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
Michele Nardelli
1
, Antonio Nardelli
Abstract
In this paper, 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 the following equation.
where K = 0.764
we obtain:
0.764 (integrate(1/(sqrt(log(x))))dx) + θ(x)
Input
Result
Plots (figures that can be related to the open strings)
4
Alternate form assuming x>0
Alternate form
Series expansion of the integral at x=0
Series expansion of the integral at x=1
5
Series expansion of the integral at x=∞
Indefinite integral assuming all variables are real
From the alternate form
we obtain:
1.35415 (erfi(sqrt(log(x))) + 0.738468 θ(x))
Input interpretation
Result
6
Plots (figures that can be related to the open strings)
Expanded form
Alternate form assuming x>0
Alternate form assuming x is positive
From the following alternate form
we obtain:
7
1.35415 erfi(sqrt(log(x))) + 0.999996
Input interpretation
Result
Plots (figures that can be related to the open strings)
Alternate form
8
Series expansion at x=0
Series expansion at x=1
Series expansion at x=∞
Derivative
Indefinite integral
9
Global minimum
From the alternate form
for x = 2+3/5 :
1.35415 (erfi(sqrt(log(2+3/5))) + 0.738468)
Input interpretation
Result
3.1435331188359…. ≈ π (Ramanujan Recurring Number)
Alternative representations
10
Series representations
11
Integral representations
12
From which, we obtain also:
1/6*(1.35415 (erfi(sqrt(log(2+3/5))) + 0.738468))^2-2(MRB const)^(1-1/(4π)+π)
Input interpretation
Result
1.6447211541…. ≈ ζ(2) = π
2
/6 = 1.644934 (trace of the instanton shape and
Ramanujan Recurring Number)
sqrt(1/(1/6*(1.35415 (erfi(sqrt(log(2+3/5))) + 0.738468))^2-2(MRB const)^(1-
1/(4π)+π))*(4/3))
Input interpretation
Result
0.900375…. ≈ 0.9003163161571…. =
(DN Constant)
Possible closed forms
13
(2π*√2(sqrt(1/(1/6*(1.35415 (erfi(sqrt(log(2+3/5))) + 0.738468))^2-2(MRB
const)^(1-1/(4π)+π))*(4/3))))^4-1
Input interpretation
Result
4096.06…. ≈ 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)
14
27*sqrt((2π*√2(sqrt(1/(1/6*(1.35415 (erfi(sqrt(log(2+3/5))) + 0.738468))^2-2(MRB
const)^(1-1/(4π)+π))*(4/3))))^4-1)+1
Input interpretation
Result
1729.01….
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)
6π*√2(sqrt(1/(1/6*(1.35415 (erfi(sqrt(log(2+3/5))) + 0.738468))^2-2(MRB
const)^(1-1/(4π)+π))*(4/3)))
Input interpretation
Result
24.0016…. ≈ 24
15
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.
We analyze the following equation.
we obtain:
integrate((1+(x/(b+1))^2)/(1+(x/a)^2 ) (1+(x/(b+2))^2)/(1+(x/(a+1))^2 )
(1+(x/(b+3))^2)/(1+(x/(a+2))^2 ))
16
Indefinite integral
We have:
(a (4 a (a^2 + 3 a + 2)^2 x - ((a + 1) (a + 2)^2 (a^6 - a^4 (3 b^2 + 12 b + 14) + a^2 (3
b^4 + 24 b^3 + 72 b^2 + 96 b + 49) - (b^3 + 6 b^2 + 11 b + 6)^2) tan^(-1)(x/a))/(2 a +
1) + (4 a (a + 1) (a + 2)^2 (a^6 + 6 a^5 + a^4 (-3 b^2 - 12 b + 1) - 12 a^3 (b^2 + 4 b +
3) + a^2 (3 b^4 + 24 b^3 + 54 b^2 + 24 b - 20) + 6 a (b + 2)^2 (b^2 + 4 b + 2) - b (b +
2)^2 (b^3 + 8 b^2 + 19 b + 12)) tan^(-1)(x/(a + 1)))/(4 a^2 + 8 a + 3) - (a (a^2 + 3 a +
2) (a^6 + 12 a^5 + a^4 (-3 b^2 - 12 b + 46) - 24 a^3 (b^2 + 4 b - 2) + a^2 (3 b^4 + 24
b^3 - 192 b - 47) + 12 a (b^4 + 8 b^3 + 16 b^2 - 5) - b (b^5 + 12 b^4 + 46 b^3 + 48
b^2 - 47 b - 60)) tan^(-1)(x/(a + 2)))/(2 a + 3)))/(4 (b^3 + 6 b^2 + 11 b + 6)^2)
that for a = 2 and b = 4 :
4 a (a^2 + 3 a + 2)^2 x - ((a + 1) (a + 2)^2 (a^6 - a^4 (3 b^2 + 12 b + 14) + a^2 (3 b^4
+ 24 b^3 + 72 b^2 + 96 b + 49)
17
8 (4 + 6 + 2)^2 x - ((3) (4)^2 (2^6 - 2^4 (3*16 + 12*4 + 14) + 2^2 (3 4^4 + 24 4^3 +
72 4^2 + 96 4 + 49)))
Input
Result
Plot
Geometric figure
Alternate form
Root
Properties as a real function
Domain
18
Range
Bijectivity
Derivative
Indefinite integral
- (b^3 + 6 b^2 + 11 b + 6)^2) tan^(-1)(x/a))/(2 a + 1) + (4 a (a + 1) (a + 2)^2 (a^6 + 6
a^5 + a^4 (-3 b^2 - 12 b + 1)
(-(4^3 + 6*16 + 11*4 + 6)^2) tan^(-1)(x/2)/(4 + 1) + (8 (3) (4)^2 (2^6 + 6 2^5 + 2^4
(-3 4^2 - 12 4 + 1)))
Input
19
Result
Plots (figures that can be related to the open strings)
Alternate forms
Roots
Properties as a real function
Domain
20
Range
Injectivity
Series expansion at x=0
Series expansion at x=-2 i
Series expansion at x=2 i
21
Series expansion at x=∞
Derivative
Indefinite integral
Limit
22
- 12 a^3 (b^2 + 4 b + 3) + a^2 (3 b^4 + 24 b^3 + 54 b^2 + 24 b - 20) + 6 a (b + 2)^2
(b^2 + 4 b + 2) - b (b + 2)^2 (b^3 + 8 b^2 + 19 b + 12))
- 12*8 (16 + 16 + 3) + 4 (3 4^4 + 24 4^3 + 54 4^2 + 24 4 - 20) + 12 (6)^2 (16 + 16 +
2) - 4 (6)^2 (4^3 + 8*16 + 19 *4+ 12))
Input
Result
-16016
tan^(-1)(x/(a + 1)))/(4 a^2 + 8 a + 3) - (a (a^2 + 3 a + 2) (a^6 + 12 a^5 + a^4 (-3 b^2 -
12 b + 46) - 24 a^3 (b^2 + 4 b - 2) + a^2 (3 b^4 + 24 b^3 - 192 b - 47) + 12 a (b^4 + 8
b^3 + 16 b^2 - 5)
tan^(-1)(x/(3))/(16 + 16 + 3) - (2 (4 + 6 + 2) (2^6 + 12 2^5 + 16 (-3*16 - 12*4 + 46) -
24*8 (16 + 16 - 2) + 2^2 (3 4^4 + 24 4^3 - 192 4 - 47) + 24 (4^4 + 8 4^3 + 16*16 -
5)))
Input
Result
23
Plots (figures that can be related to the open strings)
Alternate forms
Roots
Properties as a real function
Domain
Range
24
Injectivity
Series expansion at x=0
Series expansion at x=-3 i
Series expansion at x=3 i
25
Series expansion at x=∞
Derivative
Indefinite integral
Limit
26
Alternative representations
27
Series representations
28
Integral representations
29
- b (b^5 + 12 b^4 + 46 b^3 + 48 b^2 - 47 b - 60)) tan^(-1)(x/(a + 2)))/(2 a + 3)))/(4
(b^3 + 6 b^2 + 11 b + 6)^2)
-(((( 4 (4^5 + 12 4^4 + 46 4^3 + 48 4^2 - 47 4 - 60)) tan^(-1)(x/(4)))/(4 + 3)))/(4 (4^3
+ 6 4^2 + 11 4 + 6)^2)
Input
Result
Plots (figures that can be related to the open strings)
Alternate form
30
Root
Properties as a real function
Domain
Range
Injectivity
Parity
Series expansion at x=0
31
Series expansion at x=-4 i
Series expansion at x=4 i
Series expansion at x=∞
Derivative
32
Indefinite integral
Limit
Alternative representations
33
Series representations
Integral representations
34
In conclusion, we obtain:
(2 (1152 x - 665280) – (-8820 tan^(-1)(x/2) – 485376)+(16016)) ((1/35 tan^(-1)(x/3) -
583200))+(6/245 tan^(-1)(x/4))
Input
Result
35
Plots (figures that can be related to the open strings)
Alternate forms
36
Expanded forms
Series expansion at x=0
Series expansion at x=-2 i
37
Series expansion at x=2 i
38
Series expansion at x=-3 i
39
Series expansion at x=3 i
40
Series expansion at x=-4 i
Derivative
41
From the exact result
for x = 2, we obtain:
6/245 tan^(-1)(2/4) + (-583200 + 1/35 tan^(-1)(2/3)) (501392 + 2 (-665280 + 1152 2)
+ 8820 tan^(-1)(2/2))
Input
Exact Result
Decimal approximation
4.768434283412…*10
11
476843428341.27997543….
Alternate forms
42
Expanded form
Alternative representations
43
Series representations
44
Integral representations
45
From which, after some calculations, we obtain:
64ln(6/245 tan^(-1)(2/4) + (-583200 + 1/35 tan^(-1)(2/3)) (501392 + 2 (-665280 +
1152 2) + 8820 tan^(-1)(2/2)))+8
Input
Exact Result
Decimal approximation
1728.98905801473….
46
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
47
Series representations
48
Integral representations
49
(64ln(6/245 tan^(-1)(2/4) + (-583200 + 1/35 tan^(-1)(2/3)) (501392 + 2 (-665280 +
1152 2) + 8820 tan^(-1)(2/2)))+8)^1/15+(MRB const)^(1-1/(4π)+π)
Input
Exact Result
50
Decimal approximation
1.644937330602…. ≈ ζ(2) = π
2
/6 = 1.644934 (trace of the instanton shape and
Ramanujan Recurring Number)
Alternate forms
sqrt(1/((64ln(6/245 tan^(-1)(2/4) + (-583200 + 1/35 tan^(-1)(2/3)) (501392 + 2 (-
665280 + 1152 2) + 8820 tan^(-1)(2/2)))+8)^1/15+(MRB const)^(1-1/(4π)+π))*(4/3))
Input
51
Exact Result
Decimal approximation
0.900315422988658…. ≈ 0.9003163161571…. =
(DN Constant)
Alternate forms
52
(1/27((64ln(6/245 tan^(-1)(2/4) + (-583200 + 1/35 tan^(-1)(2/3)) (501392 + 2 (-
665280 + 1152 2) + 8820 tan^(-1)(2/2)))+8)-1))^2
Input
Exact Result
Decimal approximation
4095.948127048921…. ≈ 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)
53
Alternate forms
Expanded forms
Alternative representations
54
55
Series representations
56
Integral representations
And:
√π/2 Γ(a+1/2)/Γ(a) Γ(b+1)/Γ(b+1/2) Γ(b-a+1/2)/Γ(b-a+1)
Input
57
Exact result
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.
58
Contour plot
Roots
59
Series expansion at a=0
60
Series expansion at a=∞
Derivative
Local maximum
61
From the exact result
for a = 2 , b = 4 , we obtain:
(sqrt(π) Γ(1/2 + 2) Γ(1 + 4) Γ(1/2 - 2 + 4))/(2 Γ(2) Γ(1/2 + 4) Γ(1 - 2 + 4))
Input
Exact result
Decimal approximation
1.615676221846…. result that is a very good approximation to the value of the
golden ratio 1.618033988749… (Ramanujan Recurring Number)
Property
Alternative representations
62
Series representations
63
Integral representations
From the derivative result
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:
(((sqrt(π) Γ(a + 1/2) Γ(1 + b) Γ(-a + b + 1/2) (-polygamma(0, -a + b + 1/2) +
polygamma(0, (1 + b) - a) - polygamma(0, a) + polygamma(0, a + 1/2)))/(2 Γ(a) Γ(b
+ 1/2) Γ((1 + b) - a))))(1/3*√2*a^3)
64
Input
Exact result
Alternate form
Expanded form
65
Derivative
And:
(((sqrt(π) Γ(a + 1/2) Γ(1 + b) Γ(-a + b + 1/2) (-polygamma(0, -a + b + 1/2) +
polygamma(0, (1 + b) - a) - polygamma(0, a) + polygamma(0, a + 1/2)))/(2 Γ(a) Γ(b
+ 1/2) Γ((1 + b) - a))))(4/3*π*(a/2)^3)
Input
66
Exact result
Alternate form
Expanded form
67
Derivative
Dividing the two alternate forms
and
we obtain, simplifying:
((sqrt(Pi/2))/3)*1/((Pi^(3/2))/12)
68
Input
Exact result
Decimal approximation
0.9003163161571…. =
(DN Constant)
Property
Series representations
69
We have that:
If
Then
is a complicated function. …
;
;
;
;
;
;
;
From
70
We consider:
sinnx/(e^(2π√x)-1)
Input
Values
3D plots
Real part (figures that can be related to the D-branes/Instantons)
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.
71
Imaginary part
72
Plots (figures that can be related to the open strings)
Contour plots
Real part
73
Imaginary part
Alternate forms
Roots
Property as a real function
Domain
74
Series expansion at x=0
Derivative
Limit
From series expansion at x=0
we obtain:
(n sqrt(x))/(2 π)-(n x)/2+1/6 π n x^(3/2)-((15 n^3 + 2 π^4 n) x^(5/2))/(180π)+(n^3
x^3)/12+((π^5 n)/945-(π n^3)/36) x^(7/2)+((315 n^5+140 π^4 n^3-8 π^8 n)
x^(9/2))/(75600 π)-(n^5 x^5)/240+(x^(11/2))
75
Input
Exact result
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.
76
Contour plot
Alternate forms
77
Expanded form
Property as a real function
Domain
Series expansion at x=0
Derivative
78
Indefinite integral
We analyze also:
And we consider:
φ(n)-1/(2n)+φ(π^2/n) √((2π^3)/n^3 )
Input
Exact result
79
Alternate form
From the exact result
for
we obtain:
sqrt(2) π^(3/2) sqrt(1/((2π)/5)^3) golden ratio(π^2/((2π)/5)) + (8-3√5)/16 - 1/(2
((2π)/5))
Input
Exact result
80
Decimal approximation
70.72278114333….
Property
Alternate forms
Expanded form
81
Series representations
82
From which, after some calculations
((sqrt(2) π^(3/2) sqrt(1/((2π)/5)^3) golden ratio(π^2/((2π)/5)) + (8-3√5)/16 - 1/(2
((2π)/5)))-1/√2-6)^2-2
Input
83
Exact result
Decimal approximation
4096.006564040226…. ≈ 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)
Property
Alternate forms
84
Expanded form
Series representations
85
86
27sqrt(((sqrt(2) π^(3/2) sqrt(1/((2π)/5)^3) golden ratio(π^2/((2π)/5)) + (8-3√5)/16 -
1/(2 ((2π)/5)))-1/√2-6)^2-2)+1
Input
Exact result
Decimal approximation
1729.00138460168….
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)
87
Property
Alternate forms
Expanded form
88
Series representations
89
(27sqrt(((sqrt(2) π^(3/2) sqrt(1/((2π)/5)^3) golden ratio(π^2/((2π)/5)) + (8-3√5)/16 -
1/(2 ((2π)/5)))-1/√2-6)^2-2)+1)^1/15+(MRB const)^(1-1/(4π)+π)
Input
Exact result
Decimal approximation
1.644938111889…. ≈ ζ(2) = π
2
/6 = 1.644934 (trace of the instanton shape and
Ramanujan Recurring Number)
90
Alternate forms
Expanded form
91
sqrt(1/( (MRB const)^(1 - 1/(4 π) + π) + (27 sqrt((25/4 sqrt(5) π φ - 6 - 1/sqrt(2) +
1/16 (8 - 3 sqrt(5)) - 5/(4 π))^2 - 2) + 1)^(1/15))*(4/3))
Input
Exact result
Decimal approximation
0.9003152091797…. ≈ 0.9003163161571…. =
(DN Constant)
Alternate forms
92
All 2
nd
roots of 4/(3 (C
MRB
1-1/(4 π)+π
+(27 sqrt((25/4 sqrt(5) π ϕ-6-1/sqrt(2)+1/16
(8-3 sqrt(5))-5/(4 π))
2
-2)+1)
1/15
))
93
(6π√2)sqrt(1/((MRB const)^(1-1/(4π)+π)+(27sqrt((25/4 sqrt(5) π φ-6-1/sqrt(2)+1/16
(8-3sqrt(5))-5/(4π))^2-2)+1)^(1/15))*(4/3))
Input
Exact result
Decimal approximation
23.9999704909…. ≈ 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).
94
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
95
Expanded form
(2π√2)√(1/((MRB const)^(1-1/(4π)+π)+(27√((25/4 √(5) π φ-6-1/√(2)+1/16 (8-3√(5))-
5/(4π))^2-2)+1)^(1/15))*(4/3))
Input
96
Exact result
Decimal approximation
7.999990163657962385…. ≈ 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.
Alternate forms
97
Expanded form
98
From the two previous exact results
and
we obtain also:
1/25*1/144((-2 + (-6 - 1/sqrt(2) + 1/16 (8 - 3 sqrt(5)) - 5/(4 π) + 25/4 sqrt(5) φ
π)^2)+(27 sqrt((25/4 sqrt(5) π φ - 6 - 1/sqrt(2) + 1/16 (8 - 3 sqrt(5)) - 5/(4 π))^2 - 2) +
1))
Input
Exact result
99
Decimal approximation
1.61805776351164…. result that is a very good approximation to the value of the
golden ratio 1.618033988749… (Ramanujan Recurring Number)
Alternate forms
100
Expanded form
Series representations
101
102
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
103
Result
Decimal approximation
0.900316316157106…. =
(DN Constant)
Property
Series representations
104
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)
105
Property
Series representations
Integral representations
106
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
107
Decimal approximation
1.64529737852…. ≈ ζ(2) = π
2
/6 = 1.644934 (trace of the instanton shape and
Ramanujan Recurring Number)
Alternate forms
Expanded forms
108
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
109
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
110
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
111
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
112
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.
113
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):
114
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
115
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
116
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
117
Series representations
118
From
we obtain also:
sqrt(6(1/3*(2/(((2sqrt2)/Pi)))^2))
Input
Exact result
Decimal approximation
3.14159265358… = π
119
Property
All 2
nd
roots of π
2
Series representations
Integral representations
120
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
121
Decimal approximation
0.900316316157106…. =
(DN Constant)
Property
All 2
nd
roots of 8/π
2
Series representations
122
123
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
124
“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
125
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
126
We have, in certain cases, the following connections:
Fig. 1
Fig. 2
127
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
129
Where ζ(2+it) :
Input
Plots
Roots
130
Series expansion at t=0
Alternative representations
Series representations
131
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:
132
we obtain:
2Pi/(ln(2))
Input:
Exact result:
Decimal approximation:
9.06472028365….
Alternative representations:
133
Series representations:
Integral representations:
134
From which:
(2Pi/(ln(2)))*(1/12 π log(2))
Input:
Exact result:
Decimal approximation:
1.6449340668…. = ζ(2) =
135
From:
Modular equations and approximations to - Srinivasa Ramanujan - Quarterly
Journal of Mathematics, XLV, 1914, 350 – 372
We have that:
136
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
137
Minimal polynomial
Expanded forms
Series representations
138
139
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
140
Minimal polynomial
Expanded forms
Series representations
141
142
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
143
Minimal polynomial
Expanded forms
All 15th roots of ϕ + 27 (-2 + 2 sqrt(1093) - (sqrt(10 - 2 sqrt(5)) - 2)/(2 (sqrt(5) -
1)))
144
Series representations
145
Integral representation
146
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:
147
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:
148
Now, we have the following calculations:
= 1.6272016… * 10^-6
from which:
= 1.6272016… * 10^-6
0.000244140625
=
= 1.6272016… * 10^-6
Now:
And:
149
(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:
150
Result:
0.00666501785…
Series representations:
Now:
= 0.0066650177536
=
= 0.00666501785…
151
From:
ln(0.00666501784619)
Input interpretation:
Result:
-5.010882647757…
Alternative representations:
Series representations:
152
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:
153
(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:
154
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:
155
(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:
156
Solutions:
Alternate forms:
Expanded form:
Alternate form assuming b, M, and V are positive:
Alternate form assuming b, M, and V are real:
157
Derivative:
Implicit derivatives
Global minimum:
158
Global minima:
From:
we obtain:
(225.913 (-0.054323 M^2 + 6.58545×10^-10 sqrt(M^4)))/M^2
Input interpretation:
159
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:
160
Expanded form:
Property as a function:
Parity
Series expansion at M = 0:
Series expansion at M = ∞:
Derivative:
161
Indefinite integral:
Global maximum:
Global minimum:
Limit:
Definite integral after subtraction of diverging parts:
162
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:
163
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:
164
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:
165
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
166
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
167
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:
168
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...
169
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
170
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
171
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
172
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
173
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
174
References
Bruce Berndt – “Ramanujan: a century of inspiration”
https://www.youtube.com/watch?v=VRRRi1WcPJs&t=13s
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