1
On the analysis of some Ramanujan’s equations. New possible mathematical
connections with the Ramanujan Recurring Numbers, DN Constant and some
sectors of String Theory II
Michele Nardelli
1
, Antonio Nardelli
Abstract
In this paper (part II), we analyze some Ramanujan’s equations. We obtain new
possible mathematical connections with the Ramanujan Recurring Numbers, DN
Constant and some sectors of String Theory
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
A. Nardelli studied at the Università degli Studi di Napoli Federico II - Dipartimento di Studi
Umanistici – Sezione Filosofia - scholar of Theoretical Philosophy
3
From
Bruce Berndt – “Ramanujan: A Century of Inspiration”
https://www.youtube.com/watch?v=VRRRi1WcPJs&t=13s
We analyze again the following equation.
is a complicated function. …
;
;
;
;
;
;
;
From
For
4
(6+√5)/4-(5√10)/8-1/(2(π/5))+golden ratio(π^2/((π/5))) √((2π^3)/((π/5)^3))
Input
Exact result
Decimal approximation
401.14935569865….
Property
Expanded form
5
Alternate forms
Series representations
6
And for
1/3-√3 (3/16-1/(8π))-1/(2((2π)/3))+golden ratio(π^2/(((2π)/3))) √((2π^3)/(((2π)/3)^3))
Input
Exact result
7
Decimal approximation
19.6485833899….
Property
Alternate forms
Expanded form
8
Series representations
9
From the two exact results
and
we obtain:
(-(5 sqrt(5/2))/4 + 1/4 (6 + sqrt(5)) - 5/(2 π) + 25 sqrt(10) φ π)/(9/4 sqrt(3) π φ + 1/3 -
sqrt(3) (3/16 - 1/(8 π)) - 3/(4 π))
Input
Result
Decimal approximation
20.41619732772955….
10
Property
Alternate forms
Expanded forms
11
Series representations
12
13
From which, we obtain, after some calculations:
((-(5 sqrt(5/2))/4 + 1/4 (6 + sqrt(5)) - 5/(2 π) + 25 sqrt(10) φ π)/(9/4 sqrt(3) π φ + 1/3 -
sqrt(3) (3/16 - 1/(8 π)) - 3/(4 π)))^(1/(2Pi))
Input
Exact result
Decimal approximation
1.6161771915…. result that is a good approximation to the value of the golden ratio
1.618033988749… (Ramanujan Recurring Number)
Alternate forms
14
Expanded form
Series representations
15
Integral representation
16
And:
((-(5 sqrt(5/2))/4 + 1/4 (6 + sqrt(5)) - 5/(2 π) + 25 sqrt(10) φ π)/(9/4 sqrt(3) π φ + 1/3 -
sqrt(3) (3/16 - 1/(8 π)) - 3/(4 π)))^1/6
Input
Exact result
Decimal approximation
1.6532142590576…. result quite near to the 14th root of the following Ramanujan’s
class invariant
= 1164.2696 i.e. 1.65578...
17
Property
Alternate forms
18
Expanded forms
All 6
th
roots of (25 sqrt(10) π ϕ-(5 sqrt(5/2))/4+1/4 (6+sqrt(5))-5/(2 π))/(9/4
sqrt(3) π ϕ+1/3-sqrt(3) (3/16-1/(8 π))-3/(4 π))
19
20
Series representations
21
Integral representation
22
From the two exact results
and
we obtain:
1/2((((25 sqrt(10) π φ - (5 sqrt(5/2))/4 + 1/4 (6 + sqrt(5)) - 5/(2 π))/(9/4 sqrt(3) π φ +
1/3 - sqrt(3) (3/16 - 1/(8 π)) - 3/(4 π)))^(1/6))+1.6161771915)+9(MRB const)^(1-
1/(4π)+π)
Input interpretation
Result
1.64480088371259…. ≈ ζ(2) = π
2
/6 = 1.644934 (trace of the instanton shape and
Ramanujan Recurring Number)
23
sqrt(6(1/2((((25 sqrt(10) π φ - (5 sqrt(5/2))/4 + 1/4 (6 + sqrt(5)) - 5/(2 π))/(9/4 sqrt(3)
π φ + 1/3 - sqrt(3) (3/16 - 1/(8 π)) - 3/(4 π)))^(1/6))+1.6161771915)+9(MRB
const)^(1-1/(4π)+π)))
Input interpretation
Result
3.14146547…. ≈ π (Ramanujan Recurring Number)
sqrt(1/(1/2((((25 √(10) π φ - (5 √(5/2))/4 + 1/4 (6 + √(5)) - 5/(2 π))/(9/4 √(3) π φ + 1/3
- √(3) (3/16 - 1/(8 π)) - 3/(4 π)))^(1/6))+1.6161771915)+9(MRB const)^(1-
1/(4π)+π))*(4/3))
Input interpretation
24
Result
0.900352765713…. ≈ 0.9003163161571…. =
(DN Constant)
Possible closed forms
(2π√2)sqrt(1/(1/2((((25 √(10) π φ - (5 √(5/2))/4 + 1/4 (6 + √(5)) - 5/(2 π))/(9/4 √(3) π
φ + 1/3 - √(3) (3/16 - 1/(8 π)) - 3/(4 π)))^(1/6))+1.6161771915)+9(MRB const)^(1-
1/(4π)+π))*(4/3))
Input interpretation
25
Result
8.00032388222…. ≈ 8
value that is linked to the "Ramanujan function" (an elliptic modular function that
satisfies the need for "conformal symmetry") that has 8 "modes" corresponding to the
physical vibrations of a superstring.
(6π√2)sqrt(1/(1/2((((25 √(10) π φ - (5 √(5/2))/4 + 1/4 (6 + √(5)) - 5/(2 π))/(9/4 √(3) π
φ + 1/3 - √(3) (3/16 - 1/(8 π)) - 3/(4 π)))^(1/6))+1.6161771915)+9(MRB const)^(1-
1/(4π)+π))*(4/3))
Input interpretation
Result
24.0009716467…. ≈ 24
The value 24 is linked to the "Ramanujan function" (an elliptic modular function that
satisfies the need for "conformal symmetry") that has 24 "modes" corresponding to
the physical vibrations of a bosonic string representing a bosons.
26
From the analysis, we observe that the is no number theoretic connection with
physical vibrations of fermionic strings at extremally low entropy. This fact is
confirmed by the fact that the Higgs bosons at the moment of the big bang and
infinitesimally shortly thereafter, facilitated the creation of fermions (matter and
antimatter particles). Thus we note that the ingredients for the formation of
electromagnetic radiation from photons (a Boson), and the formation of matter from
the Higgs boson after the big bang, are intrinsic properties of the vacuum energy in
pre-big bang.
((2π√2)sqrt(1/(1/2((((25 √(10) π φ - (5 √(5/2))/4 + 1/4 (6 + √(5)) - 5/(2 π))/(9/4 √(3) π
φ + 1/3 - √(3) (3/16 - 1/(8 π)) - 3/(4 π)))^(1/6))+1.6161771915)+9(MRB const)^(1-
1/(4π)+π))*(4/3)))^4-Φ
Input interpretation
Result
4096.0453171…. ≈ 4096 = 64
2
, (Ramanujan Recurring Number) that multiplied by 2
give 8192, indeed:
27
The total amplitude vanishes for gauge group SO(8192), while the vacuum energy is
negative and independent of the gauge group. The vacuum energy and dilaton tadpole
to lowest non-trivial order for the open bosonic string. While the vacuum energy is
non-zero and independent of the gauge group, the dilaton tadpole is zero for a unique
choice of gauge group, SO(2
13
) i.e. SO(8192). (From: “Dilaton Tadpole for the Open
Bosonic String “ Michael R. Douglas and Benjamin Grinstein - September 2,1986)
27*sqrt(((2π√2)sqrt(1/(1/2((((25√(10) π φ-(5√(5/2))/4+1/4(6+√(5))-5/(2 π))/(9/4√(3)
π φ+1/3-√(3)(3/16-1/(8π))-3/(4 π)))^(1/6))+1.6161771915)+9(MRB const)^(1-
1/(4π)+π))*(4/3)))^4-Φ)+1
Input interpretation
Result
1729.009559….
28
This result is very near to the mass of candidate glueball f
0
(1710) scalar meson.
Furthermore, 1728 occurs in the algebraic formula for the j-invariant of an elliptic
curve (1728 = 8
2
* 3
3
). The number 1728 is one less than the Hardy–Ramanujan
number 1729 (taxicab number, as it can be expressed as the sum of two cubes in two
different ways (10
3
+ 9
3
= 12
3
+ 1
3
= 1729) and Ramanujan's recurring number)
Now, we have:
for αβ = π
2
, n = 1/2, m = 1/4 and k = 1 , we consider, from the left-hand side:
(4π)^-(1/2)*(1/2*ζ(2)+1/((1/4)^2*(e^(π/2)-1)))-(-4*π)^-(1/2)*(1/2*ζ(2)+
1/((1/4)^2*(e^(π/2)-1)))
Input
Exact result
29
Decimal approximation
Alternate complex forms
Polar coordinates
2.0033
Alternate forms
30
Expanded forms
Alternative representations
31
Series representations
Integral representations
32
From which, after some calculations:
-((((4π)^-(1/2)*(1/2*ζ(2)+1/((1/4)^2*(e^(π/2)-1)))-(-4*π)^-(1/2)*(1/2*ζ(2)+
1/((1/4)^2*(e^(π/2)-1)))))^12+76+4+1/√2)
Input
Exact result
33
Decimal approximation
4096.011316589…. ≈ 4096 = 64
2
, (Ramanujan Recurring Number) that multiplied
by 2 give 8192, indeed:
The total amplitude vanishes for gauge group SO(8192), while the vacuum energy is
negative and independent of the gauge group. The vacuum energy and dilaton tadpole
to lowest non-trivial order for the open bosonic string. While the vacuum energy is
non-zero and independent of the gauge group, the dilaton tadpole is zero for a unique
choice of gauge group, SO(2
13
) i.e. SO(8192). (From: “Dilaton Tadpole for the Open
Bosonic String “ Michael R. Douglas and Benjamin Grinstein - September 2,1986)
Expanded form
Alternate forms
34
Alternative representations
35
Series representations
36
Integral representations
37
27*sqrt(-((((4π)^-(1/2)*(1/2*ζ(2)+1/((1/4)^2*(e^(π/2)-1)))-(-4*π)^-(1/2)*(1/2*ζ(2)+
1/((1/4)^2*(e^(π/2)-1)))))^12+76+4+1/√2))+1
Input
38
Exact result
Decimal approximation
1729.00238709141….
This result is very near to the mass of candidate glueball f
0
(1710) scalar meson.
Furthermore, 1728 occurs in the algebraic formula for the j-invariant of an elliptic
curve (1728 = 8
2
* 3
3
). The number 1728 is one less than the Hardy–Ramanujan
number 1729 (taxicab number, as it can be expressed as the sum of two cubes in two
different ways (10
3
+ 9
3
= 12
3
+ 1
3
= 1729) and Ramanujan's recurring number)
Alternate forms
Alternative representations
39
Series representations
40
41
Integral representations
42
(27*sqrt(-((((4π)^-(1/2)*(1/2*ζ(2)+1/((1/4)^2*(e^(π/2)-1)))-(-4*π)^-(1/2)*(1/2*ζ(2)+
1/((1/4)^2*(e^(π/2)-1)))))^12+76+4+1/√2))+1)^1/15+(MRB const)^(1-1/(4π)+π)
Input
Exact result
Decimal approximation
1.644938175429…. ≈ ζ(2) = π
2
/6 = 1.644934 (trace of the instanton shape and
Ramanujan Recurring Number)
43
Alternate forms
sqrt(1/((27*sqrt(-((((4π)^-(1/2)*(1/2*ζ(2)+1/((1/4)^2*(e^(π/2)-1)))-(-4*π)^-
(1/2)*(1/2*ζ(2)+ 1/((1/4)^2*(e^(π/2)-1)))))^12+76+4+1/√2))+1)^1/15+(MRB
const)^(1-1/(4π)+π))*(4/3))
Input
Exact result
44
Decimal approximation
0.9003151917912373…. ≈ 0.9003163161571…. =
(DN Constant)
All 2
nd
roots of 4/(3 (C
MRB
1-1/(4 π)+π
+(1+27 sqrt(-80-1/sqrt(2)+(16/(e
π/2
-
1)+π
2
/12)
12
/(64 π
6
)))
1/15
))
45
(2π√2)sqrt(1/((27*sqrt(-((((4π)^-(1/2)*(1/2*ζ(2)+1/((1/4)^2*(e^(π/2)-1)))-(-4*π)^-
(1/2)*(1/2*ζ(2)+ 1/((1/4)^2*(e^(π/2)-1)))))^12+76+4+1/√2))+1)^1/15+(MRB
const)^(1-1/(4π)+π))*(4/3))
Input
Exact result
Decimal approximation
7.99999000914813…. ≈ 8
value that is linked to the "Ramanujan function" (an elliptic modular function that
satisfies the need for "conformal symmetry") that has 8 "modes" corresponding to the
physical vibrations of a superstring.
46
Alternate forms
(6π√2)sqrt(1/((27*sqrt(-((((4π)^-(1/2)*(1/2*ζ(2)+1/((1/4)^2*(e^(π/2)-1)))-(-4*π)^-
(1/2)*(1/2*ζ(2)+ 1/((1/4)^2*(e^(π/2)-1)))))^12+76+4+1/√2))+1)^1/15+(MRB
const)^(1-1/(4π)+π))*(4/3))
Input
Exact result
47
Decimal approximation
23.999970027444…. ≈ 24
The value 24 is linked to the "Ramanujan function" (an elliptic modular function that
satisfies the need for "conformal symmetry") that has 24 "modes" corresponding to
the physical vibrations of a bosonic string representing a bosons. From the analysis,
we observe that the is no number theoretic connection with physical vibrations of
fermionic strings at extremally low entropy. This fact is confirmed by the fact that the
Higgs bosons at the moment of the big bang and infinitesimally shortly thereafter,
facilitated the creation of fermions (matter and antimatter particles). Thus we note
that the ingredients for the formation of electromagnetic radiation from photons (a
Boson), and the formation of matter from the Higgs boson after the big bang, are
intrinsic properties of the vacuum energy in pre-big bang.
Alternate forms
From
for αβ = π
2
, n = 1/2, m = 1/4 and k = 1 :
(-1)^0*((BernoulliB(2))/(2)!)*(BernoulliB(1))/((1+2-2)!)*π^(1/2+1-1)*π
48
Input
Exact result
Decimal approximation
-0.2320136665346544…
Property
Alternative representations
49
Series representations
Integral representations
50
Dividing the two exact results
and
we obtain:
(((1/2 + i/2) (16/(e^(π/2) - 1) + π^2/12))/sqrt(π))/(-π^(3/2)/24)
Input
Exact result
Decimal approximation
Alternate complex forms
51
Polar coordinates
8.6342
Expanded forms
Alternate forms
Series representations
52
Integral representations
From which, after some calculations:
(((((1/2 + i/2) (16/(e^(π/2) - 1) + π^2/12))/sqrt(π))/(-π^(3/2)/24))+√3/2)^4+89+3^2
Input
53
Exact result
Decimal approximation
Alternate complex forms
54
55
56
Polar coordinates
4095.9 ≈ 4096 = 64
2
, (Ramanujan Recurring Number) that multiplied by 2 give
8192, indeed:
The total amplitude vanishes for gauge group SO(8192), while the vacuum energy is
negative and independent of the gauge group. The vacuum energy and dilaton tadpole
to lowest non-trivial order for the open bosonic string. While the vacuum energy is
non-zero and independent of the gauge group, the dilaton tadpole is zero for a unique
choice of gauge group, SO(2
13
) i.e. SO(8192). (From: “Dilaton Tadpole for the Open
Bosonic String “ Michael R. Douglas and Benjamin Grinstein - September 2,1986)
Alternate forms
57
Expanded forms
Series representations
58
27*sqrt((((((1/2 + i/2) (16/(e^(π/2) - 1) + π^2/12))/sqrt(π))/(-
π^(3/2)/24))+√3/2)^4+89+3^2)+e+(1/2 + sqrt(2)) sqrt(π)
Input
Exact result
59
Decimal approximation
Alternate complex forms
Polar coordinates
1728.93
This result is very near to the mass of candidate glueball f
0
(1710) scalar meson.
Furthermore, 1728 occurs in the algebraic formula for the j-invariant of an elliptic
curve (1728 = 8
2
* 3
3
). The number 1728 is one less than the Hardy–Ramanujan
number 1729 (taxicab number, as it can be expressed as the sum of two cubes in two
different ways (10
3
+ 9
3
= 12
3
+ 1
3
= 1729) and Ramanujan's recurring number)
Alternate forms
60
Expanded form
61
Series representations
62
63
64
(27*sqrt((((((1/2 + i/2) (16/(e^(π/2) - 1) + π^2/12))/sqrt(π))/(-
π^(3/2)/24))+√3/2)^4+89+3^2)+e+(1/2 + sqrt(2)) sqrt(π))^1/15+(MRB const)^(1-
1/(4π)+π)
Input
Exact result
Decimal approximation
Polar coordinates
1.6449296487…. ≈ ζ(2) = π
2
/6 = 1.644934 (trace of the instanton shape and
Ramanujan Recurring Number)
65
sqrt(1/((27*sqrt((((((1/2 + i/2) (16/(e^(π/2) - 1) + π^2/12))/sqrt(π))/(-
π^(3/2)/24))+√3/2)^4+89+3^2)+e+(1/2 + sqrt(2)) sqrt(π))^1/15+(MRB const)^(1-
1/(4π)+π))*(4/3))
Input
Exact result
Decimal approximation
66
Polar coordinates
0.9003175…. ≈ 0.9003163161571…. =
(DN Constant)
Possible closed forms
67
From
THE FINAL PROBLEM : AN ACCOUNT OF THE MOCK THETA
FUNCTIONS - G. N. WATSON – 14 November 1935
We analyze the following equation:
We consider:
((e^(-3/2*a*x^2))*(((cosh(5/2*a*x)+cosh(1/2*a*x)))/(cosh(3a*x))))
Input
Exact result
68
3D plot (figure that can be related to a D-brane/Instanton)
The key observation from the above plots and is that at , which is taken as the
energy density of the universe at the Big Bang, with the zero spacetime volume,
the vacuum geometry brakes / or there is symmetry breaking on the vacuum quantum
geometry. We see from the plots as the vacuum spacetime break/tear appart.
Contour plot
Alternate forms
69
Expanded form
Alternate form assuming a and x are real
Roots
70
Properties as a real function
Domain
Parity
Roots for the variable x
71
Series expansion at x=0
Derivative
From the series expansion at x = 0
we obtain:
2 - 1/4 (a (23 a + 12)) x^2 + 1/192 a^2 (3985 a^2 + 1656 a + 432) x^4 + (x^5)
Input
72
3D plot (figure that can be related to a D-brane/Instanton)
The key observation from the above plots and is that at , which is taken as the
energy density of the universe at the Big Bang, with the zero spacetime volume,
the vacuum geometry brakes / or there is symmetry breaking on the vacuum quantum
geometry. We see from the plots as the vacuum spacetime break/tear appart.
Contour plot
Alternate forms
73
Expanded form
Real root
Roots
74
Properties as a real function
Domain
Range
Surjectivity
Derivative
75
Indefinite integral
From the indefinite integral result, for (2πi*q^(((2n+1)^2)/6) cos(1/6*(2n+1)*π)) ,
we obtain:
(2πi*q^(((2n+1)^2)/6) cos(1/6*(2n+1)*π))((797 a^4 x^5)/192 + (69 a^3 x^5)/40 + (9
a^2 x^5)/20 - (23 a^2 x^3)/12 - a x^3 + x^6/6 + 2 x)
Input
Exact result
Alternate forms
76
Expanded forms
Reduced trigonometric form
Roots
77
Derivative
78
Indefinite integral
Again, from the indefinite integral result
we obtain:
1/480 i π q^(1/6 (1 + 2 n)^2) (960 x^2 - 20 a (12 + 23 a) x^4 + 1/6 a^2 (432 + 1656 a
+ 3985 a^2) x^6 + (160 x^7)/7) cos(1/6 (1 + 2 n) π)
Input
79
Exact result
Alternate forms
Expanded forms
80
Alternate form assuming a, n, q, and x are positive
Reduced trigonometric form
Roots
81
Derivative
Indefinite integral
From the indefinite integral result
82
for V = 1/3*√2*a^3 (octahedron volume) and V = (4/3*π*r^3) (sphere volume),
where r = (a/2), considering (1/3*√2*a^3) and (4/3*π*(a/2)^3) , we obtain:
(1/480 i π q^(1/6 (1 + 2 n)^2) (320 x^3 - 4 a (12 + 23 a) x^5 + 1/42 (432 a^2 + 1656
a^3 + 3985 a^4) x^7 + (20 x^8)/7) cos(1/6 (1 + 2 n) π))(1/3*√2*a^3)
Input
Exact result
Alternate forms
83
Expanded form
84
Alternate form assuming a, n, q, and x are positive
Reduced trigonometric form
Roots
85
Derivative
Indefinite integral
And:
(1/480 i π q^(1/6 (1 + 2 n)^2) (320 x^3 - 4 a (12 + 23 a) x^5 + 1/42 (432 a^2 + 1656
a^3 + 3985 a^4) x^7 + (20 x^8)/7) cos(1/6 (1 + 2 n) π)) (4/3*π*(a/2)^3)
Input
86
Exact result
Alternate forms
Expanded form
87
Alternate form assuming a, n, q, and x are positive
Reduced trigonometric form
Roots
88
Derivative
Indefinite integral
Dividing the two indefinite integral results:
and
89
we obtain, simplifying:
π/(720√2)*1/((π^2)/2880)
Input
Result
Decimal approximation
0.9003163161571…. =
(DN Constant)
Property
90
Series representations
91
On the application of the formulas of the volumes of an octahedron and a sphere
With regard to a sphere inscribed in an octahedron, we have the following formulas.
Fig: sphere inscribed in an octahedron
V
0
=
V
s
=
where r
s
= (l/2)
With regard the ratio between the two above formulas (octahedron and sphere)
(1/3*√2*l^3)/(4/3*π*(l/2)^3)
we obtain:
Input
92
Result
Decimal approximation
0.900316316157106…. =
(DN Constant)
Property
Series representations
93
From which:
1/3*(2/((1/3*√2*l^3)/(4/3*π*(l/2)^3)))^2
Input
Result
Decimal approximation
1.644934066848226… = ζ(2) = π
2
/6 = 1.644934 (trace of the instanton shape and
Ramanujan Recurring Number)
94
Property
Series representations
Integral representations
95
We note that, from the sum of the first nine numbers excluding 0, i.e.,
1+2+3+4+5+6+7+8+9 = 45 (these are the fundamental numbers, from which, through
infinite combinations, all the other numbers are obtained), we obtain the following
interesting formula:
1+1/(((φ^2+(2Pi)/3*MRB const)(1/e((1+2+3+4+5+6+7+8+9)^(1/Pi))))^1/3)
Input
Exact result
96
Decimal approximation
1.64529737852…. ≈ ζ(2) = π
2
/6 = 1.644934 (trace of the instanton shape and
Ramanujan Recurring Number)
Alternate forms
Expanded forms
97
And:
sqrt(6(1+1/(((φ^2+(2Pi)/3*MRB const)(1/e((1+2+3+4+5+6+7+8+9)^(1/Pi))))^1/3)))
Input
Exact result
Decimal approximation
3.141939571526…. ≈ π (Ramanujan Recurring Number)
Alternate forms
98
Expanded forms
All 2
nd
roots of 6 (3
-2/(3 π)
5
-1/(3 π)
(e/((2 π C
MRB
)/3+ϕ
2
))
1/3
+1)
Furthermore, we obtain also:
2π*√2((1/3*√2*l^3)/(4/3*π*(l/2)^3))
Input
99
Exact result
8
value that is linked to the "Ramanujan function" (an elliptic modular function that
satisfies the need for "conformal symmetry") that has 8 "modes" corresponding to the
physical vibrations of a superstring.
Series representations
100
6π*√2((1/3*√2*l^3)/(4/3*π*(l/2)^3))
Input
Exact result
24
The value 24 is linked to the "Ramanujan function" (an elliptic modular function that
satisfies the need for "conformal symmetry") that has 24 "modes" corresponding to
the physical vibrations of a bosonic string representing a bosons. From the analysis,
we observe that the is no number theoretic connection with physical vibrations of
fermionic strings at extremally low entropy. This fact is confirmed by the fact that the
Higgs bosons at the moment of the big bang and infinitesimally shortly thereafter,
facilitated the creation of fermions (matter and antimatter particles) [8]. Thus we note
that the ingredients for the formation of electromagnetic radiation from photons (a
Boson), and the formation of matter from the Higgs boson after the big bang, are
intrinsic properties of the vacuum energy in pre-big bang.
Series representations
101
This could imply that all matter (fermions) was preceded by bosons. That is, before
the Big Bang, from perturbations of the vacuum energy itself, bosons were created,
and successively at the Big Bang, and infinitesimally shortly after the Big Bang,
fermions, were created from the vacuum energy that underwent a violent “breaking”
that formed a hot plasma. of particle-antiparticle pairs. This therefore implies that
quantum gravity was not necessarily “dark” to some extent, because a photon (light
particle) is itself a boson. Therefore, a big bang was not necessarily the moment of
the creation of light, but of the creation of matter (fermions) from vacuum energy, as
this undergoes further "breaking" in the cosmological constant, in the hot plasma of
matter and in the energy dark.
102
Indeed:
From:
https://www.academia.edu/75787512/The_Theory_of_String_A_Candidate_for_a_G
eneralized_Unification_Model
The Einstein’s field equation and the String Theory.
The Einstein’s field equation which includes the cosmological constant is:
μν
μν
μν
GT
μν
(8)
where
μν
is the Ricci tensor, its trace, the cosmological constant,
μν
the metric
tensor of the space geometry, G the Newton’s gravitational constant and
μν
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.
The actions of bosonic string and superstring (also containing fermions) are
connected by the Palumbo-Nardelli relation (Palumbo et al. 2005):
103
The sign minus in the above equation comes from the inversion of any relationship.
(2π*√2((1/3*√2*l^3)/(4/3*π*(l/2)^3)))^4
Input
Exact result
4096 = 64
2
, (Ramanujan Recurring Number) that multiplied by 2 give 8192, indeed:
The total amplitude vanishes for gauge group SO(8192), while the vacuum energy is
negative and independent of the gauge group. The vacuum energy and dilaton tadpole
to lowest non-trivial order for the open bosonic string. While the vacuum energy is
non-zero and independent of the gauge group, the dilaton tadpole is zero for a unique
choice of gauge group, SO(2
13
) i.e. SO(8192). (From: “Dilaton Tadpole for the Open
Bosonic String “ Michael R. Douglas and Benjamin Grinstein - September 2,1986)
27*sqrt((2π*√2((1/3*√2*l^3)/(4/3*π*(l/2)^3)))^4)+1
Input
104
Exact result
1729
This result is very near to the mass of candidate glueball f
0
(1710) scalar meson.
Furthermore, 1728 occurs in the algebraic formula for the j-invariant of an elliptic
curve (1728 = 8
2
* 3
3
). The number 1728 is one less than the Hardy–Ramanujan
number 1729 (taxicab number, as it can be expressed as the sum of two cubes in two
different ways (10
3
+ 9
3
= 12
3
+ 1
3
= 1729) and Ramanujan's recurring number)
Series representations
105
We note that:
1/25*1/144(((2π*√2((1/3*√2*l^3)/(4/3*π*(l/2)^3)))^4)+(27*sqrt((2π*√2((1/3*√2*l^
3)/(4/3*π*(l/2)^3)))^4)+1))
Input
Exact result
Decimal approximation
1.61805555…. result that is a very good approximation to the value of the golden
ratio 1.618033988749… (Ramanujan Recurring Number)
Repeating decimal
106
Series representations
107
From
we obtain also:
sqrt(6(1/3*(2/(((2sqrt2)/Pi)))^2))
Input
Exact result
Decimal approximation
3.14159265358… = π
108
Property
All 2
nd
roots of π
2
Series representations
Integral representations
109
It is plausible to hypothesize that π and φ, in addition to being important
mathematical constants, are constants that also have a fundamental relevance in the
various sectors of Theoretical Physics and Cosmology
From
, we obtain:
sqrt(1/(Pi^2/6)*(4/3))
Input
Exact result
110
Decimal approximation
0.900316316157106…. =
(DN Constant)
Property
All 2
nd
roots of 8/π
2
Series representations
111
112
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 8
2
= 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
113
“Golden” Range
Finally we note how 8
2
= 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 = 8
2
, 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
114
Appendix
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 Approximations 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 e
π√22
115
We have, in certain cases, the following connections:
Fig. 1
Fig. 2
116
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
118
Where ζ(2+it) :
Input
Plots
Roots
119
Series expansion at t=0
Alternative representations
Series representations
120
Integral representations
Functional equations
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:
121
we obtain:
2Pi/(ln(2))
Input:
Exact result:
Decimal approximation:
9.06472028365….
Alternative representations:
122
Series representations:
Integral representations:
123
From which:
(2Pi/(ln(2)))*(1/12 π log(2))
Input:
Exact result:
Decimal approximation:
1.6449340668…. = ζ(2) =
124
From:
Modular equations and approximations to - Srinivasa Ramanujan - Quarterly
Journal of Mathematics, XLV, 1914, 350 – 372
We have that:
125
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
Decimal approximation
1729.0526944….
This result is very near to the mass of candidate glueball f
0
(1710) scalar meson.
Furthermore, 1728 occurs in the algebraic formula for the j-invariant of an elliptic
curve. (1728 = 8
2
* 3
3
) The number 1728 is one less than the Hardy–Ramanujan
number 1729 (taxicab number)
Alternate forms
126
Minimal polynomial
Expanded forms
Series representations
127
128
Or:
27((4096+276)^1/2-2-1/2(((√(10-2√5) -2))⁄((√5-1))))+φ
Input
Result
Decimal approximation
1729.0526944…. as above
Alternate forms
129
Minimal polynomial
Expanded forms
Series representations
130
131
From which:
(27((4372)^1/2-2-1/2(((√(10-2√5) -2))⁄((√5-1))))+φ)^1/15
Input
Exact result
Decimal approximation
1.64381856858…. ≈ ζ(2) =
Alternate forms
132
Minimal polynomial
Expanded forms
All 15th roots of ϕ + 27 (-2 + 2 sqrt(1093) - (sqrt(10 - 2 sqrt(5)) - 2)/(2 (sqrt(5) -
1)))
133
Series representations
134
Integral representation
135
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, β
E
and correspond to the exponents of e (i.e. of exp).
Thence we obtain for p = 5 and β
E
= 1/2:
136
Therefore, with respect to the exponentials of the vacuum equations, the Ramanujan’s
exponential has a coefficient of 4096 which is equal to 64
2
, while -6C+ is equal to -
. From this it follows that it is possible to establish mathematically, the dilaton
value.
For
exp((-Pi*sqrt(18)) we obtain:
Input:
Exact result:
Decimal approximation:
1.6272016… * 10
-6
Property:
Series representations:
137
Now, we have the following calculations:
= 1.6272016… * 10^-6
from which:
= 1.6272016… * 10^-6
0.000244140625
=
= 1.6272016… * 10^-6
Now:
And:
138
(1.6272016* 10^-6) *1/ (0.000244140625)
Input interpretation:
Result:
0.006665017...
Thence:
0.000244140625
=
Dividing both sides by 0.000244140625, we obtain:
=
= 0.0066650177536
((((exp((-Pi*sqrt(18)))))))*1/0.000244140625
Input interpretation:
139
Result:
0.00666501785…
Series representations:
Now:
= 0.0066650177536
=
= 0.00666501785…
140
From:
ln(0.00666501784619)
Input interpretation:
Result:
-5.010882647757…
Alternative representations:
Series representations:
141
Integral representation:
In conclusion:
and for C = 1, we obtain:
=
Note that the values of n
s
(spectral index) 0.965, of the average of the Omega mesons
Regge slope 0.987428571 and of the dilaton , are also connected to
the following two Rogers-Ramanujan continued fractions:
142
(http://www.bitman.name/math/article/102/109/)
Also performing the 512
th
root of the inverse value of the Pion meson rest mass
139.57, we obtain:
((1/(139.57)))^1/512
Input interpretation:
Result:
0.99040073.... result very near to the dilaton value = and to the
value of the following Rogers-Ramanujan continued fraction:
143
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:
We have:
144
(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
Input interpretation:
Result:
145
Solutions:
Alternate forms:
Expanded form:
Alternate form assuming b, M, and V are positive:
Alternate form assuming b, M, and V are real:
146
Derivative:
Implicit derivatives
Global minimum:
147
Global minima:
From:
we obtain:
(225.913 (-0.054323 M^2 + 6.58545×10^-10 sqrt(M^4)))/M^2
Input interpretation:
148
Result:
Plots:
Alternate form assuming M is real:
-12.2723 result very near to the black hole entropy value 12.1904 = ln(196884)
Alternate forms:
149
Expanded form:
Property as a function:
Parity
Series expansion at M = 0:
Series expansion at M = ∞:
Derivative:
150
Indefinite integral:
Global maximum:
Global minimum:
Limit:
Definite integral after subtraction of diverging parts:
151
From b that is equal to
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
Input interpretation:
Result:
152
Plots: (possible mathematical connection with an open string)
M = -0.5; M = 0.2
(possible mathematical connection with an open string)
M = 2 ; M = 3
Root:
Property as a function:
Parity
Series expansion at M = 0:
153
Series expansion at M = ∞:
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)
Input interpretation:
154
Result:
-4.38851344947*10
-16
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
Input interpretation:
Result:
7.021621519159*10
-17
155
For M = 3:
1/3 (0.0814845 ((225.913 (-0.054323 3^2 + 6.58545×10^-10 sqrt(3^4)))/3^2 ) + 1)^2
3^2
Input interpretation:
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
156
Input interpretation:
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)]
Input interpretation:
157
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:
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)))))
Input interpretation:
Result:
1.6181818182… result that is a very good approximation to the value of the golden
ratio 1.618033988749...
158
From the Planck units:
Planck Length
5.729475 * 10
-35
Lorentz-Heaviside value
Planck’s Electric field strength
1.820306 * 10
61
V*m Lorentz-Heaviside value
Planck’s Electric flux
5.975498*10
-8
V*m Lorentz-Heaviside value
Planck’s Electric potential
1.042940*10
27
V Lorentz-Heaviside value
159
Relationship between Planck’s Electric Flux and Planck’s Electric Potential
E
P
* l
P
= (1.820306 * 10
61
) * 5.729475 * 10
-35
Input interpretation:
Result:
Scientific notation:
1.042939771935*10
27
≈ 1.042940*10
27
Or:
E
P
* l
P
2
/ l
P
= (5.975498*10
-8
)*1/(5.729475 * 10
-35
)
Input interpretation:
Result:
1.042939885417*10
27
≈ 1.042940*10
27
160
Fig. 1
It is therefore possible to consider the vortices of the "quantum vacuum" schematized
as cubes or octahedrons (the + sign inside a given vortex indicates its
counterclockwise rotation, while the - sign indicates its clockwise rotation). Between
vortex and vortex there is a layer of "bubbles"-universes (or universes-spheres),
which flows, as in the simplified two-dimensional drawing, from A to B
161
Fig. 2
Proposal
Image of space-time at quantum scale: the circles in red represent the points
corresponding to the compactified dimensions and the hexagons in blue, represent the
"fluctuations" (potential universes - green circles) of the quantum vacuum (2D). In
reality, we will have n-dimensional hyperspheres in which the compactified
dimensions "roll up" and octahedrons representing the "fluctuations", containing
spheres (bubbles of potential universes), of the quantum vacuum
162
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
163
References
Bruce Berndt – “Ramanujan: A Century of Inspiration”
https://www.youtube.com/watch?v=VRRRi1WcPJs&t=13s
THE FINAL PROBLEM : AN ACCOUNT OF THE MOCK THETA
FUNCTIONS - G. N. WATSON – 14 November 1935
Modular equations and approximations to - Srinivasa Ramanujan - Quarterly
Journal of Mathematics, XLV, 1914, 350 – 372
An Update on Brane Supersymmetry Breaking - Jihad Mourad and Augusto
Sagnotti - arXiv:1711.11494v1 [hep-th] 30 Nov 2017
Properties of Nilpotent Supergravity - Emilian Dudas, Sergio Ferrara, Alex
Kehagias and Augusto Sagnotti - arXiv:1507.07842v2 [hep-th] 14 Sep 2015
See also:
The Geometry of the MRB constant by Marvin Ray Burns
https://www.academia.edu/22271085/The_Geometry_of_the_MRB_constant
(See also Page 29 the applications of the CMRB in various sectors of Theoretical
Physics (String Theory) and Cosmology )
http://xoom.virgilio.it/source_filemanager/na/ar/nardelli/michele%20and%20antonio
%20papers/Try%20to%20beat%20these%20MRB%20constant%20records!%20-
%20Online%20Technical%20Discussion%20Groups%E2%80%94Wolfram%20Com
munity%20b.pdf