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

5

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

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

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

Decimal approximation

Polar coordinates

Polar coordinates

20.578

13

Polar forms

Approximate form

Alternate forms

14

Alternative representations

Series representations

15

Integral representations

16

Continued fraction representations

17

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

Decimal approximation

19

Polar coordinates

Polar coordinates

860.67

Polar forms

20

Approximate form

Alternate forms

Alternative representations

21

Series representations

22

23

Integral representations

24

Continued fraction representations

25

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

Decimal approximation

Polar coordinates

Polar coordinates

364454

28

Polar forms

Approximate form

Alternate forms

29

Expanded form

Alternative representations

30

Series representations

31

Integral representations

32

Continued fraction representations

33

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

Decimal approximation

Polar coordinates

17691

Polar forms

36

Approximate form

Alternate forms

Alternative representations

37

Series representations

38

Integral representations

39

Continued fraction representations

40

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

log(7)/6))

Input

41

Exact Result

Decimal approximation

Polar coordinates

17731

Polar coordinates

42

Polar forms

Approximate form

Alternate forms

43

Alternative representations

Series representations

44

Integral representations

45

Continued fraction representations

46

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

Decimal approximation

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.

49

Polar forms

50

Approximate form

Alternate forms

51

Expanded form

Alternative representations

52

Series representations

53

54

Integral representations

55

Continued fraction representations

56

57

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

Decimal approximation

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

60

61

Approximate form

Alternate forms

62

Alternative representations

63

Series representations

64

Integral representations

65

Continued fraction representations

66

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

Decimal approximation

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

Alternative representations

72

Series representations

73

74

Continued fraction representations

75

76

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

Decimal approximation

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))))

81

82

83

84

Alternative representations

85

Series representations

86

Integral representations

87

88

Continued fraction representations

89

90

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

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

93

Alternate forms of the integral

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

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

Alternate forms of the integral

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

Decimal approximation

Polar coordinates

0.42871

Polar forms

96

Approximate form

Alternate forms

Alternative representations

97

Series representations

Integral representations

98

Continued fraction representations

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)))))))

Input interpretation

Result

Polar coordinates

144

102

Alternative representations

103

Series representations

104

105

Integral representations

106

107

Continued fraction representations

108

109

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

Input interpretation

Result

Polar coordinates

1729

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

111

Alternative representations

112

Series representations

113

114

115

Integral representations

116

Continued fraction representations

117

118

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

Input interpretation

Result

Polar coordinates

1.64382 ≈ ζ(2) = 𝜋2

6= 1.644934… (trace of the instanton shape)

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

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

Alternate forms

Real root

122

Complex roots

Series expansion at q=0

Series expansion at q=∞

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

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

Alternate forms

125

Expanded form

Real roots

126

Complex roots

Series expansion at q=0

Series expansion at q=∞

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….

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)

(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

Input interpretation

130

Result

1.643821533….≈ ζ(2) = 𝜋2

6= 1.644934… (trace of the instanton shape)

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….

Alternative representations

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

6= 1.644934… (trace of the instanton shape)

Alternative representations

134

135

Series representations

136

137

Integral representations

138

139

Continued fraction representations

140

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

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

142

Alternate forms

Complex roots

Polynomial discriminant

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

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

Alternate form

145

Expanded form

Complex roots

Polynomial discriminant

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)

For q = 0.7788 , we obtain:

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….

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)

148

Series representations

(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

6= 1.644934… (trace of the instanton shape)

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

For q = 0.7788 , we obtain:

(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

Series representations

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

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

154

Alternate forms

Expanded form

155

Derivative

From the expression

For q = 0.7788 , we obtain:

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

Input interpretation

Result

1728.97….

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)

157

Alternative representations

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

Input interpretation

Result

1.64381346388….≈ ζ(2) = 𝜋2

6= 1.644934… (trace of the instanton shape)

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

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

160

Alternate forms

Real root

Complex roots

161

Series expansion at q=0

Series expansion at q=∞

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

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)

(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

6= 1.644934… (trace of the instanton shape)

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

Alternative representations

166

Series representations

167

Integral representations

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

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

170

Alternate form

Partial fraction expansion

Real root

Series expansion at q=0

171

Series expansion at q=∞

Indefinite integral

172

From the alternate form

For q = 0.7788 , we obtain:

-(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

Input interpretation

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

Input interpretation

Result

1728.06….

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)

175

Alternative representation

Series representations

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

Input interpretation

Result

1.6438192746….≈ ζ(2) = 𝜋2

6= 1.644934… (trace of the instanton shape)

(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

Input interpretation

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

Input interpretation

179

Result

Polar coordinates

2.26395

Polar forms

Alternative representations

180

Integral representations

181

182

Continued fraction representations

183

184

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

Input interpretation

Result

Polar coordinates

1.64623 ≈ ζ(2) = 𝜋2

6= 1.644934… (trace of the instanton shape)

Polar forms

186

Alternative representations

187

Integral representations

188

Continued fraction representations

189

190

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

Input interpretation

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

Input interpretation

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

Input interpretation

Result

1.6449339….≈ ζ(2) = 𝜋2

6= 1.644934… (trace of the instanton shape)

Possible closed forms

Alternative representations

194

Series representations

195

Integral representations

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

Input interpretation

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)

Alternative representations

Series representations

197

Integral representations

198

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