the above two cases, we get

( )

∑

( )

∑

( )

( )

− = − = +

+

−

<

<

<

+

−

1 1 1 1 1

1

ln

2

ln

n i n i n i i n

_p

p

i

d

p

i

d

p

γ

γ

.

Dividing by n− 1 and taking limits as

n

→

∞

, we get the following result, i.e.,

∑

( )

( )

− = ∞ →

−

−

−

=

1 1

0

1

ln

1

1

lim

n i n i n

n

p

p

i

d

n

.

2. Zeta Strings, Zeta Nonlocal Scalar Fields and Some Lagrangians with Zeta Function Nonlocality [7], [8], [9]

The exact tree-level Lagrangian for effective scalar field

ϕ

which describes open p-adic string tachyon is

_











+

+

−

−

=

−2 +1 2 2

₁

1

2

1

1

1

p p

p

p

p

p

g

ϕ

ϕ

ϕ



L

, (13)

where

p

is any prime number,

₌

_{− ∂}

2

₊

_∇

2

t



is the D-dimensional d’Alambertian and we adopt metric with signature

(

−

+

...

+

)

. Now, we want to show a model which incorporates the p-adic string Lagrangians in a restricted adelic way. Let us take the following Lagrangian

∑

∑

∑

∑

≥ ≥ ≥ ≥ + −













+

+

−

=

−

=

=

1 1 1 1 1 2 2 2

₁

1

2

1

1

1

n n n n n n n n

n

n

g

n

n

C

L

φ

φ

φ



L

L

. (14)

Recall that the Riemann zeta function is defined as

( )

∑

∏

≥

−

−

=

=

1

1

1

1

n

n

s p

p

s

s

ζ

,

s

=

σ

+

i

τ

,

σ

> 1. (15)

Employing usual expansion for the logarithmic function and definition (15) we can rewrite (14) in the form

_





+

+

(

−

)

_













−

=

φ ζ

φ

φ

ln

1

φ

2

2

1

1

2



g

L

, (16) where

φ

<

1

.













2



ζ

acts as pseudodifferential operator in the following way:

( ) ( )

x

e

ixk

k

( )

k

dk

D

ζ

φ

π

φ

ζ

~

2

2

1

2

2

∫

_



−

_



=















,

−

=

2

−

2

>

2

+

ε

0 2

_k

_k

k



, (17) where

_φ

( )

k

e

( ikx)

_φ

( )

x

dx

∫

−

=

~

is the Fourier transform of

φ

( )

x

.

Dynamics of this field

φ

is encoded in the (pseudo)differential form of the Riemann zeta function. When the d’Alambertian is an argument of the Riemann zeta function we shall call such string a “zeta string”. Consequently, the above

φ

is an open scalar zeta string. The equation of motion for the zeta string

φ

is

_{( )}

∫

_{− > +}

( )

−

=









₋

=













ε

φ

φ

φ

ζ

π

φ

ζ

2 2 2 2 0

1

~

2

2

1

2

k k ixk D

k

dk

k

e





(18)

which has an evident solution

φ

=

0

.

For the case of time dependent spatially homogeneous solutions, we have the following equation of motion

( ) ( )

( )

( )

_{( )}

t

t

dk

k

k

e

t

k t ik t

φ

φ

φ

ζ

π

φ

ζ

ε



=

−







=







 ∂

−

∫

> + −

1

~

2

2

1

2

0 0 2 0 2 2 0 0 _{. (19)}

With regard the open and closed scalar zeta strings, the equations of motion are

( )

( )

( )

∫

∑

≥ −

=









−

=













n n n n ixk D

k

dk

k

e

ζ

φ

θ

φ

π

φ

ζ

1 2 1 2

_~

2

2

1

2



, (20)

_{( )}

( )

(

₍

)

₎

( )

(

)

∫

∑

_≥ − − +













−

+

−

+

=









₋

=













1 1 1 2 1 2

1

1

2

1

~

4

2

1

4

2 n n n n n ixk D

_n

n

n

dk

k

k

e

ζ

θ

θ

θ

φ

π

θ

ζ



, (21)

and one can easily see trivial solution

φ

=

θ

=

0

.

The exact tree-level Lagrangian of effective scalar field

ϕ

, which describes open p-adic string tachyon, is:

















+

+

−

−

=

2 −2 +1 2

₁

1

2

1

1

2 p m p D p p

_p

p

_p

p

g

m

_p

ϕ

ϕ

ϕ



L

, (22)

where

p

is any prime number,

₌

_{− ∂}

2

₊

_∇

2

t



is the D-dimensional d’Alambertian and we adopt metric with signature

(

−

+

...

+

)

, as above. Now, we want to introduce a model which incorporates all the above string Lagrangians (22) with

p

replaced by n∈ N. Thence, we take the sum of all Lagrangians

L

n in the

form

∑

∑

∞ + = + − ∞ + =

















+

+

−

−

=

=

1 1 2 1 2 2

₁

1

2

1

1

2 n n m n n D n n n n

_n

n

_n

n

g

m

C

C

L

φ

n

φ

φ



L

, (23)

whose explicit realization depends on particular choice of coefficients

C

n, masses

m

n and coupling

constants

g

n.

Now, we consider the following case

n _h

n

n

where

h

is a real number. The corresponding Lagrangian reads

















+

+

−

=

∑

∑

+ ∞ = + − + ∞ = − − 1 1 1 2 2

₂

₁

1

2 n n h n h m D h

_n

n

n

g

m

L

φ

φ

φ

 (25)

and it depends on parameter h. According to the Euler product formula one can write

∑

∏

_{− −} + ∞ = − −

−

=

p _m h n h m

p

n

2 2 2 1 2

1

1

  . (26)

Recall that standard definition of the Riemann zeta function is

( )

∑

∏

+ ∞ =

−

−

=

=

1

1

1

1

n

n

s p

p

s

s

ζ

,

s

=

σ

+

i

τ

,

σ

> 1, (27)

which has analytic continuation to the entire complex

s

plane, excluding the point s= 1, where it has a simple pole with residue 1. Employing definition (27) we can rewrite (25) in the form

_











+

+













₊

−

=

∑

+ ∞ = + − 1 1 2 2

₂

₂

₁

1

n n h D h

n

n

h

m

g

m

L

φ ζ



φ

φ

. (28) Here













₊

_h

m

2

2



ζ

acts as a pseudodifferential operator

( ) ( )

h

( )

k

dk

m

k

e

x

h

m

ixk D

ζ

φ

π

φ

ζ

~

2

2

1

2

2 2 2

∫









₋

₊

=















₊

, (29)

where

_φ

( )

k

e

( ikx)

_φ

( )

x

dx

∫

−

=

~

is the Fourier transform of

φ

( )

x

.

We consider Lagrangian (28) with analytic continuations of the zeta function and the power series

∑

− +

+

1

1

n h

n

n

φ

, i.e.

_











+

+













₊

−

=

∑

+ ∞ = + − 1 1 2 2

₂

₂

₁

1

n n h D h

_n

n

AC

h

m

g

m

L

φ ζ



φ

φ

, (30)

where

AC

denotes analytic continuation.

Potential of the above zeta scalar field (30) is equal to

−

L

h at



=

0

, i.e.

( )

_



( )

_



+

−

=

∑

+ ∞ = + − 1 1 2 2

₂

₁

n n h D h

_n

n

AC

h

g

m

V

φ

φ

ζ

φ

, (31)

where

h

≠

1

since

ζ

( )

1

=

∞

. The term with

ζ

-function vanishes at

h

=

−

2

,

−

4

,

−

6

,...

. The equation of motion in differential and integral form is

∑

+ ∞ = −

=













₊

1 2

2

n n h

n

AC

h

m

φ

ϕ

ζ



, (32)

( )

∫

( )

∑

+ ∞ = −

=









+

−

1 2 2

_~

2

2

1

n n h R ixk D

_m

h

k

dk

AC

n

k

e

D

ζ

φ

φ

π

, (33) respectively.

Now, we consider five values of h, which seem to be the most interesting, regarding the Lagrangian (30):

,

0

=

h

h

=

±

1

,

and h= ±2. For h= −2, the corresponding equation of motion now read:

_{( )}

∫

( )

₍

(

₎

)

−

+

=









₋

₋

=













₋

D R ixk D

_m

k

dk

k

e

m

2 3 2 2

₁

1

~

2

2

2

1

2

2

_φ

φ

φ

φ

ζ

π

φ

ζ



. (34)

This equation has two trivial solutions:

φ

( )

x

=

0

and

φ

( )

x

=

−

1

. Solution

φ

( )

x

=

−

1

can be also shown taking

_φ

~

( )

_k

₌

₋

_δ

( )( )

_k

₂

_π

D _and

_ζ

( )

₋

₂

₌

₀

_{in (34).}

For h= −1, the corresponding equation of motion is:

_{( )}

∫

( )

₍

₎

−

=









₋

₋

=













₋

D R ixk D

_m

k

dk

k

e

m

2 2 2 2

₁

~

1

2

2

1

1

2

_φ

φ

φ

ζ

π

φ

ζ



. (35) where

( )

12

1

1

=

−

−

ζ

.

The equation of motion (35) has a constant trivial solution only for

φ

( )

x

=

0

. For h= 0, the equation of motion is

_{( )}

∫

( )

−

=









₋

=













D R ixk D

_m

k

dk

k

e

m

φ

φ

φ

ζ

π

φ

ζ

1

~

2

2

1

2

2 2 2



. (36)

It has two solutions:

φ

=

0

and

φ

=

3

. The solution

φ

=

3

follows from the Taylor expansion of the Riemann zeta function operator

( )

∑

( )

( )

≥













+

=













1 2 2

_!

₂

0

0

2

n n n

m

n

m





ζ

ζ

ζ

, (37) as well as from

φ

~

( ) ( ) ( )

k

=

2

π

D

3

δ

k

. For h= 1, the equation of motion is:

_{( )}

∫

_RD



_

−

+



_

( )

=

−

(

−

)

ixk D

_m

k

dk

k

e

2 2 2

1

ln

2

1

~

1

2

2

1

φ

φ

ζ

π

, (38) where

ζ

( )

1

=

∞

gives

V

₁

( )

φ

=

∞

.

_{( )}

∫

_RD

_



−

+

_



( )

=

−

∫

(

−

)

ixk D

_w

dw

w

dk

k

m

k

e

ζ

φ

φ

π

0 2 2 2

2

1

ln

~

2

2

2

1

. (39)

Since holds equality

−

∫

1

(

−

)

=

∑

∞₌

=

( )

0 1 2

2

1

1

ln

n

_n

dw

w

w

ζ

one has trivial solution

φ

=

1

in (39).

Now, we want to analyze the following case: 2 ₂

1

n

n

C

_n

=

−

. In this case, from the Lagrangian (23), we obtain:

_











−

+

























+













₋

−

=

φ

φ

φ

ζ

ζ

φ

1

2

1

2

2

1

2 2 2 2

_m

_m

g

m

L

D

_

_

. (40)

The corresponding potential is:

( )

(

1

)

2

24

7

31

_φ

φ

φ

φ

−

−

−

=

g

m

V

D . (41)

We note that 7 and 31 are prime natural numbers, i.e. 6n± 1 with

n

=1 and 5, with 1 and 5 that are Fibonacci’s numbers. Furthermore, the number 24 is related to the Ramanujan function that has 24 “modes” that correspond to the physical vibrations of a bosonic string. Thence, we obtain:

( )

₍

₎

2

1

24

7

31

φ

φ

φ

φ

−

−

−

=

g

m

V

D

( )

























+

+









+

⋅





















⇒

− ∞ ₋

∫

4

2

7

10

4

2

11

10

log

'

142

'

cosh

'

cos

log

4

₂ ' ' 4 0 ' 2 2

w

t

itw

e

dx

e

x

txw

anti

w w t w x

φ

π

π

π π . (41b)

[

(

)

]

(

)

2 2 2 2

₁

1

1

2

1

2

−

+

−

=





















+













₋

φ

φ

φ

φ

ζ

ζ

m

m





. (42)

Its weak field approximation is:

2

0

2

1

2

2 2



=







₋













+













₋

_ζ

_φ

ζ

m

m





, (43)

which implies condition on the mass spectrum

2

2

1

2

2 2 2 2

=









+









₋

m

M

m

M

_ζ

ζ

. (44)

From (44) it follows one solution for

_M

2

_>

0

_at

_M

2

_≈

₂

_.

₇₉

_m

2 _{and many tachyon solutions when} 2

2

_38m

M

<

−

.

We note that the number 2.79 is connected with

2

1

5

−

=

φ

and

2

1

5

+

=

Φ

, i.e. the “aurea” section and the “aurea” ratio. Indeed, we have that:

2

,

772542

2

,

78

2

1

5

2

1

2

1

5

2 2

≅

=









−

+









+

_.

Furthermore, we have also that:

( )

Φ

14/7

+

( )

Φ

−25/7

=

2

,

618033989

+

0

,

179314566

=

2

,

79734

With regard the extension by ordinary Lagrangian, we have the Lagrangian, potential, equation of motion and mass spectrum condition that, when 2 ₂

1

n

n

_











−

+

+













₋













−













₋

−

=

φ

φ

φ

φ

φ

ζ

ζ

φ

1

ln

2

1

2

1

2

2

2 2 2 2 2 2 2

_m

_m

_m

g

m

L

D







, (45)

( )

_



( ) ( )

_



−

−

−

+

+

−

=

φ

φ

ζ

ζ

φ

φ

1

1

ln

1

0

1

2

2 2 2

g

m

V

D , (46)

(

)

2 2 2 2 2 2

1

2

ln

1

2

1

2

φ

φ

φ

φ

φ

φ

φ

ζ

ζ

−

−

+

+

=









₋

₊













+













₋

m

m

m







, (47) ₂ 2 2 2 2 2

2

1

2

m

M

m

M

m

M

₌









+









₋

ζ

ζ

. (48)

In addition to many tachyon solutions, equation (48) has two solutions with positive mass:

_M

2

_≈

₂

_.

₆₇

_m

2 and

_M

2

_≈

₄

_.

₆₆

_m

2_.

We note also here, that the numbers 2.67 and 4.66 are related to the “aureo” numbers. Indeed, we have that:

2

.

6798

2

1

5

5

2

1

2

1

5

2

_≅









₋

⋅

+









₊

,

4

.

64057

2

1

5

2

1

2

1

5

2

1

5

2 2

≅









₊

+









₊

+









₊

.

Furthermore, we have also that:

( )

Φ

14/7

+

( )

Φ

−41/7

=

2

,

618033989

+

0

,

059693843

=

2

,

6777278

_;

( )

Φ

22/7

+

( )

Φ

−30/7

=

4

,

537517342

+

0

,

1271565635

=

4

,

6646738

. Now, we describe the case of

( )

₂

1

n

n

n

C

_n

=

µ

−

. Here

µ

( )

n

is the Mobius function, which is defined for all positive integers and has values 1, 0, – 1 depending on factorization of

n

into prime numbers

p

. It is defined as follows:

( ) ( )











−

=

,

1

,

1

,

0

k

n

µ

(

)











=

=

≠

=

=

.

0

,

1

,

...

2 1 2

k

n

p

p

p

p

p

n

m

p

n

j i k (49)

The corresponding Lagrangian is

( )

( )

















+

+

−

+

=

∑

+∞

∑

= ∞ + = + 1 1 1 2 2 0 0

1

2

1

2 n n n m D

n

n

n

n

g

m

C

L

_µ

L

φ

µ

_

φ

µ

φ

(50)

Recall that the inverse Riemann zeta function can be defined by

_{( )}

∑

( )

+ ∞ =

=

1

1

n

n

s

n

s

µ

ζ

,

s

=

σ

+

it

,

σ

> 1. (51)

Now (50) can be rewritten as

( )

























+













−

+

=

_∫

∞ 0 2 2 0 0

2

1

2

1

φ

φ

φ

ζ

φ

µ

d

m

g

m

C

L

L

D

M



, (52) where

( )

=

∑

+ ∞₌₁

( )

=

−

2

−

3

−

5

+

6

−

7

+

10

−

11

−

_...

n n

n

φ

φ

φ

φ

φ

φ

φ

φ

φ

µ

φ

M

The corresponding potential,

equation of motion and mass spectrum formula, respectively, are:

_µ

( )

φ

=

−

_µ

(

=

)

=



_

φ

(

−

φ

)

−

φ

−

∫

φ

( )

φ

φ

_



0 2 2 2 0 2

₂

1

ln

0

C

d

g

m

L

V



D

M

, (53)

( )

2

ln

0

2

1

0 2 0 2

=

−

−

−













φ

φ

φ

φ

φ

ζ

C

m

C

m





M

, (54)

2

1

0

2

1

0 2 2 0 2 2

−

+

−

=









m

C

M

C

m

M

ζ

,

φ

< <

1

, (55)

where usual relativistic kinematic relation 2 2 2 0

2

_k

_k

_M

k

=

−

+



=

−

is used.

Now, we take the pure numbers concerning the eqs. (44) and (48). They are: 2.79, 2.67 and 4.66. We note that all the numbers are related with

2

1

5

+

=

Φ

, thence with the aurea ratio, by the following expressions:

( )

15/7

79

,

2

≅

Φ

;

( )

13/7

( )

21/7

67

,

2

≅

Φ

+

Φ

− ;

( )

22/7

( )

30/7

66

,

4

≅

Φ

+

Φ

− . (56) 3. Mathematical connections

Now we take the eqs. (5) and (12) of the Section 1, and the eqs. (18), (21) and (36) of Section 2. We obtain the following mathematical connections:

( 1) / 2 1

( )

lim

x k x x k i F _k i x x i

F

F

x

F

F

φ

+ → ∞ ₌ +

Γ

=

∏

⇒

( )



( )

=

−

⇒







₋

∫

− > +ε

φ

φ

φ

ζ

π

2 2 2 2 0

1

~

2

2

1

k k ixk D

k

dk

k

e



( )

( )

(

(

)

)

( )

(

)

_

⇒











−

+

−

+

=









₋

⇒

∫

∑

≥ + − − 1 1 1 2 1 2

1

1

2

1

~

4

2

1

2 n n n n n ixk D

_n

n

n

dk

k

k

e

ζ

θ

θ

θ

φ

π

_{( )}

∫

( )

−

=









₋

⇒

_RD ixk D

_m

k

dk

k

e

φ

φ

φ

ζ

π

1

~

2

2

1

2 2 ; (57)

⇒

_{( )}

( )

⇒

−

=









₋

⇒

∫

+ > − ε

φ

φ

φ

ζ

π

2 2 2 2 0

1

~

2

2

1

k k ixk D

k

dk

k

e



_{( )}

( )

(

₍

)

₎

( )

(

)

_

⇒











−

+

−

+

=









₋

⇒

_∫

∑

≥ + − − 1 1 1 2 1 2

1

1

2

1

~

4

2

1

2 n n n n n ixk D

_n

n

n

dk

k

k

e

ζ

θ

θ

θ

φ

π

( )

∫



( )

=

−







₋

⇒

_RD ixk D

_m

k

dk

k

e

φ

φ

φ

ζ

π

1

~

2

2

1

2 2 . (58)

Now we take the eqs. (7h) and (7i) of the Section 1, and the eqs. (18), (21) and (36) of Section 2. We obtain the following mathematical connections:

∫

( )

∑ ∫

− = +

⇒









−

=













=

−

n i i p n i p p n n n n n

dx

x

p

p

p

dx

x

p

p

1 1 0 1

1

ln

1

1

γ

( )



( )

=

−

⇒







₋

⇒

∫

+ > − ε

φ

φ

φ

ζ

π

2 2 2 2 0

1

~

2

2

1

k k ixk D

k

dk

k

e



_{( )}

( )

(

₍

)

₎

( )

(

)

_

⇒











−

+

−

+

=









₋

⇒

_∫

∑

≥ + − − 1 1 1 2 1 2

1

1

2

1

~

4

2

1

2 n n n n n ixk D

_n

n

n

dk

k

k

e

ζ

θ

θ

θ

φ

π

( )

∫



( )

=

−







₋

⇒

D R ixk D

_m

k

dk

k

e

φ

φ

φ

ζ

π

1

~

2

2

1

2 2 ; (59)

( )

∑ ∫

( )

∑

( )

− = − = + +

⇒

−

−

=









−

=

−

1 0 1 1 1 1

1

ln

1

ln

1

n i p p n i i n n n n i i

p

i

d

p

dx

x

p

p

p

γ

_{( )}

( )

⇒

−

=









₋

⇒

∫

+ > − ε

φ

φ

φ

ζ

π

2 2 2 2 0

1

~

2

2

1

k k ixk D

k

dk

k

e



( )

( )

(

(

)

)

( )

(

)

_

⇒











−

+

−

+

=









₋

⇒

∫

∑

≥ + − − 1 1 1 2 1 2

1

1

2

1

~

4

2

1

2 n n n n n ixk D

_n

n

n

dk

k

k

e

ζ

θ

θ

θ

φ

π

_{( )}

∫

( )

−

=









₋

⇒

_RD ixk D

_m

k

dk

k

e

φ

φ

φ

ζ

π

1

~

2

2

1

2 2 . (60) Acknowledgments

The co-author Nardelli Michele would like to thank Prof. Branko Dragovich of Institute of Physics of Belgrade (Serbia) for his availability and friendship.

References

[1] Sulle spalle dei giganti - Rosario Turco, Maria Colonnese, Michele Nardelli, Giovanni Di Maria, Francesco Di Noto, Annarita Tulumello

[2] C’è solo un’acca tra pi e phi - Rosario Turco, Maria Colonnese

[3] Gamma and related functions generalized for sequences -R. L. Ollerton

[4] Ward, M. (1936), un calcolo di sequenze. American Journal of Mathematics 58: 2, pp. 255-266.

[5] Sloane, NJA (2006) On-Line Encyclopedia of Integer Sequences - Disponibile online all'indirizzo:

www.research.att.com/ ~ njas / sequenze / (accessed 23 giugno 2005)

[6] Andrews, GE, Askey, R. Ranjan, R. (1999) Funzioni speciali. Encyclopedia of Mathematics and its Applications 71 , Cambridge University Press, Cambridge

[7] Branko Dragovich – “Zeta Strings” – arXiv:hep-th/0703008v1 – 1 Mar 2007.

[8] Branko Dragovich – “Zeta Nonlocal Scalar Fields” – arXiv:0804.4114v1 – [hep-th] – 25 Apr 2008.

[9] Branko Dragovich – “Some Lagrangians with Zeta Function Nonlocality” – arXiv:0805.0403 v1 – [hep-th] – 4 May 2008.