on the study of five ramanujan mock theta functions. new possible mathematical connections with some...

1

On the study of five Ramanujan Mock Theta functions. New possible

mathematical connections with some cosmological and physical parameters.

Michele Nardelli1, Antonio Nardelli

2

Abstract

In this paper, we analyze five Ramanujan Mock Theta functions. We describe the new

possible mathematical connections with some cosmological and physical parameters

1 M.Nardelli studied at Dipartimento di Scienze della Terra Università degli Studi di Napoli Federico II,

Largo S. Marcellino, 10 - 80138 Napoli, 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 2 A. Nardelli studies at the Università degli Studi di Napoli Federico II - Dipartimento di Studi Umanistici –

Sezione Filosofia - scholar of Theoretical Philosophy

2

From:

THE MOCK THETA FUNCTIONS (2) - By G. N. WATSON. [Received 3

August, 1936.—Read 12 November, 1936]

Now, we have the following two mock theta functions:

For q = 2, we obtain:

q+q^3(1+q)+q^6(1+q)(1+q^2)+q^10(1+q)(1+q^2)(1+q^3)

Input

Plots (figures that can be related to the open strings)

3

Alternate form

Expanded form

Real roots

Complex roots

4

Polynomial discriminant

Derivative

Indefinite integral

Local minimum

Definite integral

6

Definite integral area below the axis between the smallest and largest real

roots

7

From the solution of the integral

we obtain, for q = 2 :

q^17/17 + q^16/16 + q^15/15 + q^14/7 + q^13/13 + q^12/12 + q^11/11 + q^10/10 +

q^9/9 + q^8/8 + q^7/7 + q^5/5 + q^4/4 + q^2/2

2^17/17 + 2^16/16 + 2^15/15 + 2^14/7 + 2^13/13 + 2^12/12 + 2^11/11 + 2^10/10 +

2^9/9 + 2^8/8 + 2^7/7 + 2^5/5 + 2^4/4 + 2^2/2

Input

Exact result

Decimal approximation

17710.8660098….

From:

we obtain:

8

1+((q^2)/(1-q))+((q^8)/((1-q)(1-q^3)))

Input


Alternate forms

9

Complex roots

Series expansion at q=0

Series expansion at q=∞

10

Derivative

Indefinite integral

Global minimum

Series representations

11

From the solution of the integral

we obtain, for q = 2:

q^5/5 + q^4/4 + q^3/3 + q^2/2 - 1/6 log(q^2 + q + 1) + 2 q + 1/(3 - 3 q) + 4/3 log(1 -

q) + (tan^(-1)((2 q + 1)/sqrt(3)))/(3 sqrt(3)) + 3/2

2^5/5 + 2^4/4 + 2^3/3 + 2^2/2 - 1/6 log(2^2 + 2 + 1) + 2*2 + 1/(3 – 3*2) + 4/3 log(1

- 2) + (tan^(-1)((2*2 + 1)/sqrt(3)))/(3 sqrt(3)) + 3/2

Input

12

Exact Result


Polar coordinates

Polar coordinates

20.578

13

Polar forms

Approximate form

Alternate forms

14

Alternative representations


15

Integral representations

16

Continued fraction representations

18

Dividing the two exact results of the above integrals, we obtain:

((2712472262/153153))*1/((607/30 + (4 i π)/3 + (tan^(-1)(5/sqrt(3)))/(3 sqrt(3)) -

log(7)/6))

Input

Exact Result


19

Polar coordinates

Polar coordinates

860.67

Polar forms

20

Approximate form

Alternate forms


21


23


24


26

Multiplying the two exact solutions, we obtain:

((2712472262/153153))*((607/30 + (4 i π)/3 + (tan^(-1)(5/sqrt(3)))/(3 sqrt(3)) -

log(7)/6))

Input

27

Exact Result


Polar coordinates

Polar coordinates

364454

28

Polar forms

Approximate form

Alternate forms

29

Expanded form


30


31


32


34

And from the difference and sum, we obtain:

((2712472262/153153))-((607/30 + (4 i π)/3 + (tan^(-1)(5/sqrt(3)))/(3 sqrt(3)) -

log(7)/6))

Input

35

Exact Result


Polar coordinates

17691

Polar forms

36

Approximate form

Alternate forms


37


38


39


40

((2712472262/153153))+((607/30 + (4 i π)/3 + (tan^(-1)(5/sqrt(3)))/(3 sqrt(3)) -

log(7)/6))

Input

41

Exact Result


Polar coordinates

17731

Polar coordinates

42

Polar forms

Approximate form

Alternate forms

43



44


45


47

From which:

1/2((((2712472262/153153))*((607/30+(4iπ)/3+(tan^(-1)(5/sqrt(3)))/(3sqrt(3)) -

log(7)/6))))+((27155710577/1531530+(4iπ)/3+(tan^(-1)(5/sqrt(3)))/(3sqrt(3)) -

log(7)/6))-(2207+322+123+29+7)-(76+18+4)

Input

Exact Result


48

Polar coordinates

196883

196884/196883 is a fundamental number of the following j-invariant

(In mathematics, Felix Klein's j-invariant or j function, regarded as a function of

a complex variable τ, is a modular function of weight zero for SL(2, Z) defined on

the upper half plane of complex numbers. Several remarkable properties of j have to

do with its q expansion (Fourier series expansion), written as a Laurent series in

terms of q = e2πiτ

(the square of the nome), which begins:

Note that j has a simple pole at the cusp, so its q-expansion has no terms below q−1

.

All the Fourier coefficients are integers, which results in several almost integers,

notably Ramanujan's constant:

The asymptotic formula for the coefficient of qn is given by

as can be proved by the Hardy–Littlewood circle method)

Furthermore, 196884 is the coefficient of q of the partition function Z1(q) that is the

number of quantum states of the minimal black hole for the value of k equal to 1.

https://en.wikipedia.org/wiki/Mathematics

https://en.wikipedia.org/wiki/Felix_Klein

https://en.wikipedia.org/wiki/Complex_analysis

https://en.wikipedia.org/wiki/Modular_function

https://en.wikipedia.org/wiki/Upper_half-plane

https://en.wikipedia.org/wiki/Complex_number

https://en.wikipedia.org/wiki/Q-expansion

https://en.wikipedia.org/wiki/Fourier_series

https://en.wikipedia.org/wiki/Laurent_series

https://en.wikipedia.org/wiki/Nome_(mathematics)

https://en.wikipedia.org/wiki/Simple_pole

https://en.wikipedia.org/wiki/Almost_integer

https://en.wikipedia.org/wiki/Ramanujan%27s_constant

https://en.wikipedia.org/wiki/Asymptotic_formula

https://en.wikipedia.org/wiki/Hardy%E2%80%93Littlewood_circle_method

49

Polar forms

50

Approximate form

Alternate forms

51

Expanded form


52


54


55


58

Furthermore, from the ratio of the two exact results of the previous integrals

860.67

we obtain also:

59

2(((2712472262/153153))*1/((607/30 + (4 i π)/3 + (tan^(-1)(5/sqrt(3)))/(3 sqrt(3)) -

log(7)/6)))+8

Input

Exact Result


Polar coordinates

1729.2

This result is very near to the mass of candidate glueball f0(1710) scalar meson.

Furthermore, 1728 occurs in the algebraic formula for the j-invariant of an elliptic

curve. (1728 = 82

* 33) The number 1728 is one less than the Hardy–Ramanujan

number 1729 (taxicab number)

Polar forms

https://en.wikipedia.org/wiki/J-invariant

https://en.wikipedia.org/wiki/Elliptic_curve



https://en.wikipedia.org/wiki/1729_(number)

61

Approximate form

Alternate forms

62


63


64


65


67

(1/27(2(((2712472262/153153))*1/((607/30 + (4 i π)/3 + (tan^(-1)(5/sqrt(3)))/(3

sqrt(3)) - log(7)/6)))+7))^2-1

Input

68

Exact Result


Polar coordinates

4096 = 642 where 4096 and 64 are fundamental values indicated in the Ramanujan

paper ―Modular equations and Approximations to π‖

69

Approximate form

70

Alternate forms

Expanded forms

71


72


74


77

(2(((2712472262/153153))*1/((607/30 + (4 i π)/3 + (tan^(-1)(5/sqrt(3)))/(3 sqrt(3)) -

log(7)/6)))+8)^1/15

Input

Exact Result


Polar coordinates

1.6438 ≈ ζ(2) = 𝜋2

6= 1.644934… (trace of the instanton shape)

78

Polar forms

79

Approximate form

Alternate forms

80

All 15th roots of 8 + 5424944524/(153153 (607/30 + (4 i π)/3 - log(7)/6 + (tan^(-

1)(5/sqrt(3)))/(3 sqrt(3))))

84


85


86


88


91

From the initial expression, we calculate the following integrals:

integrate(2^5/5 + 2^4/4 + 2^3/3 + 2^2/2 - 1/6 log(2^2 + 2 + 1) + 2*2 + 1/(3 – 3*2) +

4/3 log(1 - 2) + (tan^(-1)((2*2 + 1)/sqrt(3)))/(3 sqrt(3)) + 3/2)x

Indefinite integral

Plot of the integral (figure that can be related to an open string)

Alternate forms of the integral

92

From the result:

integrate(2/3 i π x^2 + (607 x^2)/60 - 1/12 x^2 log(7) + (x^2 tan^(-1)(5/sqrt(3)))/(6

sqrt(3)))x

Indefinite integral


93


Expanded form of the integral

From the result:

integrate(1/720 x^4 (1821 + 120 i π - 15 log(7) + 10 sqrt(3) tan^(-1)(5/sqrt(3))))x

Indefinite integral

94



From the result, for x = 1, we obtain:

(1/36 i π + (607)/1440 - 1/288 log(7) + (tan^(-1)(5/sqrt(3)))/(144 sqrt(3)))

Input

95

Exact Result


Polar coordinates

0.42871

Polar forms

96

Approximate form

Alternate forms


97



98


99

From the exact result

we obtain:

607/1440 + (i π)/36 + (tan^(-1)(5/sqrt(3)))/(x* sqrt(3)) - log(7)/288 =

0.419732040151544 + 0.08726646259971647i

Input interpretation

100

Result

Alternate form assuming x is real

Alternate forms

Real solution

Complex solution

144 (Fibonacci number)

101

Thence, we obtain:

1/((((sqrt(3))*((0.41973204015 + 0.08726646259i-(607/1440 + (i π)/36)+

(log(7)/288))) *1/((tan^(-1)(5/sqrt(3)))))))


Result

Polar coordinates

144

102


103


105


107


110

From which, we obtain:

12*1/((((sqrt(3))*((0.41973204015 + 0.08726646259i-(607/1440 + (i π)/36)+

(log(7)/288))) *1/((tan^(-1)(5/sqrt(3)))))))+1


Result

Polar coordinates

1729



curve. (1728 = 82



Polar forms






111


112


115


116


119

(12*1/((((sqrt(3))*((0.41973204015 + 0.08726646259i-(607/1440 + (i π)/36)+

(log(7)/288))) *1/((tan^(-1)(5/sqrt(3)))))))+1)^1/15


Result

Polar coordinates

1.64382 ≈ ζ(2) = 𝜋2


120

From:

S. Ramanujan to G.H. Hardy - 12 January 1920 - University of Madras

Now we have the following three mock theta functions:

We analyze these functions and consider the following data:

; t = 0.25 ; q = 0.7788

From:

we obtain:

1+(q/(1+q^2))+((q^4)/((1+q)(1+q^2)))+((q^9)/((1+q)(1+q^2)(1+q^3)))

Input

121


Alternate forms

Real root

122

Complex roots



Derivative

123

Indefinite integral

Local maxima

Local minimum

From the derivative of

124

we obtain:

derivative((3 q^13 + q^12 + 5 q^11 + 9 q^10 + 12 q^8 + q^7 + 5 q^6 + q^5 + q^4 + 5

q^3 - q^2 + (1 + q))/((1 + q)^3 (1 + q^2)^2 ((1 + q^2) - q)^2))

Derivative


Alternate forms

125

Expanded form

Real roots

126

Complex roots



Indefinite integral

127

Local maxima

Local minimum

From the result of the above alternate form :

For q = 0.7788 , we obtain:

(2*0.7788+3)/(0.7788^2+1)^2-(4(2*0.7788+1))/(0.7788^2+1)^3-

(2*0.7788)/((0.7788-1) 0.7788+1)^3+6*0.7788+(2(0.7788+1))/(3((0.7788-1)

0.7788+1)^2)+10/(3 (0.7788+1)^3)-1/(0.7788+1)^4 – 2

128

Input

Result

1.4479789895….

From which:

((2*0.7788+3)/(0.7788^2+1)^2-(4(2*0.7788+1))/(0.7788^2+1)^3-

(2*0.7788)/((0.7788-1) 0.7788+1)^3+6*0.7788+(2(0.7788+1))/(3((0.7788-1)

0.7788+1)^2)+10/(3 (0.7788+1)^3)-1/(0.7788+1)^4 - 2)^23-(843+47-3)

Input

Result

4096.47156853429….. ≈ 4096 = 642, where 4096 and 64 are fundamental values

indicated in the Ramanujan paper ―Modular equations and Approximations to π‖

129

27√(((2*0.7788+3)/(0.7788^2+1)^2-(4(2*0.7788+1))/(0.7788^2+1)^3-

(2*0.7788)/((0.7788-1)0.7788+1)^3+6*0.7788+(2(0.7788+1))/(3((0.7788-

1)0.7788+1)^2)+10/(3(0.7788+1)^3)-1/(0.7788+1)^4-2)^23-(843+47-3))+1

Input

Result

1729.10….



curve. (1728 = 82



(27√(((2*0.7788+3)/(0.7788^2+1)^2-(4(2*0.7788+1))/(0.7788^2+1)^3-

(2*0.7788)/((0.7788-1)0.7788+1)^3+6*0.7788+(2(1.7788))/(3((0.7788-

1)0.7788+1)^2)+10/(3(1.7788)^3)-1/(1.7788)^4-2)^23-(843+44))+1)^1/15







130

Result

1.643821533….≈ ζ(2) = 𝜋2


Now, from the solution of the above integral:

we obtain:

1/36 (3 (3 q^4 - 4 q^3 + 6 q^2 + 6 log(q^2 + 1) + 2 log(q^2 - q + 1) - 12 q + 2/(q + 1)

+ 20 log(q + 1)) + 18 tan^(-1)(q) + 4 sqrt(3) tan^(-1)((2 q - 1)/sqrt(3)))

for q = 0.7788 :

1/36 (3 (3 0.7788^4 - 4 0.7788^3 + 6 0.7788^2 + 6 log(0.7788^2 + 1) + 2

log(0.7788^2 – 0.7788 + 1) - 12 0.7788 + 2/(0.7788 + 1) + 20 log(0.7788 + 1)) + 18

tan^(-1)(0.7788) + 4 sqrt(3) tan^(-1)((2 0.7788 - 1)/sqrt(3)))

131

Input

Result

1.108879289166….


132

1/((1/36(3(3 0.7788^4-4 0.7788^3+6 0.7788^2+6log(0.7788^2+1)+2log(0.7788^2–

0.7788+1)-12 0.7788+2/(0.7788+1)+20log(0.7788+1))+18tan^(-

1)(0.7788)+4sqrt(3)tan^(-1)((2 0.7788-1)/sqrt(3))))-0.5)

133

Input

Result

1.6423616598…..≈ ζ(2) = 𝜋2



135


137


139


141

Now, we analyze the second mock theta function:

1+q(1+q)+q^4(1+q)(1+q^3)+q^9(1+q)(1+q^3)(1+q^5)

Input


142

Alternate forms

Complex roots


143

Derivative

Indefinite integral

Local minimum

From:

Perform the derivative, we obtain:

derivative( 18 q^17 + 17 q^16 + 15 q^14 + 14 q^13 + 13 q^12 + 12 q^11 + 10 q^9 +

9 q^8 + 8 q^7 + 7 q^6 + 5 q^4 + 4 q^3 + 2 q + 1)

144

Derivative


Alternate form

145

Expanded form

Complex roots


Indefinite integral

146

Local maximum

Local minima

From:

2 (153 q^16 + 136 q^15 + 105 q^13 + 91 q^12 + 78 q^11 + 66 q^10 + 45 q^8 + 36

q^7 + 28 q^6 + 21 q^5 + 10 q^3 + 6 q^2 + 1)


2 (153 0.7788^16 + 136 0.7788^15 + 105 0.7788^13 + 91 0.7788^12 + 78 0.7788^11

+ 66 0.7788^10 + 45 0.7788^8 + 36 0.7788^7 + 28 0.7788^6 + 21 0.7788^5 + 10

0.7788^3 + 6 0.7788^2 + 1)

147

Input

Result

117.9583107512….

From which:

16(2 (153 0.7788^16 + 136 0.7788^15 + 105 0.7788^13 + 91 0.7788^12 + 78

0.7788^11 + 66 0.7788^10 + 45 0.7788^8 + 36 0.7788^7 + 28 0.7788^6 + 21

0.7788^5 + 10 0.7788^3 + 6 0.7788^2 + 1)-Pi^2)-√2

Input

Result

1728.01….



curve. (1728 = 82








148


(1/27(16(2 (153 0.7788^16 + 136 0.7788^15 + 105 0.7788^13 + 91 0.7788^12 + 78

0.7788^11 + 66 0.7788^10 + 45 0.7788^8+36 0.7788^7+28 0.7788^6+21

0.7788^5+10 0.7788^3+6 0.7788^2+1)-Pi^2)-√2))^2

149

Input

Result

4096.02….≈ 4096 = 642 where 4096 and 64 are fundamental values indicated in the

Ramanujan paper ―Modular equations and Approximations to π‖

((16(2 (153 0.7788^16+136 0.7788^15+105 0.7788^13+91 0.7788^12+78

0.7788^11+66 0.7788^10 + 45 0.7788^8 + 36 0.7788^7 + 28 0.7788^6 + 21 0.7788^5

+ 10 0.7788^3 + 6 0.7788^2 + 1)-Pi^2)-√2)+1)^1/15

Input

150

Result

1.64382….≈ ζ(2) = 𝜋2


Now, from the solution of the above integral:

q^19/19 + q^18/18 + q^16/16 + q^15/15 + q^14/14 + q^13/13 + q^11/11 + q^10/10 +

q^9/9 + q^8/8 + q^6/6 + q^5/5 + q^3/3 + q^2/2 + q


(0.7788^19/19+0.7788^18/18+0.7788^16/16+0.7788^15/15+0.7788^14/14+0.7788^1

3/13+0.7788^11/11+0.7788^10/10+0.7788^9/9+0.7788^8/8+0.7788^6/6+0.7788^5/5

+0.7788^3/3+0.7788^2/2+0.7788)

Input

Result

1.3855807917….

151

1+1/(3(1/e(0.7788^19/19+0.7788^18/18+0.7788^16/16+0.7788^15/15+0.7788^14/14

+0.7788^13/13+0.7788^11/11+0.7788^10/10+0.7788^9/9+0.7788^8/8+0.7788^6/6+

0.7788^5/5+0.7788^3/3+0.7788^2/2+0.7788)))

Input

Result

1.65394522514…. result very near to the 14th root of the following Ramanujan’s

class invariant 𝑄 = 𝐺505/𝐺101/5 3 = 1164.2696 i.e. 1.65578...

Alternative representation

152


From the sum of the two alternate form of the first two mock, i.e.

and

153

we obtain:

((1 + (q (1 + q + 2 q^3 + q^4 + q^6 + q^8))/(1 + q + q^2 + 2 q^3 + q^4 + q^5 +

q^6)))+((1 + q (1 + q) (1 + q^3 + q^6 + q^8 + q^11 + q^13 + q^16)))

Input

Result


154

Alternate forms

Expanded form

155

Derivative

From the expression


q (q + 1) (q^16 + q^13 + q^11 + q^8 + q^6 + q^3 + 1) + (q (q^8 + q^6 + q^4 + 2 q^3

+ q + 1))/(q^6 + q^5 + q^4 + 2 q^3 + q^2 + q + 1) + 2

0.7788 (0.7788+1)

(0.7788^16+0.7788^13+0.7788^11+0.7788^8+0.7788^6+0.7788^3+1)+(0.7788(0.77

88^8+0.7788^6+0.7788^4+2

0.7788^3+0.7788+1))/(0.7788^6+0.7788^5+0.7788^4+2

0.7788^3+0.7788^2+0.7788+1)+2

Input

156

Result

5.3425012228441….

11((0.7788(1.7788)(0.7788^16+0.7788^13+0.7788^11+0.7788^8+0.7788^6+1.47236

)+(0.7788(0.7788^8+0.7788^6+0.7788^4+2

0.47236+1.7788)/(0.7788^6+0.7788^5+0.7788^4+2

0.47236+0.7788^2+1.7788)+2))^3)+47+2+φ^2


Result

1728.97….



curve. (1728 = 82








157


158

(11((0.7788(1.7788)(0.7788^16+0.7788^13+0.7788^11+0.7788^8+0.7788^6+1.4723

6)+(0.7788(0.7788^8+0.7788^6+0.7788^4+2.72352)/(0.7788^6+0.7788^5+0.7788^4

+0.94472+0.7788^2+1.7788)+2))^3)+7^2+φ^2)^1/15


Result

1.64381346388….≈ ζ(2) = 𝜋2


159

Now, we analyze the third mock theta function:

We obtain:

1+[q/(1-q)+(q^3)/((1-q^2)(1-q^3))+(q^5)/((1-q^3)(1-q^4)(1-q^5))]

Input

Result


160

Alternate forms

Real root

Complex roots

161



Derivative

Indefinite integral

162

Limit

From

for q = 0.7788 :

1 + 0.7788/(1 - 0.7788) + 0.7788^3/((1 - 0.7788^2) (1 - 0.7788^3)) + 0.7788^5/((1 -

0.7788^3) (1 - 0.7788^4) (1 - 0.7788^5))

Input

Result

7.999997103998…. ≈ 8

From which:

(1 + 0.7788/(1 - 0.7788) + 0.7788^3/((1 - 0.7788^2) (1 - 0.7788^3)) + 0.7788^5/((1 -

0.7788^3) (1 - 0.7788^4) (1 - 0.7788^5)))^4

163

Input

Result

4095.99406899….≈ 4096 = 642 where 4096 and 64 are fundamental values indicated

in the Ramanujan paper ―Modular equations and Approximations to π‖

27sqrt((1 + 0.7788/(1 - 0.7788) + 0.7788^3/((1 - 0.7788^2) (1 - 0.7788^3)) +

0.7788^5/((1 - 0.7788^3) (1 - 0.7788^4) (1 - 0.7788^5)))^4)+1

Input

164

Result

1728.9987489…..≈ 1729



curve. (1728 = 82



(27sqrt((1 + 0.7788/(1 - 0.7788) + 0.7788^3/((1 - 0.7788^2) (1 - 0.7788^3)) +

0.7788^5/((1 - 0.7788^3) (1 - 0.7788^4) (1 - 0.7788^5)))^4)+1)^1/15

Input

Result

1.643815149453…..≈ ζ(2) = 𝜋2


We have also:

((1 + 0.7788/(1 - 0.7788) + 0.7788^3/((1 - 0.7788^2) (1 - 0.7788^3)) + 0.7788^5/((1 -

0.7788^3) (1 - 0.7788^4) (1 - 0.7788^5))))^136*(tan^2(17/(5^2*3)))

where






165

Input

Result

0.351600…*10

122 ≈ ΛQ

The observed value of ρΛ or Λ today is precisely the classical dual of its quantum

precursor values ρQ , ΛQ in the quantum very early precursor vacuum UQ as

determined by our dual equations


166


167


168

Multiple-argument formulas

From the derivative of

Performing:

second derivative of (1 + (q/(1 - q) + q^3/((1 - q^2) (1 - q^3)) + q^5/((1 - q^3) (1 -

q^4) (1 - q^5))))

169

we obtain:

Derivative


170

Alternate form

Partial fraction expansion

Real root


171


Indefinite integral

172

From the alternate form


-(2 (0.7788^29 + 4*0.7788^28 + 16*0.7788^27 + 41*0.7788^26 + 80*0.7788^25 +

113*0.778^24 + 133*0.7788^23 + 88*0.7788^22 – 44*0.7788^21 – 307*0.7788^20 -

655 q^19 - 988 q^18 - 1078 q^17 - 781 q^16 + 14 q^15 + 1149 q^14 + 2379 q^13 +

3327 q^12 + 3841 q^11 + 3817 q^10 + 3385 q^9 + 2691 q^8 + 1932 q^7 + 1233 q^6

+ 695 q^5 + 332 q^4 + 131 q^3 + 40 q^2 + 10 q + 1))

Input

Result

-(2 (-1.004035093647- 655 0.7788^19 - 988 0.7788^18 - 1078 0.7788^17 - 781

0.7788^16 + 14 0.7788^15 + 1149 0.7788^14 + 2379 0.7788^13 + 3327 0.7788^12 +

3841 0.7788^11 + 3817 0.7788^10 + 3385 q^9 + 2691 q^8 + 1932 q^7 + 1233 q^6 +

695 q^5 + 332 q^4 + 131 q^3 + 40 q^2 + 10 q + 1))

173

Input

Result

-(2 (-1.004035093647- 816.77764665+3385 0.7788^9 + 2691 0.7788^8 + 1932

0.7788^7 + 1233 0.7788^6 + 695 0.7788^5 + 332 0.7788^4 + 131 0.7788^3 + 40

0.7788^2 + 10 0.7788 + 1))

/((0.7788 - 1)^5 (0.7788 + 1)^3 (0.7788^2 + 1)^3 (0.7788^2 + 0.7788 + 1)^3

(0.7788^4 + 0.7788^3 + 0.7788^2 + 0.7788 + 1)^3)

Input

Result

Thence, in conclusion:

-(2 (-1.004035093647- 816.77764665+3385 0.7788^9 + 2691 0.7788^8 + 1932

0.7788^7 + 1233 0.7788^6 + 695 0.7788^5 + 332 0.7788^4 + 131 0.7788^3 + 40

0.7788^2 + 10 0.7788 + 1))/(-5.629095041)

174


Result

330.4996005811….

From which:

2e(-(2 (-1.004035093647- 816.77764665+3385 0.7788^9 + 2691 0.7788^8 + 1932

0.7788^7 + 1233 0.7788^6 + 695 0.7788^5 + 332 0.7788^4 + 131 0.7788^3 + 40

0.7788^2 + 10 0.7788 + 1))/(-5.629095041))-64-2-e


Result

1728.06….



curve. (1728 = 82








175

Alternative representation


176

Performing the 15th root of 1729.063853…, we obtain:

(2e(-(2 (-1.004035093647- 816.77764665+3385 0.7788^9 + 2691 0.7788^8 + 1932

0.7788^7+1233 0.7788^6+695 0.7788^5+332 0.7788^4+131 0.7788^3+40

0.7788^2+10 0.7788+1))/(-5.629095041))-64-2-e+1)^1/15


Result

1.6438192746….≈ ζ(2) = 𝜋2


(1/27(2e(-(2 (-1.004035093647- 816.77764665+3385 0.7788^9 + 2691 0.7788^8 +

1932 0.7788^7+1233 0.7788^6+695 0.7788^5+332 0.7788^4+131 0.7788^3+40

0.7788^2+10 0.7788+1))/(-5.629095041))-64-2-e))^2

177


Result

4096.3….≈ 4096 = 642 where 4096 and 64 are fundamental values indicated in the

Ramanujan paper ―Modular equations and Approximations to π‖

From the above integral

178

(-144 sqrt(5) log(-2 q^2 + (sqrt(5) - 1) q - 2) + 450 log(q^2 + 1) - 400 log(q^2 + q +

1) + 144 sqrt(5) log(2 q^2 + sqrt(5) q + q + 2) - 570/(q - 1) + 30/(q - 1)^2 - 3475

log(1 - q) + 900 log(q - 1) - 1125 log(q + 1) + 800 sqrt(3) tan^(-1)((2 q +

1)/sqrt(3)))/3600

-144 sqrt(5) log(-2 0.7788^2 + (sqrt(5) - 1) 0.7788 - 2) + 450 log(0.7788^2 + 1) - 400

log(0.7788^2 + 0.7788 + 1) + 144 sqrt(5) log(2 0.7788^2 + sqrt(5) 0.7788 + 0.7788 +

2)

Input

Result

Polar coordinates

1025.22

(((166.722 -1011.57 i) - 570/(0.7788 - 1) + 30/(0.7788 - 1)^2 - 3475 log(1 – 0.7788)

+ 900 log(0.7788 - 1) - 1125 log(0.7788 + 1) + 800 sqrt(3) tan^(-1)((2 0.7788 +

1)/sqrt(3))))/3600


179

Result

Polar coordinates

2.26395

Polar forms


180


182


185

From which:

((((166.722 -1011.57 i) - 570/(0.7788 - 1) + 30/(0.7788 - 1)^2 - 3475 log(1 – 0.7788)

+ 900 log(0.7788 - 1) - 1125 log(0.7788 + 1) + 800 sqrt(3) tan^(-1)((2 0.7788 +

1)/sqrt(3))))/3600 )-64/10^2


Result

Polar coordinates

1.64623 ≈ ζ(2) = 𝜋2


Polar forms

186


187


188


191

From the sum of the first two mock 5.3425012228441…. subtracting the result of

the third mock 7.999997103998…. ≈ 8 R3 , we obtain:

(7.999997103998 - 5.3425012228441)-1.0018674362

where


Result

1.6556284449539 result that is very near to the 14th root of the following

Ramanujan’s class invariant 𝑄 = 𝐺505/𝐺101/5 3 = 1164.2696 i.e. 1.65578...

Indeed, from:

113+5 505

8+ 105+5 505

8

314

= 1,65578…

192

And:

(7.999997103998 - 5.3425012228441)-(1/0.9568666373)

where


Result

1.6124181649…. result that is a very good approximation to the value of the golden

ratio 1.618033988749...

And again:

1/2(((7.999997103998 - 5.3425012228441) -(((24*8-((8*2)-4)) π)/(24^2+1)))+

((7.999997103998 - 5.3425012228441)-(1/0.9568666373)))

where

193


Result

1.6449339….≈ ζ(2) = 𝜋2


Possible closed forms


194


195


(5.3425012228441 + 7.999997103998)^108 * (Catalan + 2 - π + π log(3/2))


Result

0.35159968099…*10

122 ≈ ΛQ

The observed value of ρΛ or Λ today is precisely the classical dual of its quantum

precursor values ρQ , ΛQ in the quantum very early precursor vacuum UQ as

determined by our dual equations.

196

Fundamental are the following values: Λ = 2.846 * 10-122

that is the actual value of

the Cosmological Constant that is precisely, the classical dual of its quantum

precursor value ΛQ = 0.3516 * 10122

in the quantum very early precursor vacuum.

(New Quantum Structure of the Space-Time - Norma G. SANCHEZ - arXiv:1910.13382v1

[physics.gen-ph] 28 Oct 2019)



197


199

References

THE MOCK THETA FUNCTIONS (2) - By G. N. WATSON. [Received 3

August, 1936.—Read 12 November, 1936]

S. Ramanujan to G.H. Hardy - 12 January 1920 - University of Madras

1

Mathematical connections with some sectors of String Theory

Observations

We note that, from the number 8, we obtain as follows:

We notice how from the numbers 8 and 2 we get 64, 1024, 4096 and 8192, and that 8

is the fundamental number. In fact 82 = 64, 8

3 = 512, 8

4 = 4096. We define it

"fundamental number", since 8 is a Fibonacci number, which by rule, divided by the

previous one, which is 5, gives 1.6 , a value that tends to the golden ratio, as for all

numbers in the Fibonacci sequence

2

“Golden” Range

Finally we note how 82 = 64, multiplied by 27, to which we add 1, is equal to 1729,

the so-called "Hardy-Ramanujan number". Then taking the 15th root of 1729, we

obtain a value close to ζ(2) that 1.6438 ..., which, in turn, is included in the range of

what we call "golden numbers"

Furthermore for all the results very near to 1728 or 1729, adding 64 = 82, one obtain

values about equal to 1792 or 1793. These are values almost equal to the Planck

multipole spectrum frequency 1792.35 and to the hypothetical Gluino mass

We have that:

3

From: A. Sagnotti – AstronomiAmo, 23.04.2020

In the above figure, it is said that: “why a given shape of the extra dimensions?

Crucial, it determines the predictions for α”.

We propose that whatever shape the compactified dimensions are, their geometry

must be based on the values of the golden ratio and ζ(2), (the latter connected to 1728

or 1729, whose fifteenth root provides an excellent approximation to the above

mentioned value) which are recurrent as solutions of the equations that we are going

to develop. It is important to specify that the initial conditions are always values

belonging to a fundamental chapter of the work of S. Ramanujan "Modular equations

and Appoximations to Pi" (see references). These values are some multiples of 8 (64

and 4096), 276, which added to 4096, is equal to 4372, and finally eπ√22

4

We obtain, in certain cases, the following connections:

Fig. 1

Fig. 2

5

Fig. 3

Stringscape - a small part of the string-theory landscape showing the new de Sitter solution as a local

minimum of the energy (vertical axis). The global minimum occurs at the infinite size of the extra

dimensions on the extreme right of the figure.

Fig. 4

6

With regard the Fig. 4 the points of arrival and departure on the right-hand side of the

picture are equally spaced and given by the following equation:

we obtain:

2Pi/(ln(2))

Input:

Exact result:

Decimal approximation:

9.06472028365….

Alternative representations:

7

Series representations:

Integral representations:

8

From which:

(2Pi/(ln(2)))*(1/12 π log(2))

Input:

Exact result:


1.6449340668…. = ζ(2) = 𝜋2

6= 1.644934…

9

The Einstein’s field equation and the theory of string.

The Einstein‟s field equation which includes the cosmological constant is:

GTgRgR 8

2

1

(8)

where R is the Ricci tensor, R its trace, the cosmological constant, g

the

metric tensor of the space geometry, G the Newton‟s gravitational constant and T

the tensor representing the properties of energy, matter and momentum.

The left hand-side of (8) represents the gravitational field and, consequently, the

warped space-time, while the right hand-side represents the matter, i.e. the sources of

the gravitational field.

In string theory the gravity is related to the gravitons which are bosons, whereas the

matter is related to fermions. It follows that the left and right hand of (8) may be

respectively related to the action of bosonic and of superstrings.

From (4) that describes the parallelism between the Palumbo‟s model and the theory

of string, we may thus write:

gfGGTrgg

G

Rgxd

2

1

8

1

1626

0

2

2210

210

2

3210

210

~

2

14

2

1FTr

gHReGxd

(9)

10

The sign minus in the above equation comes from the inversion of any relationship,

like the newtonian one, when one examines it outside the range of its validity.

Let us analyze p. e. the orbits of the gravitational equation F = G x m1 x m2/ r2

, for m1

= m2 = m: i.e. F = G m2/r

2.

for r2 > G m

2 F(r) => 0, the orbits are attracted by zero,

for r2 = G m

2 F(r) = 1 are constant and equal to 1,

for r2 < G m

2 F(r) => the orbits are attracted by infinite.

The point r2 = k m

2 is a critical point since a small variation of r implies that the orbits

may tend to zero or to infinite.

Moreover, from F =G x m2/r

2, for ΔF and Δr extremely small, such as inside a black

hole or a proton, or, in the case of (9) that represents the perturbation of the quantum

dominium of strings, ΔF/F = 2Δm/m - 2Δr/r, and assuming Δm = 0 one obtains:

ΔF/F = - 2Δr/r (10)

where the sign minus indicates that F decreases when r increases, implying that 0 < ΔF/F

< 1. Let us examine this relationship outside the above range and indicate F1 at the

distance r, and F2 at the distance r + Δr .

- ΔF/F > 1 => ΔF > F => (F1 – F2) > F1 => F2 < 0 indicating that F becomes repulsive

at the distance r + Δr.

- ΔF/F < 0, since F > 0, => ΔF < 0 => (F1 – F2) < 0 => F1 < F2 indicating that F

decreases when r increases, in other words that the attraction increases with the

distance between two masses.

The same holds for Δr, whose analysis indicates that when Δr > r, F becomes repulsive

and increases with the distance between the two masses.

The sign minus that appears in (4) is thus consistent with the (i) observed repulsive

forces between quark inside a proton and the corresponding strings, (ii) repulsive force

of strings inside a black hole, and (iii) relationship (9) which relates the repulsive actions

of bosonic and supersymmetric strings in their extremely narrow dominium.

11

Now, we note that the number 8, and thence the numbers 2864 and 8232 2 , are

connected with the “modes” that correspond to the physical vibrations of a

superstring by the following Ramanujan function:

4

2710

4

21110log

'

142

'

cosh

'cos

log4

3

18

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

. (11)

Furthermore, with regard the number 24 (12 = 24 / 2 and 32 = 24 + 8) they are

related to the “modes” that correspond to the physical vibrations of the bosonic

strings by the following Ramanujan function:

4

2710

4

21110log

'

142

'

cosh

'cos

log4

24

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

. (12)

Palumbo (2001) ha proposed a simple model of the birth and of the evolution of the

Universe. Palumbo and Nardelli (2005) have compared this model with the theory of

the strings, and translated it in terms of the latter obtaining:

12

, (13)

A general relationship that links bosonic and fermionic strings acting in all natural

systems.

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 =

0,618033 = (Peitgen et al. 1986). Such a factor appears also in the famous

fractal Ramanujan identity (Hardy 1927):

, (14)

and , (15)

gfGGTrgg

G

Rgxd

2

1

8

1

1626

0

2

2210

210

2

322/110

210

~

2

14

2

1FTr

gHReGxd

/1

2

15

q

t

dt

tf

tfqR

0 5/45/1

5

)(

)(

5

1exp

2

531

5)(

2

15/1618033,0

q

t

dt

tf

tfqR

0 5/45/1

5

)(

)(

5

1exp

2

531

5)(

20

32

13

where .

Furthermore, we remember that arises also from the following identities

(Ramanujan‟s paper: “Modular equations and approximations to π” Quarterly Journal

of Mathematics, 45 (1914), 350-372.):

, (16)

and

. (17)

From (17), we have that

4

2710

4

21110log

14224

. (18)

2

15

2

13352log

130

12

4

2710

4

21110log

142

24

14

But is equal also to

π =

'

4

'

cosh

'cos

log2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

(19)

Thence:

4

2710

4

21110log

'

142

'

cosh

'cos

log4

24

2

'

'4

0

'

2

2

wtitwe

dxex

txw

anti

w

wt

wx

. (20)

Furthermore, we have the following equation:

q

t

dt

tf

tfqR

0 5/45/1

5

5

1exp

2

531

5

20

32

4

2710

4

21110log

142

24

(21)

from which we can to obtain both 24 and .

The introduction of (14) and (15) in (13) provides:

15

=

, (22)

which is the translation of (13) in the terms of the Theory of the Numbers,

specifically the possible connection between the Ramanujan identity and the

relationship concerning the Palumbo-Nardelli model.

In the work of Ramanujan, [i.e. the modular functions,] the number 24 (8 x 3) appears

repeatedly. This is an example of what mathematicians call magic numbers, which

continually appear where we least expect them, for reasons that no one understands.

Ramanujan„s function also appears in string theory. Modular functions are used in the

mathematical analysis of Riemann surfaces. Riemann surface theory is relevant to

describing the behavior of strings as they move through space-time. When strings

fGGTrgg

t

dt

tf

tfqR

G

Rgxd

q

8

1

)(

5

1exp

2

531

5)(

20

32

1

16

0 5/45/1

5

26

g

2

1

q

t

dt

tf

tfqR

R

0 5/45/1

50 211

)(

)(

5

1exp

2

531

5)(

20

32

Tr

Rg

t

dt

tf

tfqR

HReGxd

q

210

0 5/45/1

5

211

2

322/110

2

)(

)(

5

1exp

2

531

5)(

20

32

~

2

14

2

2F

http://en.wikipedia.org/wiki/Riemann_surface

16

move they maintain a kind of symmetry called "conformal invariance". Conformal

invariance (including "scale invariance") is related to the fact that points on the

surface of a string's world sheet need not be evaluated in a particular order. As long

as all points on the surface are taken into account in any consistent way, the physics

should not change. Equations of how strings must behave when moving involve the

Ramanujan function. When a string moves in space-time by splitting and

recombining a large number of mathematical identities must be satisfied. These are

the identities of Ramanujan's modular function. The KSV loop diagrams of

interacting strings can be described using modular functions. The "Ramanujan

function" (an elliptic modular function that satisfies the need for "conformal

symmetry") has 24 "modes" that correspond to the physical vibrations of a bosonic

string. When the Ramanujan function is generalized, 24 is replaced by 8 (8 + 2 = 10)

for fermionic strings.

http://en.wikipedia.org/wiki/Conformal_field_theory

http://en.wikipedia.org/wiki/Conformal_symmetry



http://en.wikipedia.org/wiki/scale_invariance

http://en.wikipedia.org/wiki/World_sheet

http://en.wikipedia.org/wiki/Superstring_theory

http://en.wikipedia.org/wiki/Ramanujan

http://en.wikipedia.org/wiki/Modular_form

http://en.wikipedia.org/wiki/Boson

http://en.wikipedia.org/wiki/Fermion

17

From:

Modular equations and approximations to 𝝅 - Srinivasa Ramanujan

Quarterly Journal of Mathematics, XLV, 1914, 350 – 372

We have that:

18

We note that, with regard 4372, we can to obtain the following results:

27((4372)^1/2-2-1/2(((√(10-2√5) -2))⁄((√5-1))))+φ

Input

Result


1729.0526944….



curve. (1728 = 82



Alternate forms






19

Minimal polynomial

Expanded forms


20

Or:

27((4096+276)^1/2-2-1/2(((√(10-2√5) -2))⁄((√5-1))))+φ

21

Input

Result


1729.0526944…. as above

Alternate forms

22

Minimal polynomial

Expanded forms


24

From which:

(27((4372)^1/2-2-1/2(((√(10-2√5) -2))⁄((√5-1))))+φ)^1/15

Input

Exact result


1.64381856858…. ≈ ζ(2) = 𝜋2

6= 1.644934…

Alternate forms

25

Minimal polynomial

Expanded forms

All 15th roots of ϕ + 27 (-2 + 2 sqrt(1093) - (sqrt(10 - 2 sqrt(5)) - 2)/(2 (sqrt(5) -

1)))

26


27

Integral representation

28

From:

An Update on Brane Supersymmetry Breaking

J. Mourad and A. Sagnotti - arXiv:1711.11494v1 [hep-th] 30 Nov 2017

From the following vacuum equations:

we have obtained, from the results almost equals of the equations, putting

instead of

a new possible mathematical connection between the two exponentials. Thence, also

the values concerning p, C, βE and 𝜙 correspond to the exponents of e (i.e. of exp).

Thence we obtain for p = 5 and βE = 1/2:

𝑒−6𝐶+𝜙 = 4096𝑒−𝜋 18

Therefore, with respect to the exponentials of the vacuum equations, the Ramanujan‟s

exponential has a coefficient of 4096 which is equal to 642, while -6C+𝜙 is equal to -

𝜋 18. From this it follows that it is possible to establish mathematically, the dilaton

value.

29

For

exp((-Pi*sqrt(18)) we obtain:

Input:

Exact result:


1.6272016… * 10-6

Property:


Now, we have the following calculations:

30

𝑒−6𝐶+𝜙 = 4096𝑒−𝜋 18

𝑒−𝜋 18 = 1.6272016… * 10^-6

from which:

1

4096𝑒−6𝐶+𝜙 = 1.6272016… * 10^-6

0.000244140625 𝑒−6𝐶+𝜙 = 𝑒−𝜋 18 = 1.6272016… * 10^-6

Now:

ln 𝑒−𝜋 18 = −13.328648814475 = −𝜋 18

And:

(1.6272016* 10^-6) *1/ (0.000244140625)

Input interpretation:

Result:

0.006665017...

31

Thence:

0.000244140625 𝑒−6𝐶+𝜙 = 𝑒−𝜋 18

Dividing both sides by 0.000244140625, we obtain:

0.000244140625

0.000244140625𝑒−6𝐶+𝜙 =

1

0.000244140625𝑒−𝜋 18

𝑒−6𝐶+𝜙 = 0.0066650177536

((((exp((-Pi*sqrt(18)))))))*1/0.000244140625


Result:

0.00666501785…


32

Now:

𝑒−6𝐶+𝜙 = 0.0066650177536

=

= 0.00666501785…

From:

ln(0.00666501784619)


Result:

-5.010882647757…

33

Alternative representations:


Integral representation:

In conclusion:

−6𝐶 + 𝜙 = −5.010882647757…

and for C = 1, we obtain:

34

𝜙 = −5.010882647757 + 6 = 𝟎.𝟗𝟖𝟗𝟏𝟏𝟕𝟑𝟓𝟐𝟐𝟒𝟑 = 𝝓

Note that the values of ns (spectral index) 0.965, of the average of the Omega mesons

Regge slope 0.987428571 and of the dilaton 0.989117352243, are also connected to

the following two Rogers-Ramanujan continued fractions:

(http://www.bitman.name/math/article/102/109/)

The mean between the two results of the above Rogers-Ramanujan continued

fractions is 0.97798855285, value very near to the ψ Regge slope 0.979:

http://www.bitman.name/math/article/102/109/

35

Also performing the 512th root of the inverse value of the Pion meson rest mass

139.57, we obtain:

((1/(139.57)))^1/512


Result:

0.99040073.... result very near to the dilaton value 𝟎.𝟗𝟖𝟗𝟏𝟏𝟕𝟑𝟓𝟐𝟐𝟒𝟑 = 𝝓 and to

the value of the following Rogers-Ramanujan continued fraction:

36

From

AdS Vacua from Dilaton Tadpoles and Form Fluxes - J. Mourad and A. Sagnotti

- arXiv:1612.08566v2 [hep-th] 22 Feb 2017 - March 27, 2018

We have:

For

ξ = 1

we obtain:

(2*e^(0.989117352243/2)) / (1+sqrt(((1-1/3*16/(Pi)^2*e^(2*0.989117352243)))))


Result:

37

Polar coordinates:

1.65919106525….. result very near to the 14th root of the following Ramanujan‟s

class invariant 𝑄 = 𝐺505 /𝐺101/5 3 = 1164.2696 i.e. 1.65578...


From

we obtain:

38

e^(4*0.989117352243) / (((1+sqrt(1-1/3*16/(Pi)^2*e^(2*0.989117352243)))))^7

[42(1+sqrt(1-

1/3*16/(Pi)^2*e^(2*0.989117352243)))+5*16/(Pi)^2*e^(2*0.989117352243)]


Result:

Polar coordinates:

54.76072411…..


39

From which:

e^(4*0.989117352243) / (((1+sqrt(1-1/3*16/(Pi)^2*e^(2*0.989117352243)))))^7

[42(1+sqrt(1-

1/3*16/(Pi)^2*e^(2*0.989117352243)))+5*16/(Pi)^2*e^(2*0.989117352243)]*1/34

40


Result:

Polar coordinates:

1.610609533…. result that is a good approximation to the value of the golden ratio

1.618033988749...


42

Now, we have:

For:

ξ = 1

𝜙 = 0.989117352243

From

we obtain:

((2*e^(-0.989117352243/2))) /

((((1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243))))))))


43

Result:

0.382082347529….


From which:

1+1/(((4((2*e^(-0.989117352243/2))) /

((((1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243)))))))))))


44

Result:

1.6543092….. We note that, the result 1.6543092... is very near to the 14th root of the

following Ramanujan‟s class invariant 𝑄 = 𝐺505 /𝐺101/5 3 = 1164.2696 i.e.

1.65578...

Indeed:

113+5 505

8+ 105+5 505

8

314

= 1,65578…


45

And from

we obtain:

e^(-4*0.989117352243) / [1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243)))]^7 *

[42(1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243)))-

13*(4Pi^2)/25*e^(2*0.989117352243)]

46


Result:

-0.034547055658…


47

From which:

47 *1/(((-1/(((((e^(-4*0.989117352243) /

[1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243))))]^7 *

[42(1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243))))-

13*(4Pi^2)/25*e^(2*0.989117352243))]))))))))

48


Result:

1.6237116159…. result that is an approximation to the value of the golden ratio

1.618033988749...


50

And again:

32((((e^(-4*0.989117352243) /

[1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243))))]^7 *

[42(1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243))))-

13*(4Pi^2)/25*e^(2*0.989117352243))]))))


Result:

-1.1055057810….

We note that the result -1.1055057810…. is very near to the value of Cosmological

Constant, less 10-52

, thence 1.1056, with minus sign

51


53

And:

-[32((((e^(-4*0.989117352243) /

[1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243))))]^7 *

[42(1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243))))-

13*(4Pi^2)/25*e^(2*0.989117352243))]))))]^5


Result:

1.651220569…. result very near to the 14th root of the following Ramanujan‟s class

invariant 𝑄 = 𝐺505 /𝐺101/5 3 = 1164.2696 i.e. 1.65578...

54


56

We obtain also:

-[32((((e^(-4*0.989117352243) /

[1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243))))]^7 *

[42(1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243))))-

13*(4Pi^2)/25*e^(2*0.989117352243))]))))]^1/2


Result:

Polar coordinates:

1.05143035007

57


59

1 / -[32((((e^(-4*0.989117352243) /

[1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243))))]^7 *

[42(1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243))))-

13*(4Pi^2)/25*e^(2*0.989117352243))]))))]^1/2


Result:

Polar coordinates:

0.95108534763

We know that the primordial fluctuations are consistent with Gaussian purely

adiabatic scalar perturbations characterized by a power spectrum with a spectral

index ns = 0.965 ± 0.004, consistent with the predictions of slow-roll, single-field,

inflation.

Thence 0.95108534763 is a result very near to the spectral index ns , to the mesonic

Regge slope, to the inflaton value at the end of the inflation 0.9402 and to the value

of the following Rogers-Ramanujan continued fraction:

60


62

From the previous expression

= -0.034547055658…

we have also:

63

1+1/(((4((2*e^(-0.989117352243/2))) /

((((1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243))))))))))) + (-0.034547055658)


Result:

1.61976215705….. result that is a very good approximation to the value of the golden

ratio 1.618033988749...


64

From

Properties of Nilpotent Supergravity

E. Dudas, S. Ferrara, A. Kehagias and A. Sagnotti - arXiv:1507.07842v2 [hep-th] 14

Sep 2015

We have that:

We analyzing the following equation:

65

We have:

(M^2)/3*[1-(b/euler number * k/sqrt6) * (φ- sqrt6/k) * exp(-(k/sqrt6)(φ- sqrt6/k))]^2

i.e.

V = (M^2)/3*[1-(b/euler number * k/sqrt6) * (φ- sqrt6/k) * exp(-(k/sqrt6)(φ-

sqrt6/k))]^2

For k = 2 and φ = 0.9991104684, that is the value of the scalar field that is equal to

the value of the following Rogers-Ramanujan continued fraction:

we obtain:

V = (M^2)/3*[1-(b/euler number * 2/sqrt6) * (0.9991104684- sqrt6/2) * exp(-

(2/sqrt6)(0.9991104684- sqrt6/2))]^2


Result:

66

Solutions:

Alternate forms:

Expanded form:

Alternate form assuming b, M, and V are positive:

Alternate form assuming b, M, and V are real:

Derivative:

67

Implicit derivatives

Global minimum:

68

Global minima:

From:

we obtain

(225.913 (-0.054323 M^2 + 6.58545×10^-10 sqrt(M^4)))/M^2


Result:

69

Plots:

Alternate form assuming M is real:

-12.2723 result very near to the black hole entropy value 12.1904 = ln(196884)

Alternate forms:

70

Expanded form:

Property as a function:

Parity

Series expansion at M = 0:

Series expansion at M = ∞:

Derivative:

Indefinite integral:

71

Global maximum:

Global minimum:

Limit:

Definite integral after subtraction of diverging parts:

From b that is equal to

72

From:

we obtain:

1/3 (0.0814845 ((225.913 (-0.054323 M^2 + 6.58545×10^-10 sqrt(M^4)))/M^2 ) +

1)^2 M^2


Result:

Plots: (possible mathematical connection with an open string)

M = -0.5; M = 0.2

73

(possible mathematical connection with an open string)

M = 2 ; M = 3

Root:

Property as a function:

Parity

Series expansion at M = 0:

Series expansion at M = ∞:

74

Definite integral after subtraction of diverging parts:

For M = - 0.5 , we obtain:

1/3 (0.0814845 ((225.913 (-0.054323 (-0.5)^2 + 6.58545×10^-10 sqrt((-0.5)^4)))/(-

0.5)^2 ) + 1)^2 * (-0.5^2)


Result:

-4.38851344947*10-16

75

For M = 0.2:

1/3 (0.0814845 ((225.913 (-0.054323 0.2^2 + 6.58545×10^-10 sqrt(0.2^4)))/0.2^2 ) +

1)^2 0.2^2


Result:

7.021621519159*10-17

For M = 3:

76

1/3 (0.0814845 ((225.913 (-0.054323 3^2 + 6.58545×10^-10 sqrt(3^4)))/3^2 ) + 1)^2

3^2


Result:

1.57986484181*10-14

For M = 2:

1/3 (0.0814845 ((225.913 (-0.054323 2^2 + 6.58545×10^-10 sqrt(2^4)))/2^2 ) + 1)^2

2^2


77

Result:

7.021621519*10-15

From the four results

7.021621519*10^-15 ; 1.57986484181*10^-14 ; 7.021621519159*10^-17 ;

-4.38851344947*10^-16

we obtain, after some calculations:

sqrt[1/(2Pi)(7.021621519*10^-15 + 1.57986484181*10^-14 +7.021621519*10^-17 -

4.38851344947*10^-16)]


Result:

5.9776991059*10-8

result very near to the Planck's electric flow 5.975498 × 10−8

that

is equal to the following formula:

78

We note that:

1/55*(([(((1/[(7.021621519*10^-15 + 1.57986484181*10^-14 +7.021621519*10^-17

-4.38851344947*10^-16)])))^1/7]-((log^(5/8)(2))/(2 2^(1/8) 3^(1/4) e log^(3/2)(3)))))


Result:

1.6181818182… result that is a very good approximation to the value of the golden

ratio 1.618033988749...

From the Planck units:

Planck Length

5.729475 * 10-35

Lorentz-Heaviside value

79

Planck‟s Electric field strength

1.820306 * 1061

V*m Lorentz-Heaviside value

Planck‟s Electric flux

5.975498*10-8

V*m Lorentz-Heaviside value

Planck‟s Electric potential

1.042940*1027

V Lorentz-Heaviside value

80

Relationship between Planck’s Electric Flux and Planck’s Electric Potential

EP * lP = (1.820306 * 1061

) * 5.729475 * 10-35


Result:

Scientific notation:

1.042939771935*1027

≈ 1.042940*1027

Or:

EP * lP2 / lP = (5.975498*10

-8)*1/(5.729475 * 10

-35)


Result:

1.042939885417*10

27 ≈ 1.042940*10

27

81

Acknowledgments

We would like to thank Professor Augusto Sagnotti theoretical physicist at Scuola

Normale Superiore (Pisa – Italy) for his very useful explanations and his availability

82

Appendix References

Modular equations and approximations to 𝝅 - Srinivasa Ramanujan

Quarterly Journal of Mathematics, XLV, 1914, 350 – 372

An Update on Brane Supersymmetry Breaking

J. Mourad and A. Sagnotti - arXiv:1711.11494v1 [hep-th] 30 Nov 2017

AdS Vacua from Dilaton Tadpoles and Form Fluxes - J. Mourad and A. Sagnotti

- arXiv:1612.08566v2 [hep-th] 22 Feb 2017 - March 27, 2018

Properties of Nilpotent Supergravity

E. Dudas, S. Ferrara, A. Kehagias and A. Sagnotti - arXiv:1507.07842v2 [hep-th] 14

Sep 2015

on the study of five ramanujan mock theta functions. new possible mathematical connections with some...

Science

number theory

string theory

ramanujan modular equations

ramanujan mock theta functions

supersymmetry

supersymmetry breaking