On some equations concerning quantum electrodynamics coupled to quantum gravity, the gravitational contributions to the gauge couplings and quantum effects in the theory of gravitation: mathematical connections with some sector of String Theory and Number

(1)

On some equations concerning quantum electrodynamics coupled to quantum gravity, the gravitational contributions to the gauge couplings and quantum effects in the theory of gravitation: mathematical connections with some sector of String Theory and Number Theory

Michele Nardelli1,2

1Dipartimento di Scienze della Terra

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

2Dipartimento 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

This paper is principally a review, a thesis, of principal results obtained from various authoritative theoretical physicists and mathematicians in some sectors of theoretical physics and mathematics. In this paper in the Section 1, we have described some equations concerning the quantum electrodynamics coupled to quantum gravity. In the Section 2, we have described some equations concerning the gravitational contributions to the running of gauge couplings. In the Section 3, we have described some equations concerning some quantum effects in the theory of gravitation. In the Section 4, we have described some equations concerning the supersymmetric Yang-Mills theory applied in string theory and some lemmas and equations concerning various gauge fields in any non-trivial quantum field theory for the pure Yang-Mills Lagrangian. Furthermore, in conclusion, in the Section 5, we have described various possible mathematical connections between the argument above mentioned and some sectors of Number Theory and String Theory, principally with some equations concerning the Ramanujan’s modular equations that are related to the physical vibrations of the bosonic strings and of the superstrings, some Ramanujan’s identities concerning π and the zeta strings.

1. On some equations concerning the quantum electrodynamics coupled to quantum gravity. [1]

The key equations that govern the behaviour of the coupling constants in quantum field theory are the renormalisation group Callan-Symanzik equations. If we let g denote a generic coupling constant, then the value of g at energy scale E , the running coupling constant g

( )

E , is determined by

( )

E g dE E dg E =β , , (1)

(2)

where

β

( )

E,g is the renormalisation group β-function. Asymptotic freedom is signaled by

( )

E →0

g as E→∞, requiring β <0 in this limit.

With regard the Einstein gravity with a cosmological constant coupled to quantum electrodynamics in four spacetime dimensions, a standard calculation shows that the effective action to one-loop order is given by

Γ( ) = lndet∆i_j −lndetQ_αβ −lndet

(

iγµ∂_µ +eγµA_µ −im

)

2

1 1

. (2)

The last term (with A_µ the background gauge field) is the result of performing a functional integral over the Dirac field. The middle term is the contribution from the ghost fields required to remove the unphysical degrees of freedom of the gravity and electromagnetic fields. The first term is the result of integrating over the spacetime metric and electromagnetic fields. For operator ∆ij the heat

kernel Ki_j

(

x, x';

τ

)

is defined by

(

τ

)

(

τ

)

τ K x,x'; Kkj x,x'; i k i j =∆ ∂ ∂ − (3)

with boundary condition K

(

x,x'; 0

)

ij

( )

x,x'

i

j

τ

= =

δ

.

τ

is called the proper time. The Green

function Gij

( )

x, x' for the operator

i j

∆ is

∆i_kGk_j

( )

x,x' =

δ

i_j

δ

( )

x,x' . (4) It follows that the Green function and heat kernel are related by

( )

∫

(

)

∞ = 0 ; ' , ' ,x d

τ

K x x

τ

x Gi_j i_j . (5)

The importance of the heat kernel for quantum field theory arises from the existence of an asymptotic expansion as τ →0: K

(

x x

) ( )

E

( )

x r i j r r n i j

∑

∞ = − ≈ 0 2 / 4 ; , τ πτ τ (6)

where n is the spacetime dimension (chosen as 4 here) and the heat kernel coefficients E_ri_j

( )

x depend only locally on the details of coefficients of the differential operator ∆ij. The divergent part

of the effective action, as well as the Green function, can be related to the heat kernel coefficients. Formally

∫

(

)

∞ ∆ = ∆ =− 0 ; , 2 1 det ln 2 1

τ

trK x x d x d L ij n i j . (7)

The one-loop effective action (2) is then given by

(3)

As with the Green function (4) the divergent part of (7) comes from the τ ≈0 limit of the proper time integral. The divergent part of L_∆ is

divp

( )

∫

      ₊ ₊ − = ∆ 1 2 2 2 0 4 4 2 ln 2 1 32 1 c c

ctrE E trE trE E

E x d L

π . (9)

The lower limit on the proper time integration can be kept as τ =0 and the divergent part of the effective action L_∆ contains a simple pole as ε →0 given by

divp

( )

_∆ =

∫

2 4 2 16 1 xtrE d L ε π . (10)

The general form of ∆i_j is

∆i_j =

( )

Aαβ i_j∂_α∂_β +

( )

Bα i_j∂_α +

( )

C i_j (11) for coefficients

( )

A ij

αβ

,

( )

B ij

α

and

( )

C that depend on the spacetime coordinates through the ij

background field. Normal coordinates are introduced at x' with xµ = x'µ+yµ and all of the coefficients in (11) are expanded about yµ =0. This gives

( ) ( )

∑

(

)

∞ = + = 1 ... 0 ... 1 1 n i j i j i j n n y y A A Aαβ αβ αβµ µ µ µ (12)

with similar expansions for

( )

Bα i_j and

( )

C i_j. The Green function is Fourier expanded as usual, Gi_j

( ) ( )

x x =

∫

_n dnpeip⋅yGi_j

( )

p π 2 1 ' , , (13)

except that the Fourier coefficient Gij

( )

p can also have a dependence on the origin of the coordinate system 'x that is not indicated explicitly. If

G_ji

( )

p =G₀i_j

( )

p +G₁i_j

( )

p +G₂i_j

( )

p +... (14)

where G_ri_j

( )

p is of order p−2−r as p→∞ it is easy to see that to calculate the pole part of Gi_j

( )

x,x as n→4 only terms up to and including G2i_j

( )

p are needed.

The gravity and gauge field contributions result in

tr  +

(

+ + +

)

Λ      + − + − + + − = 2 2 2 2 2 2 1 12 8 3 32 1 32 1 8 3 2 1 8 3 8 1 4 3 8 3 _ω _ω _ξ _ωξ _ωζ _ωξζ _ω _ζ _ξ _ζ κ F v v E . (15)

The overall result for the quadratically divergent part of the complete one-loop effective action (8) that involves F2 is

(4)

( ) 

∫

     ₋ ₊ ₊ ₋ ₊ ₋ ₊ − = Γ 2 2 4 2 2 2 2 1 32 1 32 1 8 3 2 1 8 3 8 5 4 3 8 3 32 d xF E_c quad

π

ω

ξ

ωξ

ωζ

ωξζ

ω

ζ

κ

. (16) ( ) 

∫

     + − + − + + − − = Γ 2 2 4 2 2 2 2 1 32 1 32 1 8 3 2 1 8 3 8 5 4 3 8 3 2 16 1 F x d E_c quad

ω

ξ

ωξ

ωζ

ωξζ

ω

ζ

π

κ

. (16b)

If ξ →0, ζ →0, ω→1 are taken to obtain the gauge condition independent result, the non-zero result Γ( ) =− ₂

∫

4 2 2 2 1 128 d xF Ec quad

π

κ

(17) is found.

The net result for the divergent part of the effective action that involves F and therefore 2 contributes to charge renormalization is

divp

( )

( ) _

∫

     + Λ − − = Γ 2 4 2 2 2 2 2 2 2 2 2 1 ln 48 ln 256 3 128 E d xF e E E c c c

π

κ

π

κ

. (18) divp

( )

( ) _

∫

     + Λ − − = Γ 2 4 2 2 2 2 2 2 2 2 2 1 ln 3 ln 16 3 8 16 1 F x d E e E E c c c

π

κ

π

κ

. (18b)

From this the renormalization group function in (1) that governs the running electric charge to be calculated to be

( )

E e e E e      ₊ _Λ − = 2 3 32 12 , ₂ 2 2 2 3

π

κ

π

β

. (19)

The first term on the right hand side of (19) is that present in the absence of gravity and results in the electric charge increasing with the energy. The second term on the right hand side of (19) represents the correction due to quantum gravity. For pure gravity with no cosmological constant, or for small cosmological constant Λ, the quantum gravity contribution to the renormalization group

β

-function is negative and therefore tends to result in asymptotic freedom.

2. On some equations concerning the gravitational contributions to the running of gauge couplings. [2]

The action of Einstein-Yang-Mills theory is

S =

∫

d x −g_ R− gµαgνβ _µνa _αβa_ κ 4 F F 1 1 2 4 (20)

where R is Ricci scalar and F_µνa _{is the Yang-Mills fields strength}F_µν =∇_µΑ_ν −∇_νΑ_µ −_ig

[

Α_µ_,Α_ν

]

_. It is hard to quantize this lagrangian because of gravity-part’s non-linearity and minus-dimension coupling constant κ = 16πG. Usually, one expands the metric tensor around a background metric

(5)

µν

g and treats graviton field as quantum fluctuation h_µν propagating on the background space-time determined by g_µν,

g_µν =g_µν +κh_µν . (21) Furthermore, we can rewrite the lagrangian (20) also as follows:

∫

    − − = a a g g R G g x d S µα νβ _µν _αβ π 4 F F 1 16 1 4 . (20b)

Let us set g_µν =η_µν, where η_µν is the Minkowski metric. h is interpreted as graviton field, _µν fluctuating in flat space-time. The lagrangian can be arranged to different orders of h or _µν κ. The free part of gravitation is of order unit and gives the graviton propagator

µνρσ

( )

[

gνρgµσ gµρgνσ gµνgρσ

]

k

i k

P_G = ₂ + − (21b)

in the harmonic gauge 0

2 1 = ∂ − ∂ = ν ν µ µν ν µ h h

C . For simplicity, the metric gµν is understood as

µν

η

. The interactions of gauge field and gravity field are determined by expanding the second term of the lagrangian (20). We can compute the β function by evaluating two and three point functions of gauge fields. These Green functions are general divergent, so counter-terms are needed to cancel these divergences. The relevant counter-terms to the β function are

Tµν =iδ_abQµνδ₂, Tµνρ =gfabcV_qkpµνρ

δ

₁, Qµν ≡qµqν −q2gµν, V_qkpµνρ ≡gνρ

(

q−k

)

µ +gρµ

(

k−p

)

ν +gµν

(

p−q

)

ρ. (22)

The β function is defined as

( )



     − ∂ ∂ = 2 1 2 3_δ _δ µ µ

β g g . With the consistency condition

R R I g I₂ ₂ 2 1 µν

µν = , we obtain for two and three point functions

( + ) =

∫

_−

( )

+

( ) (

+ −

) ( )

0 2 _ 2 2 2 2 2 2 3 2 0 2 3 2 q R q R R b a LR Q dx I I q x x I T µν κ µν M M ₍₂₃₎ ( )

∫

( )

(

) ( ) (

) ( )

  + − + − + = + 2 2 2 2 2 2 0 2 1 2 k R q R R qkp e d LR ig dx V I g q q g I g k k g I T µνρ κ µνρ µν ρ µ νρ M νρ µ ν ρµ M +

(

gρµpν − pρgµν

) ( )

I₂R M_p2

]

₍₂₄₎

from which we can directly read off the two-point and three-point counter-terms δ2κ and

κ δ1 respectively                       + + − − = 2 2 2₂ 2 2 2 2 2 2 ln 1 16 1 c s w s c s c M y M M γ µ µ µ π κ δκ ;

(6)

                      + + − − = 2 2 2₂ 2 2 2 2 2 1 ln 1 16 1 c s w s c s c M y M M

γ

µ

π

κ

δ

κ . (25)

Putting

δ

₁κ and

δ

₂κ in the following equation

      − ∂ ∂ = ∆ κ _δκ _δκ µ µ β 2 1 2 3 g , (26)

we obtain the gravitational corrections to the gauge β function

            + − + − = ∆ 2 0 2₂ 2 2 2 2 1 ln 16 c s w s c s M y M g

γ

µ

π

µ

κ

β

κ (26b)

In general, the total

β

function of gauge field theories including the gravitational effects may be written as follows             + − + − + − = 2 0 2₂ 2 2 2 2 3 0 2 ln 1 16 16 1 c s w s c s M y M g g b

γ

µ

π

µ

κ

π

β

κ . (27)

The interesting feature of gauge theory interactions is the possible gauge couplings unification at ultra-high energy scale when the gravitational effects are absent. Where the running of gauge coupling in the Model Standard Super Symmetric (MSSM) without gravitational contributions is known to be

( )

µ

α

( )

_π

_µ

α

_e _e M lnM 10 33 1 1 = − + − _;

( )

µ

π

α

µ

α

_w _w M lnM 2 1 1 1 = − + − _;

( )

µ

π

α

µ

α

_s _s M lnM 2 3 1 1 = − − − ₍₂₈₎

with experimental input at MZ

αe−1

( )

MZ =58.97±0.05;

( )

29.61 0.05 1 = ± − Z w M α ; αs−1

( )

MZ =8.47±0.22 (29) We note that these values are equal to the following values connected with the aurea ratio: 38,12461180 + 20,56230590 = 58,68691770; 29,12461180; 8,49844719. Indeed, we have:

[

( )

Φ 35/7 +

( )

Φ 7/7

]

×3=

(

11,09016994+1,61803399

)

×3=12,70820393×3≅38,12461180;

[

( )

Φ 28/7

]

×3=6,85410197×3≅20,56230590;

[

( )

Φ 28/7 +

( )

Φ14/7 +

( )

Φ −21/7

]

×3=

(

6,85410197+2,61803399+0,23606798

)

×3= =9,70820394×3≅29,12461180;

[

( )

Φ14/7 +

( )

Φ −28/7 +

( )

Φ −42/7 +

( )

Φ −63/7

]

×3=

(

2,61803399+0,145898033+0,055728089+

(7)

+0,013155617

)

×3=2,832815729×3≅8,49844719. As usual, 1,61803399 2 1 5+ _≅ =

Φ , i.e. the value of the aurea ratio.

3. On various equations concerning some quantum effects in the theory of gravitation [3]

The general spherical static metric is given by

ds2 =−f

( )

r dt2 +h

( )

r −1dr2 +r2dΩ2, (30)

where f

( )

r and h

( )

r are arbitrary functions of the coordinate r . The angular part of the metric is diagonal and given by

∑

( )

∏

− = − + =         = Ω 2 1 2 1 2 2 2 sin d i d i j j i d d

θ

. (31)

Consider an interesting classical scalar field

φ

( )

x living in a spacetime with the metric (30). This field has an action given by:

∫

∑

      − ∇ − − = ∞ =2 2 n n n d g x d S

φ

λ

φ

. (32)

The

λ

_n are a set of arbitrary coupling constants. In particular,

λ

₂ ≡m2 gives the mass of a weakly coupled excitation of this field. We will expand φ in the eigenmodes of the free, classical wave equation such that

φ

( )

x =

∫

dµ_pa_pφ_p

( )

x , (33)

φ

p m

φ

p

2

2 =

∇ . (34)

A sufficiently large set of quantum numbers p label the eigenbasis. The abstract formal expression

p

dµ simply represents an appropriate measure over the modes under which

∫

d

( ) ( )

x y = d

(

x−y

)

p p pφ φ δ µ , and

∫

− p

( ) ( )

q = pq d x x g x d φ φ δ .

Now we want to consider the energy density, ρ, of a massless scalar field in a infinitely large hypercubic blackbody cavity at temperature T . Consider a real scalar field I

( )

x t

,

φ , where I is some kind of p -dimensional polarization index representing p internal degrees of freedom (for example, in a well-chosen gauge, the transverse polarization of an Abelian vector field behaves essentially like an internal index on a scalar field with p=d−2). Further assume that the field is sufficiently weakly coupled that each polarization component can be treated as an independent field

(8)

obeying an action similar to eq. (32) with all interaction coefficients higher than 2 2 ≡M

λ

set equal

to zero. Then each field component obeys the classical equation of motion ∇2

φ

I

( )

x =M2

φ

I

( )

x . (35)

The solutions to equation (35) may be expressed as a sum over modes labelled by a wave vector k _a obeying k2 =−M2:

( )

a a I k a a I k I k = A sin k x +B cosk x φ , (36)

for arbitrary real coefficients A and kI I k

B . We take the state to be labelled by the d−1 spatial components of k and fix the frequency of each mode by _a ω_k2 ≡k₀2 =M2+k_iki. We now confine the field to live in a cubic box of side length L by demanding Dirichlet boundary conditions at

L

xi =0, for i=1...d−1. This demands BkI =0 and

i mi

L

k =π , (37)

where m is a spatial vector whose components are non-negative integers. The total energy in the _i hypercube is given by

∑ ∑ ∑

= ∞ = ∞ = − = p I m m mI k d n U 1 1 0 1 0 ...

ω

, (38)

where we understand that

ω

_k is given by

∑

− = + = 1 1 2 2 2 2 d i i k m L M

π

ω

. (39)

The set of integers n defining the quantum state must obey the appropriate statistics for the field mI I

φ

. We have described a pure quantum state of the theory. At a finite temperature T and zero chemical potential, the system will be in a mixed state governed by the partition function

{ }

∑

− = mI k n e Q βω , (40)

where β ≡1/T. This can be evaluated to give

∑ ∑

(

)

∞ = ∞ = − − − − = 0 0 1 1 1 ln ... ln m md k e p Q ξ ξ βω , (41)

where ξ =1 for bosons and ξ =−1 for fermions. The overall factor of p occurs because the energy is independent of p , so each polarization mode contributes equally. The average occupation number of a given momentum mode in the thermal state is then given by

(9)

∑

= − = ∂ ∂ − = = p I k mI m _k e p Q n n 1 ln 1

ξ

ω

β

βω . (42)

The total energy in the hypercube can now be found by combining the expressions (38) and (42), or by

∑ ∑

∞ = ∞ = ∞ = ∞ = − − − = = ∂ ∂ − = 0 0 0 0 1 1 1 1 ... ... ln m m m m k m k d d k e p n Q U

ξ

ω

β

βω . (43)

We should pass from a state labelling in terms of quantum numbers m to a labelling in terms of i

physical momenta k , with a mode density determined by the differential limit of equation (37). i

The sums over m then become integrals over i k as i

( )

∫

₋ →→∞ − −₋

ξ

ω

π

_eβωk p L k d U k d d d L 1 1 1 2 , (44)

where

ω

_k is understood as M2+k_iki . The factors of 2 in the denominator of the measure arise because the integrals over the k run over both positive and negative values, whereas the i m were i

only summed over non-negative values. Equation (44) scales properly with the volume, so that even in the infinite volume limit we can define the spectral energy density over k modes. Using the _i spherical symmetry of the infinite volume limit and defining k = k_iki = k_i , eq. (44) becomes

( )

∫

∞ + − − − − + = = 0 2 2 2 1 2 2 2 2

π

ξ

ρ

_β k M d d d e k M k dk S Vol p V U . (45)

This defines the spectral energy density over the magnitude of the spatial momentum, via

( )

∫

≡ dkuk k ρ , as

( )

π β ₋ξ + = + − − − 2 2 2 2 2 1 2 2 M k d d d k e k M k S Vol p k u . (46)

Similarly, we can define the spectral energy density over the frequency as

( )

(

)

( )

ξ

ω

π

ω

βω ω = − ₋ − − − e M S Vol p u d d d 2 2 2 3/2 1 2 2 , (47)

where

ω

runs over

[

M,∞

]

.

The total energy density can now be evaluated using either eq. (46) or eq. (47) match. Simple analytic results can be found for the case M =0, which will also apply when T >>M. In this case,

k =

ω and equations (46) and (47) match. They give

( )

∫

∞ − − − − = 0 1 1 2 2π ξ ρ x d d d d e x dx S Vol pT . (48)

(10)

The integral can be evaluated by pulling the exponential into the numerator, performing a Taylor series in e−x, doing the integral, and resumming the Taylor series. The result for the general definite integral is given by ( )

( )(

)

∫

∑

− = − − − − ₋ −       − = − b a d n b n a n d x d n e b e a d d e dx x 1 0 2 / 1 1 ! ! 1 2 2 1

ζ

ξ

ξ , (49)

where

ζ

( )

s is the Riemann zeta function, which can be defined for real s>1 as

( )

∑

∞ = = 1 1 n s n s ζ . (50)

This series arises in the evaluation of the integral with the ξ =1, for bosons. For fermions, ξ =−1, the corresponding series is

( )

∑

( )

∞ = − − ≡ 1 1 1 1 n s n n s R _{. (51)}

This series can be evaluated using

ζ

( )

s

( )

s _s _s _s _sζ

( )

s 2 2 ... 6 2 4 2 2 2 ₊ ₊ ₊ ₌ = −R _{, (52)} so that

( )

s _sζ

( )

s      ₋ = 2 2 1 R _{, (53)}

which is the origin of this factor in eq. (49). So, eq. (48) becomes

( )

_{( )(}

₎

( ) d d d d _d T d d p 2 / 1 2 2 / 1 2 1 2 ! 1 2 2 1 − − −       − Γ −       ₋ =

π

ζ

ρ

ξ . (54)

Furthermore, we can rewrite the eq. (48) also as follows:

( )

− = = ₋−

_∫

∞ − 0 1 1 2 2

π

ξ

ρ

x d d d d e x dx S Vol pT ( )

_{( )(}

₎

( ) d d d d T d d d p 2 / 1 2 2 / 1 2 1 2 ! 1 2 2 1 − − −       − Γ −       ₋

π

ζ

ξ . (54b)

Before, we have considered the energy density, ρ, of a massless scalar field in a infinitely large hypercubic cavity at temperature T . We now want to calculate the energy flux, Φ, emitted from a blackbody with this same temperature. For the flux calculation, instead of

( )

d−2

S Vol we will encounter

∫

( )

=      − Θ Ω − − −2 2 2 2 cos d d d d θ π θ

(11)

( )

(

( )

)

( )

(

)

( )

∫

∏

= − =       − Γ − =       = − − − = − − − − π π π π

π

θ

2 0 0 0 2 / 0 3 2 / 2 3 2 1 2 2 3 2 1 2 1 2 2 2 2 sin cos ... d d d i i i d d d S Vol d d d d d d d

( )

(

)

( )

2 2 2 2 2 1 2 1 ₋ −       − Γ       − Γ − = = d d S Vol d d d B Vol

π

, (55)

where Θ

( )

x is the step function and B is the n-dimensional unit ball: the compact subspace of n R n bounded by Sn−1(for example, B3 is the unit 3-ball of volume 4π/3 bounded by the sphere S of 2 area 4 ). The fact that the expression for the energy density becomes that for the flux when π

( )

d−2 S

Vol is replaced by Vol

( )

Bd−2 makes physical sense, since B is the projection of n S onto n R . n We are left with the relationship of flux to energy density as

(

)

( )

ρ

π

2 2 2 2 2 2 1 − − =       − Γ −       − Γ = Φ _dd S Vol B Vol d d d . (56)

Thus, the d-dimensional Stefan-Boltzmann law is given by

( )

_{( )(}

₎

(

)

d d d d T d d d d p 2 / 2 2 / 1 2 2 2 2 ! 1 2 2 1

π

ζ

ξ       − Γ − −       − = Φ − − . (57)

4. On some equations concerning the supersymmetric Yang-Mills theory applied in string theory and some lemmas and equations concerning various gauge fields in any non-trivial quantum field theory for the pure Yang-Mills Lagrangian. [4] [5]

The fields of the minimal N =2 supersymmetric Yang-Mills theory are the following: a gauge field

m

A , fermions λ_αi and λ_α&_i transforming as

(

1/2,0,1/2

)

and

(

0,1/2,1/2

)

under

( )

L SU

( )

R SU

( )

I

SU 2 × 2 × 2 , and a complex scalar B - all in the adjoint representation of the gauge group. Covariant derivatives are defined by

DmΦ=

(

∂m+iAm

)

Φ, (58)

and the Yang-Mills field strength is

Fmn =∂mAn−∂nAm+

[

Am,An

]

. (59)

The supersymmetry generators transform as

(

1/2,0,1/2

) (

⊕ 0,1/2,1/2

)

; introducing infinitesimal parameters η_αi and η_α&_i, furthermore D=

[ ]

B,B . The minimal Lagrangian is

(12)

[ ]

[

]

[

]

∫

      + − − − − − = M i i ij j i ij m m i m m i mn mn B i B i B B B D B D D i F F xTr d e

L

λ

α

σ

_α_α&

λ

α

ε

λ

α

λ

_α

ε

λ

_α&

λ

α&

& , 2 , 2 , 2 1 4 1 1 ₄ 2 2 (60) Here Tr is an invariant quadratic form on the Lie algebra which for G=SU

( )

N we can conveniently take to be the trace in the N dimensional representation.

It is possible to realize a mass term for the N =1 matter multiplet (which consists of B and 2&

α

λ

ψ

= ) by adding to the Lagrangian a term of the form I

( ) {

ω

+ Q₁,...

}

, where Q₁ is the charge corresponding to the

ρ

₁ transformation. Furthermore, Q₁ is the only essential symmetry. We obtain: = +

( ) {

+

}

= −

∫

(

+

)

−

∫

M M B B mTr m x d e m m xTr d L Q I L L 2 2 1 ,... ˆ 2 4 2 2 2 2 4 1 _&& & & & & σ σ α αλ λ λ λ ω , (61) with m=

σ

_mn_2&&₂

ω

_kl

ε

mnkl. (62)

The mass is proportional to the holomorphic two-form

ω

. The N =1 gauge multiplet, consisting of the gauge field A and the gluino _m λ_α =λ_αi , remains massless.

With regard the Λ16-amplitude in the type IIB description, the classical action for the operator

( )

Λ16 in the AdS5×S5 supergravity action is

[ ]

∫

∏

(

)

(

)

( )

= Λ −       − Λ         − + = 16 1 ˆ 0 5ˆ 0 0 0 0 4 16 5 0 0 0 4 12 1 2 ₁ ; , 1 5 0 p p p p S s i g x J x x x x K t d x d V g e J S s µ µ χ π

γ

ρ

. (63)

This result is in agrees with the following expression obtained in the Yang-Mills calculation:

( )

∫

∏

_[

₍

₎

_]

(

)

= + −         − + − + = 16 1 0 0 0 4 2 0 2 0 4 0 8 8 5 0 0 0 4 8 8 16 1 1 2 2 p A p A p i g YM p p p p p p p YM YM _x _x x x d d d x d e g x G _α _µ _αµ_α α θ π

ξ

σ

η

ρ

ξ

η

ρ

& & (64) i.e. the correlation function in the super Yang-Mills description.

The low energy effective action for type IIB superstring theory in ten dimensions includes the interaction (in the string frame)

∫

( )

( ) ( )

( )

      ₊ ₊ ₊ ₊ − ≈ 4 10 2 −2 2 4 6 4 ... 6 9 2 4 6 2 3 2 3 _e _e _e _D R g x d l S _s

ζ

φ

ζ

φ

ζ

φ , (65)

where the … involve contributions form D-instantons.

We consider the perturbative contributions to the _D6R4_{interaction. The sum of the contributions to} the four graviton amplitude at tree level and at one loop in type II string theory compactified on T2 is proportional to

(

₂

) (

₂

) (

₂

)

4 2 2 2 2 2 2 4 / 1 4 / 1 4 / 1 4 / 4 / 4 / R       + + Γ + Γ + Γ − Γ − Γ − Γ − − I u l t l s l u l t l s l e V s s s s s s

π

φ , (66)

(13)

where V₂ is the volume of T in the string frame, 2 s ,,t u are the Mandelstam variables, and I is obtained from the one loop amplitude. The amplitude is the same for type IIA and type IIB string theories. Now I is given by

∫

( )

Ω Ω Ω Ω = F 2 , 2 2 F Z d I lat (67)

where F is the fundamental domain of SL ,

( )

2 Z , and d2Ω=dΩdΩ/2. In the above expression, the lattice factor Z which depends on the moduli is given by lat

(

)

(

)

( )

∑

× ∈              Ω Ω − − = Z Mat A lat U A U T A iT V Z , 2 2 2 2 2 2 2 1 1 det 2 exp

π

. (68)

Expanding eq. (67) to sixth order in the momenta, we get that

(

3 3 3

)

[

₁ ₂

]

6 ˆ ˆ 3 s t u I I l I = s + + + _{, (69)} where

∫

∏

(

) (

)

= Ω − Ω − Ω − Ω Ω Ω = L _i i lat d Z d I F T 3 1 3 2 3 1 2 1 2 2 2 2 2 1 4 ln ˆ ; ln ˆ ; ˆ ; ˆ

ν

_χ

_ν

_χ

_ν

_χ

_ν

_{, (70)} and Iˆ₂ =

∫

∏

[

(

)

]

= Ω − Ω Ω Ω L _i i lat d Z d F T 3 1 3 2 1 2 2 2 2 2 ; ˆ ln

χ

ν

. (71)

We have also that

( )

∫

( )

Ω Ω ( )= + + Ω Ω = L I I I E Z d I _lat SL Z F 3 3 2 1 1 1 , 2 3 2 2 2 1 3 ˆ ˆ ˆ , ˆ 4 4

π

, (72) where 2 1 1 1,ˆ ˆ I I , and 3 1 ˆ

I are the contributions from the zero orbit, the non-degenerate orbits and the degenerate orbits of SL ,

( )

2 Z respectively. In the eq. (72), we have used the expression

( )

(

)

1 2 1 2 1 2 2 1 2 2 / 5 2 / 5 0 , 0 2 1 2 3 2 2 3 2 , 2 3 5 2 4 3 6 2 , Ω ≠ ≠ Ω Ω + Ω + Ω = Ω Ω

_∑

imm m m Z SL e m m K m m E

ζ

π

ζ

π

π . (73)

Thence, we can rewrite the eq. (72) also as follows:

( )

_∫

Ω Ω = L lat Z d I F 2 2 2 1 3 ˆ 4 4

π

_×

_{( )}

(

)

₁ ₂ ₁ 2 1 2 2 1 2 2 / 5 2 / 5 0 , 0 2 1 2 3 2 2 3 2 5 2 4 3 6 2 Ω ≠ ≠ Ω Ω + Ω + Ω

_∑

imm m m e m m K m m

_π

π

ζ

π

ζ

. (74)

(14)

∫

( )

Ω Ω ( )= Ω Ω = L Z SL E d V I F , 0 ˆ 2, 3 2 2 2 2 1 1 . (75)

The contribution from the non-degenerate orbits gives

∫

∞

∫

( )

∑

∞ − ∞ ≠ ≥ > + + Ω Ω − − = Ω Ω Ω Ω Ω = 0 0 , 0 2 , 2 3 2 2 2 1 2 2 1 2 2 2 2 , 2 ˆ p j k pU j k U T iTkp Z SL e E d d V I π π

(

)

( )

∑

(

)

≠ ≠ = 0 , 0 2 2 2 / 5 2 / 5 , 2 3 2 1 2 , 2 k p ipkT Z SL e pk T K k p U U E T

π

π . (76)

Furthermore, the contribution from the degenerate orbits gives

( )

_{( )}

( )

₍

₎

( ) ( ) ( )

∫

₋

∫

∞

∑

≠ + Ω −       + = Ω Ω Ω Ω Ω = 1/2 2 / 1 0 0 , 0 , , 2 3 2 2 3 2 3 , 2 3 2 2 2 1 2 3 1 , 4 5 3 6 2 2 , ˆ 2 2 2 2 p j Z SL pU j U T Z SL U U E T T e E d d V I

ζ

πζ

π

π . (77)

Now we want to show two lemmas and equations concerning the gauge fields as described in the

Jormakka’s paper “Solutions to Yang-Mills equations [5]”. Thence, in the next Section, we describe some possible mathematical connections between some equations concerning this interesting argument and some equations concerning the Ramanujan’s modular equations that are related to the physical vibrations of the bosonic strings and of the superstrings, and some Ramanujan’s identities concerning π.

Lemma 1

Let the gauge field satisfy

(

)

= − ∑= 3 1 2 2 3 2 1 , , , , 0 j j e c s x x x A_µaR _a _µ β ρ (78)

where ρj and c are as in the following expressions: µ j j j i r =ρ + σ , ρ_j,σ_j∈R;

ρ

₁ =x₁−x₂ + 2x₃;

ρ

₂ =x₁−x₂− 2x₃; ₃ ₁ ₃ 2 1 x x + − = ρ ;

σ

₁=

σ

₂ =0; ₃ ₀ 2 1 x = σ ; h

( ) (

r_j =u ρ_j,σ_j

) (

+iv ρ_j,σ_j

)

(79) c₀ = 2; c₁ =c₂ =1; c₃ =0; e₀ =− 2; e₁ =e₂ =1; e₃ =0, (80) and

β

,s_a∈R. Then

∫

(

)

      = ₃ 2 / 3 2 2 2 3 2 1 3 1 2 , , , 0

β

π

k a a k x x x s c A x d . (81)

(15)

The expression (79) describes a localized gauge field, gauge boson, which moves in the x₁, x₂ direction with the speed of light as a function of x . We select a concrete case that gives easy ₀ calculations. Let f

( )

ρ

_j =−

β

2

ρ

2_j and extend it to h

( )

r_j =−

β

2r_j2. The real and imaginary parts of

µ µ

µ c ie

d = + are described from (80). We evaluate the gauge potential at x0 =0 and take the real part.

We change the variables to y₁,y₂,y₃

₁ ₁ ₂ ₃ 6 1 3 2 3x x x y = − − ; ₂ ₂ ₃ 3 1 3 2 x x y = − ; y₃ =2x₃ (82) Then

∑

= + + = 3 1 2 3 2 2 2 1 2 j j y y y

ρ

(83)

As y2 and y3 are not functions of x1 we can change the order of integration

(

)

( )

(

)

( )

∫

− ∑= =

∫

− ∑= = − + − = − + − = 2 1 2 2 3 2 2 2 2 1 2 2 3 2 2 2 3 1 2 2 3 1 2 2 ₂ 2 1 1 2 2 2 1 1 2 3 3 3 1 y y y y y y y e dy xe d e dx xe d xe d xe d β j ρj β j j β β β β =

∫

2 −

(

+

)

2

( )

2 −1 3 1 2 3 2 2 2

β

π

β y y xe d (84)

As y3 is not a function of x2 we can change the order of integration

∫

−

(

+

)

∫

− ( ) 2 1 2 2 3 2 2 2 2 2 1 1 2 3 1 y y y e dy xe d β β =

∫

2 −

(

+

)

2

( )

2 −1= 3 1 2 3 2 2 2

β

π

β y y xe d =

( )

−

∫

−

∫

− ( ) =

( )

−

∫

−

∫

− ( ) = 2 2 2 2 3 2 2 2 2 2 3 2 2 2 1 2 3 1 2 2 1 2 3 1 2 2 2 3 3 1 2 2 3 1 _y y _y y e dy e dx e dx e dx β β

π

β

β β

β

π

( )

3 2 3 3 2 3 2 2 2 2 1 2 3 3 1 2 2 2 1 2 3 3 1 2 2 2 3 3 1 2 3 2 2 3 2 − ₋ − − − = = =

π

β

_∫

β y

π

β

_∫

β y

π

β

e dy e dx . (85) Thus 2 ₃ 3 2 3 1 2 2 2

β

π

ρ β

∫

      = ∑ − j xe d  ⇒      = ∑ ⇒

_∫

− ₃ 3 2 3 1 2 2 2

β

π

ρ β j xe d

∫

(

)

      = ⇒ ₃ 2 / 3 2 2 2 3 2 1 3 1 2 , , , 0

β

π

k a a k x x x s c A x d . (86)

Thence, we can conclude that the eq. (81) is true. We can rewrite the eq. (85) also as follows:

∫

−

(

+

)

∫

− ( ) 2 1 2 2 3 2 2 2 2 2 1 1 2 3 1 y y y e dy xe d β β =

∫

2 −

(

+

)

2

( )

2 −1= 3 1 2 3 2 2 2

β

π

β y y xe d

( )

2

( )

−3 = 3 2 2 2 1 2 3 3 1

_π

_β

(16)

(

)

( )

6 3 2 1 2 2 1 4 8 2 2 3 1 2 3 2 2 2

β

π

β

π

β ₌ _×       =

_∫

− y +y − xe d . (86b) Lemma 2

Let the gauge field satisfy

(

)

= − ∑= 3 1 2 2 3 2 1 , , , , 0 j j e c s x x x A_µaR _a _µ β ρ (87)

where ρ_j and c are as in (79), (80) and _µ

β

,s_a∈R. Then

∫

=−

∑

a aB s d 2 3 3 16 1

π

β

R L ₍₈₈₎

where in Minkowski’s metric at x₀ =0, we have B=0. In the negative definite metric

( )

            − − − − = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ,ν µ µν g , (88b) we have that 4 3 2 3 13₊ ₊ = B .

From (79) and (82) follows that

1 1 2 3 2 1 6 1 3 1 y y y − + =

ρ

; 2 1 2 3 2 1 6 1 3 1 y y y − − =

ρ

; 3 1 2 3 2 3 1 y y − − =

ρ

. (89)

For Minkowski’s metric

P

( )

ρ

=0=B₁y₁2 +B₂y₂2 +B₃y₃2+B₄y₁y₂+B₅y₁y₃+B₆y₂y₃ (90) where B_k =0 for all k . For the metric (88b)

P

( )

ρ

=4

ρ

₁2+4

ρ

₂2 +

ρ

₃2−4

ρ

₂

ρ

₃ =B₁y₁2 +B₂y₂2 +B₃y₃2+B₄y₁y₂+B₅y₁y₃+B₆y₂y₃ (91) where 3 13 1= B ; 3 2 2 = B ; B₃ =4; 2 3 4 4 =− B ; 6 4 5 =− B ; 3 4 6 =− B . (92)

(17)

( )

(

)

( )

(

)

∫

− + + =

∫

− + + 32 2 2 2 1 2 2 3 2 2 2 1 2 ₂ 2 1 3 2 2 1 3 y y y xe d P xe d β ρ ρ ρ

ρ

β ( 4 1 2 5 1 3 6 2 3) 2 3 3 2 2 2 2 1 1y B y B y B y y B y y B y y B + + + + + . (93) As y₂ and y are not functions of ₃ x₁ we can change the order of integration and change the integration parameter x₁ to y₁ ( )

(

)

( )

(

)

∫

− + + =

∫

− + + 32 2 2 2 1 2 2 3 2 2 2 1 2 ₂ 2 1 3 2 2 1 3 y y y xe d P xe d β ρ ρ ρ

ρ

β ( 4 1 2 5 1 3 6 2 3) 2 3 3 2 2 2 2 1 1y B y B y B y y B y y B y y B + + + + + =

∫

( )

(

)

_

∫

( )

(

)

∫

( )   + + + + = − + − − 12 2 2 1 2 2 3 2 2 2 ₂ 2 1 1 3 2 6 2 3 3 2 2 2 2 2 1 2 1 1 1 2 2 1 2 y y y y e dx y y B y B y B e y dx B xe d β β β

(

)

( ) _=   + +

_∫

− 12 2 2 2 1 1 1 3 5 2 4 y e y dx y B y B β ( )

(

)

_ ( )

(

)

( )   + + + + =

_∫

− +

_∫

−

_∫

− 12 2 2 1 2 2 3 2 2 2 ₂ 2 1 1 3 2 6 2 3 3 2 2 2 2 2 1 2 1 1 1 2 2 1 2 3 1 y y y y e dy y y B y B y B e y dy B xe d β β β

(

)

( ) _=   + +

_∫

− 12 2 2 2 1 1 1 3 5 2 4 y e y dy y B y B β ( )

(

)

( )

(

)

∫

      + + + = − +

β

π

β

π

β 2 1 2 2 1 2 3 1 3 2 6 2 3 3 2 2 2 3 1 2 2 1 2 2 3 2 2 2 y y B y B y B B xe d y y . (94)

As y is not a function of ₃ x₂ we can change the order of integration and change the integration parameter x₂ to y₂: ( )

(

)

( )

(

)

∫

− 2 12+ 22+ 32 =

∫

− 2 2 12+ 22+ 32 2 1 3 2 2 1 3 y y y xe d P xe d β ρ ρ ρ

ρ

β (B₁y₁2 +B₂y₂2 +B₃y₃2+B₄y₁y₂+B₅y₁y₃+B₆y₂y₃)= ( )

(

)

( )

(

)

=      + + + =

_∫

− +

β

π

β

π

β 2 1 2 2 1 2 3 1 3 2 6 2 3 3 2 2 2 3 1 2 2 1 2 2 3 2 2 2 y y B y B y B B xe d y y ( )

( )

( ) ( )

∫



∫

  + + = − − − 22 2 2 2 2 2 3 2 ₂ 2 1 2 2 2 2 2 2 1 2 3 1 2 2 1 3 2 1 2 1 2 2 3 3 1 y y y e y dy B e dy B e dx β β β

β

π

( ) ( ) _=   + +

_∫

−

_∫

− 22 2 2 2 2 ₂ 2 1 2 2 3 6 2 2 1 2 2 3 3 2 1 2 1 y y e y dy y B e dy y B β β

β

( )

∫

_=      + + = −

β

π

β

π

β

π

β

π

β 2 1 2 2 1 2 1 2 2 1 2 1 2 2 1 2 2 3 3 1 2 3 3 3 2 3 1 2 2 1 3 2 3 2 y B B B e dx y ( )

( )

=      + + =

_∫

− 2 2 3 3 4 2 4 1 2 2 1 3 2 1 2 1 2 1 2 2 1 2 3 3 1 2 3 2

β

π

dye β y B B B y

( ) (

)( )

( ) (

5 1 2 3

)

2 3 5 3 2 1 2 3 2 1 2 1 2 2 1 2 3 3 1 B B B B B B + + = + + =

β

π

β

π

. (95)

(18)

( )

(

)

(

)

( )

₅ 2 3 2 3 2 6 3 1 5 2 1 4 2 3 3 2 2 2 2 1 1 2 2 1 3 16 81 4 2 3 2 2 2 1 2

β

π

β × =       + + + + +

∫

d xe− y +y +y By B y B y B y y B y y B y y . (95b)

Thence, we can conclude that the eq. (88) is true. Indeed, we have that:

( )

(

)

_{( )}

( )

(

)

∫

− + + =

∫

− 12+ 22+ 32 2 2 3 2 2 2 1 2 ₂ 2 1 3 2 2 1 3 y y y xe d P xe d β ρ ρ ρ

ρ

β (B₁y₁2 +B₂y₂2+B₃y₃2+B₄y₁y₂+B₅y₁y₃+B₆y₂y₃)=

( ) (

5 1 2 3

)

2 3 2 1 B B B + + =

β

π

=

_∫

=−

∑

a aB s d 2 3 3 16 1

π

β

R L _{. (96)}

5. Ramanujan’s equations, zeta strings and mathematical connections

Now we describe some mathematical connections with some sectors of String Theory and Number Theory, principally with some equations concerning the Ramanujan’s modular equations that are related to the physical vibrations of the bosonic strings and of the superstrings, the Ramanujan’s identities concerning π and the zeta strings.

3.1 Ramanujan’s equations [6] [7]

With regard the Ramanujan’s modular functions, we note that the number 8, and thence the numbers 64=82 and 32=22×8, are connected with the “modes” that correspond to the physical vibrations of a superstring by the following Ramanujan function:

( )

                ₊ +         ₊ ⋅           = − ∞ ₋

∫

4 2 7 10 4 2 11 10 log ' 142 ' cosh ' cos log 4 3 1 8 2 ' ' 4 0 ' 2 2 w t itw e dx e x txw anti w w t w x

φ

π

π π . (97)

Furthermore, with regard the number 24 (12 = 24 / 2 and 32 = 24 + 8) this is related to the physical vibrations of the bosonic strings by the following Ramanujan function:

(19)

( )

                ₊ +         ₊ ⋅           = − ∞ ₋

∫

4 2 7 10 4 2 11 10 log ' 142 ' cosh ' cos log 4 24 2 ' ' 4 0 ' 2 2 w t itw e dx e x txw anti w w t w x

φ

π

π π . (98)

It is well-known that the series of Fibonacci’s numbers exhibits a fractal character, where the forms

repeat their similarity starting from the reduction factor 1/φ = 0,618033 = 2 1 5−

(Peitgen et al. 1986). Such a factor appears also in the famous fractal Ramanujan identity (Hardy 1927):

      − − + + + = − = =

∫

q t dt t f t f q R 0 1/5 4/5 5 ) ( ) ( 5 1 exp 2 5 3 1 5 ) ( 2 1 5 / 1 618033 , 0

φ

, (99) and                     − − + + + − Φ =

∫

q t dt t f t f q R 0 1/5 4/5 5 ) ( ) ( 5 1 exp 2 5 3 1 5 ) ( 20 3 2

π

, (100) where 2 1 5+ = Φ .

Furthermore, we remember that

π

arises also from the following identities (Ramanujan’s paper: “Modular equations and approximations to π” Quarterly Journal of Mathematics, 45 (1914), 350-372.):

(

)(

)

_      ₊ ₊ = 2 13 3 5 2 log 130 12

π

, (100a) and                 ₊ +         ₊ = 4 2 7 10 4 2 11 10 log 142 24

π

. (100b)

From (100b), we have that

                ₊ +         ₊ = 4 2 7 10 4 2 11 10 log 142 24

π

. (100c)

(20)

( )

, ... 1 1 1 1 : : 3 2 5 / 1 + + + + = =u q q q q q u q <1 (101)

and set v=u

( )

q2 . Recall that

ψ

( )

q is defined by the following equation

( )

∑

( )

(

( )

)

∞ = ∞ ∞ + ₌ = = 0 2 2 2 2 / 1 3 ; ; , : n n n q q q q q q q f q

ψ

. (102) Then

∫

( )

(

)

_       + − − + + = 2 3 ₂2 5 5 2 5 1 2 5 1 log 5 log 5 8 uv uv v u q dq q q

ψ

. (103)

We note that 1+

(

5−2

)

=2⋅0,61803398 and that 1−

(

5+2

)

=2⋅1,61803398, where 61803398

, 0 =

φ and Φ=1,61803398 are the aurea section and the aurea ratio respectively. Let

( )

2

:

: k q uv

k = = . Then from page 326 of Ramanujan’s second notebook, we have

2 5 1 1       + − = k k k u and       − + = k k k v 1 1 2 5 . (104) It follows that

( )

      + − = k k k v u 1 1 log 5 1 log 2 3 8 . (105)

If we set ε =

(

5+1

)

/2=1,61803398, i.e. the aurea ratio, we readily find that ε3 = 5+2 and 2

5

3 = −

−

ε . Then, with the use of (105), we see that (103) is equivalent to the equality

∫

( )

_      − + +       + − = − k k k k k q dq q q 3 3 8 5 5 1 1 log 5 1 1 log 5 1 5 8

ε

ψ

. (106)

Now from Entry 9 (vi) in Chapter 19 of Ramanujan’s second notebook,

( )

(

( )

₄

)

3 2 5 3 2 5 5 , , log 5 1 25 q q f q q f dq d q q q q q q ₌

_ψ

₊ ₋

ψ

. (107)

By the Jacobi triple product identity

f

( ) (

a,b = −a;ab

) (

_∞ −b;ab

) (

_∞ ab;ab

)

_∞, (108) we have

(

( )

)

(

) (

)

(

( ) (

) (

)

( )

q v q u q q q q q q q q q q q q q q q q q q q q q q q q q q q f q q f 1/5 10 8 10 2 5 3 5 2 10 6 10 4 5 4 5 5 4 5 5 3 5 2 4 3 2 ; ; ; ; ; ; ; ; ; ; ; ; , , = = − − − − = ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ _{, (109)}

by the following expression

( )

(

( ) (

) (

)

∞ ∞ ∞ ∞ = 1/5 ₂ 5₅ 4₃ 5₅ ; ; ; ; q q q q q q q q q q u . (110)

(21)

Using (109) in (107), we find that

∫

( )

=

∫

( )

+

∫

−

∫

(

q u v

)

dq= dq d dq q dq q q q q dq q q / log 8 5 8 40 5 8 3 5 1/5 5 5

ψ

∫

( )

∫

( )

+ − − + = − = k k k dq q q q v u dq q q q 1 1 log 5 24 log 5 8 40 / log 8 40 ψ ψ3 5 ψ ψ3 5 , (111)

where (104) has been employed. We note that we can rewrite the eq. (111) also as follows:

∫

( )

∫

( )

+ − − + = k k k dq q q q q dq q q 1 1 log 5 24 log 5 8 40 5 8 3 5 5 5

ψ

. (112)

Multiplying both sides for 64

5

, we obtain the following identical expression:

∫

( )

∫

( )

+ − ⋅ − + ⋅ = k k k dq q q q q dq q q 1 1 log 8 1 3 log 8 1 8 1 25 8 1 3 5 5 5

ψ

, (112b) or

∫

( )

∫

( )

      + − − + = k k k dq q q q q dq q q 1 1 log 3 log 25 8 1 8 1 3 5 5 5

ψ

. (112c)

In the Ramanujan’s notebook part IV in the Section “Integrals” are examined various results on integrals appearing in the 100 pages at the end of the second notebook, and in the 33 pages of the third notebook. Here, we have showed some integrals that can be related with some arguments above described.

∫

∞

( )

∑

(

)(

)

∫

(

)(

)

∞ = ∞ − + − − + − + = 0 1 0 4 4 2 2 4 4 2 2 2 2 4 1 4 4 1 4 8 1 2 k x k n a x a e xdx a k a e k a a dn n e _π _π π ψ ; (113)

(

)(

)

(

)

∫

∞

∑

∞ = + + + − = + − 0 1 2 2 4 4 2 2 1 4 4 1 4 1 4 k x k a a a a x a e xdx a _π

π

. (114)

Multiplying both sides for π2, we obtain the equivalent expression:

(

)(

)

(

)

∫

∞

∑

∞ = + + + − = + − ⋅ 0 1 2 2 2 3 2 4 4 2 2 2 1 4 4 4 1 4 k x k a a a a x a e xdx a

π

π . (114b) Let n≥0. Then

( )

(

)

( )

_{

₍

₎

_}

(

)

{

(

)

}

∫

∞

∑

∞ = + − + + + − − = + 0 0 1 2 2 / 1 2 cosh 1 2 1 2 cos 1 2 4 cos cosh 2 sin k n k k k k n k e x x x dx nx