1
On the possible mathematical connections between the κ formula regarding the
Riemann Zeta Function, some Ramanujan equations, the study of other Riccati
equations and various topics of String Theory, in particular the Supersymmetry
Breaking. III
Michele Nardelli
1
, Antonio Nardelli
2
Abstract
In this research thesis (part III), we describe the possible mathematical connections
between the κ formula regarding the Riemann Zeta Function, some Ramanujan
equations, the study of other Riccati equations and various topics of String Theory, in
particular the Supersymmetry Breaking.
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
Conte Jacopo Riccati
Mathematician
(1676 – 1754)
Vesuvius landscape with gorse – Naples
4
5
From:
On the Zeros of the Davenport Heilbronn Function
S. A. Gritsenko - Received May 15, 2016 - ISSN 0081-5438, Proceedings of the
Steklov Institute of Mathematics, 2017, Vol. 296, pp. 65–87.
We have:
(
10 2
5 2) (
5 1 ) =
6
Input:
Decimal approximation:
0.28407904384…. = κ
Alternate forms:
Minimal polynomial:
Expanded forms:
7
For ((((√(10-2√5) -2))⁄((√5-1)))) = 8πG; G = 0.011303146014
Indeed:
((((√(10-2√5) -2))⁄((√5-1))))/(8π)
Input:
Result:
Decimal approximation:
0.01130314…. = g (gravitational coupling constant)
Property:
8
Alternate forms:
Expanded forms:
Series representations:
9
We note that:
(((√(10-2√5) -2))⁄((√5-1)))*((2 i (sqrt(5) - 1) t + sqrt(5) - 1)/(2 (sqrt(2 (5 - sqrt(5))) -
2)))
Input:
Exact result:
10
Plot:
Alternate form assuming t>0:
Alternate forms:
11
1/2+it = real part of every nontrivial zero of the Riemann zeta function
Derivative:
Indefinite integral:
And again:
(((√(10-2√5) -2))⁄((2x)))*((2 i (sqrt(5) - 1) t + sqrt(5) - 1)/(2 (sqrt(2 (5 - sqrt(5))) - 2)))
= (1/2+it)
Input:
12
Exact result:
Alternate form assuming t and x are real:
Alternate form:
Alternate form assuming t and x are positive:
Expanded forms:
13
Solutions:
Input:
Decimal approximation:
0.6180339887…. =
1
Solution for the variable x:
Implicit derivatives:
14
From:
Opere, vol. 3, Jacopo Riccati, In Lucca, presso Iacopo Giusti, 1764.
15
16
17
18
From:
Subtracting the second integral to the first, we obtain:
19
3
2
3
2
3=
2
Where
πr = APQ* ((π r)/(A P Q))
3((APQ* ((π r)/(A P Q)))) =
2
3((APQ* ((π r)/(A P Q)))) = 3πr
Input:
Result:
3((APQ* ((π r)/(A P Q))))
Input:
Result:
3πr
20
Plot:
Geometric figure:
Property as a function:
Parity
Derivative:
Indefinite integral:
Definite integral over a hypercube of edge length 2 L:
For r = 4/3, we obtain:
3((APQ* ((π*4/3)/(A P Q))))
21
Input:
Result:
4π
Decimal approximation:
12.566370614…. = S
BH
black hole entropy
Property:
Alternative representations:
22
Series representations:
Integral representations:
23
From:
For:
3
2
3
2
3=
2
y*x = (sqrt(3 π) sqrt(r))
Input:
Geometric figure:
Solutions:
And:
24
-3/2x+y^3/z+2Pi*r – ((sqrt(3 π) sqrt(r))/x*( (sqrt(3 π) sqrt(r))/y))^2
Input:
Exact result:
Alternate form:
Root:
Properties as a function:
Domain
Range
25
Derivative:
Indefinite integral:
Limit:
Series representations:
26
Integral representations:
From:
For x = 2, y = 3, r = 4/3:
27
2 π *(4/3) - (3 *2)/2 - (9 π^2 (4/3)^2)/(2^2 *3^2) + 3^3/z
Input:
Result:
Plots:
Complex map:
28
Alternate forms:
Root:
Branch points:
Derivative:
Indefinite integral:
29
Limit:
Alternative representations:
Series representations:
30
Integral representations:
For z = 11.9:
2 π *(4/3) - (3 *2)/2 - (9 π^2 (4/3)^2)/(2^2 *3^2) + 3^3/11.9
Input:
Result:
3.25999712766….
31
Alternative representations:
Series representations:
32
Integral representations:
For z = 92.1:
2 π *(4/3) - (3 *2)/2 - (9 π^2 (4/3)^2)/(2^2 *3^2) + 3^3/92.1
Input:
33
Result:
1.284249173….
Alternative representations:
Series representations:
34
Integral representations:
35
From the difference between the two previous expressions, after some calculations,
we obtain:
1+1/3[((2 π *(4/3) - (3 *2)/2 - (9 π^2 (4/3)^2)/(2^2 *3^2) + 3^3/11.9))-((2 π *(4/3) -
(3 *2)/2 - (9 π^2 (4/3)^2)/(2^2 *3^2) + 3^3/92.1))]
Input:
Result:
1.6585826513…. result very near to the 14th root of the following Ramanujan’s class
invariant =
505
/
101/5
3
= 1164.2696 i.e. 1.65578...
Rational approximation:
Alternative representation:
36
And:
sqrt[[((2 π *(4/3) - (3 *2)/2 - (9 π^2 (4/3)^2)/(2^2 *3^2) + 3^3/11.9))*1/((2 π *(4/3) -
(3 *2)/2 - (9 π^2 (4/3)^2)/(2^2 *3^2) + 3^3/92.1))]+1/4((((√(10-2√5) -2))⁄((√5-1))))]
Input:
Result:
1.61538407868…. result that is a very good approximation to the value of the golden
ratio 1.618033988749...
All 2nd roots of 2.60947:
37
Series representations:
38
((sqrt[[((2 π *(4/3) - (3 *2)/2 - (9 π^2 (4/3)^2)/(2^2 *3^2) + 3^3/11.9))*1/((2 π *(4/3)
- (3 *2)/2 - (9 π^2 (4/3)^2)/(2^2 *3^2) + 3^3/92.1))]+1/4((((√(10-2√5) -2))⁄((√5-
1))))]-1))^1/32
Input:
Result:
0.9849424…. result very near to the value of the following Rogers-Ramanujan
continued fraction:
and to the Omega mesons ( ) Regge
slope value (0.988) connected to the dilaton scalar field . =
1
above the two low-lying pseudo-scalars. (bound states of gluons, or ’glueballs’)
(Glueball Regge trajectories - Harvey Byron Meyer, Lincoln College -Thesis
submitted for the degree of Doctor of Philosophy at the University of Oxford Trinity
Term, 2004)
Note that
39
= 0.9863870313564812915… (π
2
/6 – 1)^1/32
From:
3
2
3
2
3=
2
A = ((sqrt(3 π) sqrt(r))/2 ; a = ((sqrt(3 π) sqrt(r))/3))
=
2
/3
((((sqrt(3 π) sqrt(r))/2 * ((sqrt(3 π) sqrt(r))/3))))^2 * 1/(3*(((sqrt(3 π)
sqrt(r))/2)^2*((sqrt(3 π) sqrt(r))/3)))
Input:
Exact result:
40
1/3 sqrt(π/3) sqrt(r)
A = ((sqrt(3 π) sqrt(r))/2 ; a = ((sqrt(3 π) sqrt(r))/3))
=
2
/3 = 1/3 sqrt(π/3) sqrt(r)
-2 * ((sqrt(3 π) sqrt(r))/3))^2 * (1/3 sqrt(π/3) sqrt(r))+ (((sqrt(3 π) sqrt(r))/2)^2* ((1/3
sqrt(π/3) sqrt(r))/Q)
Input:
Exact result:
-(2 π^(3/2) r^(3/2))/(9 sqrt(3)) + (π^(3/2) r^(3/2))/(4 sqrt(3) Q)
+(((sqrt(3 π) sqrt(r))/3)))^3* ((1/3 sqrt(π/3) sqrt(r))/Q)^3 * 1/(((sqrt(3 π) sqrt(r))/2 *
((sqrt(3 π) sqrt(r))/3)))
Input:
Exact result:
41
(2 π^2 r^2)/(729 Q^3)
- (((sqrt(3 π) sqrt(r))/2) (1/3 sqrt(π/3) sqrt(r))* (((sqrt(3 π) sqrt(r))/2 * ((sqrt(3 π)
sqrt(r))/3)))
Input:
Exact result:
-1/12 π^2 r^2
-(2 π^(3/2) r^(3/2))/(9 sqrt(3)) + (π^(3/2) r^(3/2))/(4 sqrt(3) Q)+(2 π^2 r^2)/(729
Q^3)-1/12 π^2 r^2
Input:
Exact result:
42
We have:
-(2 π^(3/2) (4/3)^(3/2))/(9 sqrt(3)) +(2 π^2 (4/3)^2)/(729 Q^3) + (π^(3/2)
(4/3)^(3/2))/(4 sqrt(3) Q) - (2 π^(3/2) (4/3)^(3/2))/(9 sqrt(3)) - (π^2
(4/3)^2)/12(π^(3/2) (4/3)^(3/2))/(4 sqrt(3) Q)
Input:
Exact result:
Plots:
43
Alternate forms:
Real root:
Complex roots:
Derivative:
44
Indefinite integral:
Local maximum:
Local minimum:
45
Limit:
Series representations:
Integral representations:
46
(2 π^2 (4/3)^2)/(729 Q^3)-1/12 π^2 (4/3)^2
Input:
Result:
47
Plots:
Alternate forms:
Real root:
48
Complex roots:
Derivative:
Indefinite integral:
Limit:
Alternative representations:
49
Series representations:
Integral representations:
50
(32 π^2)/(6561 Q^3) - (8 π^(7/2))/(243 Q) + (2 π^(3/2))/(9 Q) - (32 π^(3/2))/81 + (32
π^2)/(6561 Q^3) - (4 π^2)/27
Input:
Exact result:
Plots:
Alternate forms:
51
Real root:
Complex roots:
Derivative:
52
Indefinite integral:
Local maximum:
Local minimum:
Limit:
53
Alternative representations:
Series representations:
54
Integral representations:
From:
55
we obtain, for Q = 5.3:
(32 π^2)/(6561 5.3^3) - (8 π^(7/2))/(243 *5.3) + (2 π^(3/2))/(9 *5.3) - (32 π^(3/2))/81
+ (32 π^2)/(6561 5.3^3) - (4 π^2)/27
Input:
Result:
-3.7692528951….
Alternative representations:
56
Integral representations:
For Q = 1.1:
(32 π^2)/(6561 1.1^3) - (8 π^(7/2))/(243 *1.1) + (2 π^(3/2))/(9 *1.1) - (32 π^(3/2))/81
+ (32 π^2)/(6561 1.1^3) - (4 π^2)/27
Input:
57
Result:
-4.10955937166….
Alternative representations:
Integral representations:
58
From the two results, after some calculations:
1/2((3.7692528951448984*4.1095593716602807-ln(196883)))
Note that 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 = e
2πiτ
(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:
60
Series representations:
Integral representations:
61
((1/2((3.7692528951448984*4.1095593716602807-ln(196883)))-1))^1/32
Input interpretation:
Result:
0.986618838…. result very near to the value of the following Rogers-Ramanujan
continued fraction:
and to the Omega mesons ( ) Regge
slope value (0.988) connected to the dilaton scalar field . =
1
above the two low-lying pseudo-scalars. (bound states of gluons, or ’glueballs’)
(Glueball Regge trajectories - Harvey Byron Meyer, Lincoln College -Thesis
submitted for the degree of Doctor of Philosophy at the University of Oxford Trinity
Term, 2004)
62
Note that
= 0.9863870313564812915… (π
2
/6 – 1)^1/32
We have:
63
64
From:
2/3*cg * u^2/2
i.e.
2/3*a*v * b^2/2
65
Input:
Result:
Geometric figure:
Roots:
Polynomial discriminant:
Property as a function:
Parity
Derivative:
66
Indefinite integral:
Definite integral over a sphere of radius R:
Definite integral over a cube of edge length 2 L:
From:
(2/3*(cx)^(3/2))/sqrt(v) *b^2/2
Input:
Result:
Expanded form:
67
Real roots:
Series expansion at x = 0:
Series expansion at x = ∞:
Derivative:
68
Indefinite integral:
Global minimum:
Limit:
Series representations:
69
From:
b = 2; c = 3; x = 1; v = 16
(4*3^1.5)/(3sqrt16)
Input:
Result:
1.7320508075…. = √3
Possible closed forms:
From:
70
integrate(((2/3*(cx)^(3/2))/sqrt(v) *b^2/2 ))
Indefinite integral:
Alternate form assuming b, c, v, and x are positive:
Expanded form of the integral:
Series expansion of the integral at x = 0:
Series expansion of the integral at x = ∞:
From:
71
(2*4*3^1.5*1)/(15*sqrt16)
Input:
Result:
0.692820….
For the value 92.1 as previously calculated:
((2*4*3^1.5*1)/(15*sqrt16))92.1
Input:
Result:
63.8088…. ≈ 64
2/3*3*16*2
Input:
Exact result:
64
72
(2/3*3*16*2)^2
Input:
Exact result:
4096 = 64
2
(2/3*3*16*2)^2+276
Input:
Exact result:
4372
Where 4372 is a value indicated in the fundamental Ramanujan paper “Modular
equations and Approximations to π”
73
And adding the value ((((√(10-2√5) -2))⁄((√5-1)))) = κ:
((2*4*3^1.5*1)/(15*sqrt16))92.1+((((√(10-2√5) -2))⁄((√5-1))))
Input:
Result:
64.0928…. ≈ 64
27((((2*4*3^1.5*1)/(15*sqrt16))92.1+((((√(10-2√5) -2))⁄((√5-1))))))-3/2
Input:
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. The number 1728 is one less than the Hardy–Ramanujan number 1729
(taxicab number)
74
((27((((2*4*3^1.5*1)/(15*sqrt16))92.1+((((√(10-2√5) -2))⁄((√5-1))))))-3/2))^1/15
Input:
Result:
1.643815636…. ≈ ζ(2) =
2
6
= 1.644934
75
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
76
“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
77
Mathematical connections with some sectors of String Theory
From:
Modular equations and approximations to - Srinivasa Ramanujan
Quarterly Journal of Mathematics, XLV, 1914, 350 – 372
We have that:
78
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:
6+
= 4096
18
Therefore, with respect to the exponentials of the vacuum equations, the Ramanujan’s
exponential has a coefficient of 4096 which is equal to 642, while -6C+ is equal to -
18. From this it follows that it is possible to establish mathematically, the dilaton
value.
For
79
exp((-Pi*sqrt(18)) we obtain:
Input:
Exact result:
Decimal approximation:
1.6272016… * 10
-6
Property:
Series representations:
Now, we have the following calculations:
6+
= 4096
18
18
= 1.6272016… * 10^-6
80
from which:
1
4096
6+
= 1.6272016… * 10^-6
0.000244140625
6+
=
18
= 1.6272016… * 10^-6
Now:
ln
18
= 13.328648814475 =
18
And:
(1.6272016* 10^-6) *1/ (0.000244140625)
Input interpretation:
Result:
0.006665017...
Thence:
0.000244140625
6+
=
18
81
Dividing both sides by 0.000244140625, we obtain:
0.000244140625
0.000244140625
6+
=
1
0.000244140625
18
6+
= 0.0066650177536
((((exp((-Pi*sqrt(18)))))))*1/0.000244140625
Input interpretation:
Result:
0.00666501785…
Series representations:
Now:
82
6+
= 0.0066650177536
=
= 0.00666501785…
From:
ln(0.00666501784619)
Input interpretation:
Result:
-5.010882647757…
Alternative representations:
83
Series representations:
Integral representation:
In conclusion:
6+ = 5.010882647757
and for C = 1, we obtain:
= 5.010882647757 + 6 = . =
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 0.989117352243, are also connected to
the following two Rogers-Ramanujan continued fractions:
84
(http://www.bitman.name/math/article/102/109/)
The mean between the two results of the above Rogers-Ramanujan continued
fractions is 0.97798855285, value very near to the ψ Regge slope 0.979:
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:
85
Result:
0.99040073.... result very near to the dilaton value . = and to
the value of the following Rogers-Ramanujan continued fraction:
From
AdS Vacua from Dilaton Tadpoles and Form Fluxes - J. Mourad and A. Sagnotti
- arXiv:1612.08566v2 [hep-th] 22 Feb 2017 - March 27, 2018
We have:
For
ξ = 1
we obtain:
(2*e^(0.989117352243/2)) / (1+sqrt(((1-1/3*16/(Pi)^2*e^(2*0.989117352243)))))
86
Input interpretation:
Result:
Polar coordinates:
1.65919106525….. result very near to the 14th root of the following Ramanujan’s
class invariant =
505
/
101/5
3
= 1164.2696 i.e. 1.65578...
Series representations:
87
From
we obtain:
e^(4*0.989117352243) / (((1+sqrt(1-1/3*16/(Pi)^2*e^(2*0.989117352243)))))^7
[42(1+sqrt(1-
1/3*16/(Pi)^2*e^(2*0.989117352243)))+5*16/(Pi)^2*e^(2*0.989117352243)]
Input interpretation:
Result:
Polar coordinates:
54.76072411…..
88
Series representations:
89
From which:
e^(4*0.989117352243) / (((1+sqrt(1-1/3*16/(Pi)^2*e^(2*0.989117352243)))))^7
[42(1+sqrt(1-
1/3*16/(Pi)^2*e^(2*0.989117352243)))+5*16/(Pi)^2*e^(2*0.989117352243)]*1/34
Input interpretation:
Result:
Polar coordinates:
1.610609533…. result that is a good approximation to the value of the golden ratio
1.618033988749...
90
Series representations:
91
Now, we have:
For:
ξ = 1
= 0.989117352243
92
From
we obtain:
((2*e^(-0.989117352243/2))) /
((((1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243))))))))
Input interpretation:
Result:
0.382082347529….
Series representations:
93
From which:
1+1/(((4((2*e^(-0.989117352243/2))) /
((((1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243)))))))))))
Input interpretation:
Result:
1.6543092….. We note that, the result 1.6543092... is very near to the 14th root of the
following Ramanujan’s class invariant =
505
/
101/5
3
= 1164.2696 i.e.
1.65578...
Indeed:
94
113+5
505
8
+
105+5
505
8
3
14
= 1,65578
Series representations:
95
And from
we obtain:
e^(-4*0.989117352243) / [1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243)))]^7 *
[42(1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243)))-
13*(4Pi^2)/25*e^(2*0.989117352243)]
Input interpretation:
96
Result:
-0.034547055658…
Series representations:
97
From which:
47 *1/(((-1/(((((e^(-4*0.989117352243) /
[1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243))))]^7 *
[42(1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243))))-
13*(4Pi^2)/25*e^(2*0.989117352243))]))))))))
98
Input interpretation:
Result:
1.6237116159…. result that is an approximation to the value of the golden ratio
1.618033988749...
Series representations:
99
100
And again:
32((((e^(-4*0.989117352243) /
[1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243))))]^7 *
[42(1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243))))-
13*(4Pi^2)/25*e^(2*0.989117352243))]))))
Input interpretation:
Result:
-1.1055057810….
We note that the result -1.1055057810…. is very near to the value of Cosmological
Constant, less 10
-52
, thence 1.1056, with minus sign
101
Series representations:
102
103
And:
-[32((((e^(-4*0.989117352243) /
[1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243))))]^7 *
[42(1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243))))-
13*(4Pi^2)/25*e^(2*0.989117352243))]))))]^5
Input interpretation:
Result:
1.651220569…. result very near to the 14th root of the following Ramanujan’s class
invariant =
505
/
101/5
3
= 1164.2696 i.e. 1.65578...
104
Series representations:
105
106
We obtain also:
-[32((((e^(-4*0.989117352243) /
[1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243))))]^7 *
[42(1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243))))-
13*(4Pi^2)/25*e^(2*0.989117352243))]))))]^1/2
Input interpretation:
Result:
Polar coordinates:
1.05143035007
107
Series representations:
108
109
1 / -[32((((e^(-4*0.989117352243) /
[1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243))))]^7 *
[42(1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243))))-
13*(4Pi^2)/25*e^(2*0.989117352243))]))))]^1/2
Input interpretation:
Result:
Polar coordinates:
0.95108534763
We know that the primordial fluctuations are consistent with Gaussian purely
adiabatic scalar perturbations characterized by a power spectrum with a spectral
index n
s
= 0.965 ± 0.004, consistent with the predictions of slow-roll, single-field,
inflation.
Thence 0.95108534763 is a result very near to the spectral index n
s
, to the mesonic
Regge slope, to the inflaton value at the end of the inflation 0.9402 and to the value
of the following Rogers-Ramanujan continued fraction:
110
Series representations:
111
112
From the previous expression
= -0.034547055658…
we have also:
113
1+1/(((4((2*e^(-0.989117352243/2))) /
((((1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243))))))))))) + (-0.034547055658)
Input interpretation:
Result:
1.61976215705….. result that is a very good approximation to the value of the golden
ratio 1.618033988749...
Series representations:
114
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
115
References
Complex Analysis in Number Theory – 22.11.1994 - Anatoly A. Karatsuba
On the Zeros of the Davenport Heilbronn Function
S. A. Gritsenko - Received May 15, 2016 - ISSN 0081-5438, Proceedings of the
Steklov Institute of Mathematics, 2017, Vol. 296, pp. 65–87.
Opere, vol. 3, Jacopo Riccati, In Lucca, presso Iacopo Giusti, 1764.
Modular equations and approximations to - Srinivasa Ramanujan
Quarterly Journal of Mathematics, XLV, 1914, 350 – 372
An Update on Brane Supersymmetry Breaking
J. Mourad and A. Sagnotti - arXiv:1711.11494v1 [hep-th] 30 Nov 2017
March 27, 2018
AdS Vacua from Dilaton Tadpoles and Form Fluxes
J. Mourad and A. Sagnotti - arXiv:1612.08566v2 [hep-th] 22 Feb 2017