1
On various equations concerning some topics of Field Theory and Gravity and
the Partition Functions. New possible mathematical connections with some
sectors of Number Theory, String Theory and Supersymmetry Breaking.
Michele Nardelli
1
, Antonio Nardelli
2
Abstract
In this research thesis, we analyze various equations concerning some topics of
Field Theory and Gravity and the Partition Functions. We describe new possible
mathematical connections with some sectors of Number Theory, String Theory and
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
3
We want to highlight that the development of the various equations was carried
out according an our possible logical and original interpretation
From:
Course of Field Theory and Gravity - Prof. Augusto Sagnotti (SNS Pisa-Italy)
We have that:
-i*(5e-8 * 0.510998950) [5.610e-28+ ((1729*i)/2)*(80379+91187.6) –
((1729*i)/2)*4180]
Input interpretation:
Result:
4
Alternate form:
3.69722
From which:
(((-i*(5e-8 * 0.510998950) [5.610e-28+ ((1729*i)/2)*(80379+91187.6) –
((1729*i)/2)*4180])))^1/neper number
Input interpretation:
Result:
Alternate form:
1.61774 result that is a very good approximation to the value of the golden ratio
1.618033988749...
From:
5
((1/sqrt2 * 80379 – 1/sqrt2 * 4180) / (1/sqrt2 * 91187.6 + 1/sqrt2 * 4180))
Input interpretation:
Result:
0.799003015699….
From which:
1+1/(((1/3((-i*(5e-8 * 0.510998950) [5.610e-28+ ((1729*i)/2)*(80379+91187.6) –
((1729*i)/2)*4180] 1/((((1/sqrt2 * 80379 – 1/sqrt2 * 4180) 1/ (1/sqrt2 * 91187.6 +
1/sqrt2 * 4180)))))))))
Input interpretation:
Result:
Alternate form:
6
1.64833 ≈ ζ(2) =
2
6
= 1.644934
((1/sqrt2 * 91187.6 + 1/sqrt2 * 5.610e-28) / (1/sqrt2 * 5.610e-28 - 1/sqrt2 *
91187.6))
Input interpretation:
Result:
-1
1-1/2[((1/sqrt2 * 91187.6 + 1/sqrt2 * 5.610e-28) / (1/sqrt2 * 5.610e-28 - 1/sqrt2 *
91187.6))]*1/((1/sqrt2 * 80379 – 1/sqrt2 * 4180) / (1/sqrt2 * 91187.6 + 1/sqrt2 *
4180))
Input interpretation:
Result:
1.625779865877…. result very near to the mean between ζ(2) =
2
6
= 1.644934
and the value of golden ratio 1.61803398…, i.e. 1.63148399
7
From:
-i/2 * sqrt(1729^2+1729^2) [91187.6-2*1/sqrt2*1/sqrt2*5.610e-28]
Input interpretation:
Result:
Polar coordinates:
1.11485*10
8
From which:
30 / ln(((-i/2 * sqrt(1729^2+1729^2) [91187.6-2*1/sqrt2*1/sqrt2*5.610e-28])))
Input interpretation:
Result:
8
Polar coordinates:
1.61326 result that is a very good approximation to the value of the golden ratio
1.618033988749...
From:
-i*(2.2*4.7) [(5.610e-28+i/2*1729*(80379+91187.6)+((1729*i)/6)*4180]-
i*2.2(5.610e-28+2/3*i*1729*4180)*2.2+i*1729*8-i*4.7(5.610e-28-
i/3*1729*4180)*4.7
Input interpretation:
Result:
Polar coordinates:
1.51618*10
9
9
From which:
22 / ((([-i*(2.2*4.7) [(5.610e-28+i/2*1729*(80379+91187.6)+((1729*i)/6)*4180)]-
i*2.2(5.610e-28+2/3*i*1729*4180)*2.2+i*1729*8-i*4.7(5.610e-28-
i/3*1729*4180)*4.7] / (1.11485×10^8 ))))
Where 22 = 2 * 11
Input interpretation:
Result:
Polar coordinates:
1.61766 result that is a very good approximation to the value of the golden ratio
1.618033988749...
From:
10
-i*1/4-i*1/4-i*0.510998950(1/4+1/4)+i/2*105.66(1/4+1/4)+i/2*105.66(1/4+1/4)
Input interpretation:
Result:
Polar coordinates:
52.0745
From which:
89/((((-i*1/4-i*1/4-
i*0.510998950(1/4+1/4)+i/2*105.66(1/4+1/4)+i/2*105.66(1/4+1/4)+3i))))
Input interpretation:
Result:
Polar coordinates:
1.61599 result that is a very good approximation to the value of the golden ratio
1.618033988749...
11
From:
We have:
(x+a)(x+b)-t^2
Input:
Geometric figure:
Alternate forms:
12
Real root:
Roots:
Polynomial discriminant:
Derivative:
Indefinite integral:
x^2+(a+b)x+(ab-t^2) = 0
Input:
Geometric figure:
13
Alternate forms:
Expanded form:
Real solution:
Solutions:
From:
14
1/2 (sqrt(a^2 - 2 a b + b^2 + 4 t^2) - a - b)
Input:
Alternate form:
Expanded form:
Real roots:
Properties as a real function:
Domain
Range
15
Parity
Roots for the variable t:
Series expansion at t = 0:
Series expansion at t = ∞:
Derivative:
Indefinite integral:
16
Global minimum:
Series representations:
RESULT
17
-1/2(a+b) + sqrt((1/4(a-b)^2+t^2))
Input:
Result:
Alternate forms:
Expanded form:
18
Real roots:
Properties as a real function:
Domain
Range
Parity
Roots for the variable t:
Series expansion at t = 0:
19
Series expansion at t = ∞:
Derivative:
Indefinite integral:
Global minimum:
Series representations:
20
RESULT
21
1/2((sqrt(105.658^2-2*105.658*0.13+0.13^2+4*0.510998950^2))-105.658-0.13)
Input interpretation:
Result:
-0.127525644….
From which, after some calculations, and considering κ = ((√(10-2√5) -2))⁄((√5-1)),
we obtain:
7/(((3Pi[-1/((((√(10-2√5) -2))⁄((√5-1)))) * ((1/2((sqrt(105.658^2-
2*105.658*0.13+0.13^2+4*0.510998^2))-105.658-0.13)))])))
Input interpretation:
Result:
1.65450678414…. result that 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, from:
22
113+5
505
8
+
105+5
505
8
3
14
= 1,65578
Series representations:
23
From:
24
-105.658/2-105.658/2*sqrt(1+4*((0.510998950^2)/(105.658^2)))
Input interpretation:
Result:
-105.6604713….
-105.658/2+105.658/2*sqrt(1+4*((0.510998950^2)/(105.658^2)))
Input interpretation:
Result:
0.00247131139….
1/((3((-105.658/2+105.658/2*sqrt(1+4*((0.510998950^2)/(105.658^2)))))))
Input interpretation:
25
Result:
134.8811541186…. result practically equal to the rest mass of Pion meson 134.9766
MeV
233 1/((1/((neper number((-
105.658/2+105.658/2*sqrt(1+4*((0.510998950^2)/(105.658^2)))))))-5))
Input interpretation:
Result:
1.6196300743…. result that is a very good approximation to the value of the golden
ratio 1.618033988749...
Series representations:
26
Now, we study the equation:
-1/2(a+b) + sqrt((1/4(a+b)^2-ab+t^2))
Input:
Result:
Alternate forms:
27
Expanded forms:
Real roots:
Properties as a real function:
Domain
Range
Parity
28
Roots for the variable t:
Series expansion at t = 0:
Series expansion at t = ∞:
Derivative:
Indefinite integral:
29
Global minimum:
Series representations:
30
From:
(2*5)/sqrt(1^2 - 2 *1 * 3 + 3^2 + 4 * 5^2)
Input:
Result:
Decimal approximation:
0.980580675…. 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’)
31
(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
= 0.9863870313564812915… (π
2
/6 – 1)^1/32
Alternate form:
From:
1/8 (2 t (sqrt(a^2 - 2 a b + b^2 + 4 t^2) - 2 a - 2 b) + (a - b)^2 log(sqrt(a^2 - 2 a b +
b^2 + 4 t^2) + 2 t)) + constant
1/8 (2*5 (sqrt(1^2 - 2*1*3 + 3^2 + 4 *5^2) - 2 - 2*3) + (1 - 3)^2 log(sqrt(1 - 2*1*3 +
3^2 + 4*5^2) + 2*5))
32
Input:
Exact result:
Decimal approximation:
4.250341544….
Property:
From which:
((1/8 (10 (-8 + 2 sqrt(26)) + 4 log(10 + 2 sqrt(26)))))^1/3
Input:
Exact result:
33
Decimal approximation:
1.61984929…. result that is a very good approximation to the value of the golden
ratio 1.618033988749...
Property:
Alternate forms:
All 3rd roots of 1/8 (10 (2 sqrt(26) - 8) + 4 log(10 + 2 sqrt(26))):
34
Alternative representations:
Series representations:
35
Integral representations:
36
4/((1/8 (10 (-8 + 2 sqrt(26)) + 4 log(10 + 2 sqrt(26)))))
Input:
Exact result:
Decimal approximation:
0.9411008404…. result 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:
From:
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. (Astronomy
& Astrophysics manuscript no. ms c ESO 2019 - September 24, 2019 - Planck 2018
results. VI. Cosmological parameters)
37
We know that α’ is the Regge slope (string tension). With regard the Omega mesons,
the values are:
Property:
Alternate forms:
38
Alternative representations:
Series representations:
39
Integral representation:
Now, we have:
40
=
From x – x’ = 11 – 7 = 4, we obtain:
e^(((i*0.510998950)/(2*6.582119569×10^-16))(11-7)^2)
sqrt(0.510998950/(2Pi*i*6.582119569×10^-16))
Input interpretation:
Result:
Polar coordinates:
1.11157*10
7
ln((e^(((i*0.510998950)/(2*6.582119569×10^-16))(11-7)^2)
sqrt(0.510998950/(2Pi*i*6.582119569×10^-16))))
Input interpretation:
41
Result:
Polar coordinates:
16.2306
1/Pi^2((ln((e^(((i*0.510998950)/(2*6.582119569×10^-16))(11-7)^2)
sqrt(0.510998950/(2Pi*i*6.582119569×10^-16))))))
Input interpretation:
Result:
Polar coordinates:
1.6445 ≈ ζ(2) =
2
6
= 1.644934
42
Or:
exp(((i*0.510998950)/(2*1.054571817e-34))(11-7)^2)
sqrt(0.510998950/(2Pi*i*1.054571817e-34))
Input interpretation:
Result:
Polar coordinates:
2.7770390663798312 × 10
16
Also for x – x’ = 13 – 5 = 8, we note that:
exp(((i*0.510998950)/(2*1.054571817e-34))(13-5)^2)
sqrt(0.510998950/(2Pi*i*1.054571817e-34))
Input interpretation:
Result:
Polar coordinates:
43
Scientific notation:
2.7770390663798312 × 10
16
as the above result
ln(((exp(((i*0.510998950)/(2*1.054571817e-34))(13-5)^2)
sqrt(0.510998950/(2Pi*i*1.054571817e-34)))))
Input interpretation:
Result:
Polar coordinates:
37.8702
Polar forms:
From which:
(76+2)/(((ln(((exp(((i*0.510998950)/(2*1.054571817e-34))(13-5)^2)
sqrt(0.510998950/(2Pi*i*1.054571817e-34)))))+sqrt89)))
44
Input interpretation:
Result:
Polar coordinates:
1.64895 ≈ ζ(2) =
2
6
= 1.644934
From:
ON THE PARTITION FUNCTION p(n).
By HANS RADEMACHER. - [Received 30 November, 1936.—Read 10 December,
1936.]
We have the following Partition formula:
45
1/(Pi*sqrt2) sum (e^-(2Pi*i*24)/k)*k^0.5 d/dn (((sinh((Pi*sqrt(2/3)*sqrt(24-
1/24))/k)*1/(sqrt(24-1/24)))), k = 1..infinity
1/(Pi*sqrt2) sum (e^-(2Pi*i*24)/k)*k^0.5 * 5.78696/24
(((sinh((Pi*sqrt(2/3)*sqrt(24-1/24))/k)*1/(sqrt(24-1/24))))), k = 1..infinity
Input interpretation:
Result:
1575
46
1/(Pi*sqrt2) sum (e^-(2Pi*i*48)/k)*k^0.5 * 8.38915/48
(((sinh((Pi*sqrt(2/3)*sqrt(48-1/24))/k)*1/(sqrt(48-1/24))))), k = 1..infinity
Input interpretation:
Result:
147273
1/(Pi*sqrt2) sum (e^-(2Pi*i*72)/k)*k^0.5 * 10.38564/72
(((sinh((Pi*sqrt(2/3)*sqrt(72-1/24))/k)*1/(sqrt(72-1/24))))), k = 1..infinity
Input interpretation:
Result:
5.39278*10
6
Or:
1/(Pi*sqrt2) sum (e^-(2Pi*i*72)/k)*k^0.5 * 24*(0.432735/72)
(((sinh((Pi*sqrt(2/3)*sqrt(72-1/24))/k)*1/(sqrt(72-1/24))))), k = 1..infinity
47
Input interpretation:
Result:
5.39278*10
6
1/(Pi*sqrt2) sum (e^-(2Pi*i*48)/k)*k^0.5 * 24*(0.349548/48)
(((sinh((Pi*sqrt(2/3)*sqrt(48-1/24))/k)*1/(sqrt(48-1/24))))), k = 1..infinity
Input interpretation:
Result:
147273
48
1/(Pi*sqrt2) sum (e^-(2Pi*i*24)/k)*k^0.5 * 24*(0.2411223/24)
(((sinh((Pi*sqrt(2/3)*sqrt(24-1/24))/k)*1/(sqrt(24-1/24))))), k = 1..infinity
Input interpretation:
Result:
1575
1/(1+0.97177972872462)((24*(0.2411223/24) + 24*(0.349548/48) +
24*(0.432735/72))) =((√(10-2√5) -2))⁄((√5-1))
Input interpretation:
Result:
1/(1+0.97177972872462)((24*(0.2411223/24) + 24*(0.349548/48) +
24*(0.432735/72)))
Where 0.97177972872462 is a possible slow-roll inflation parameter (spectral index)
Input interpretation:
49
Result:
0.28407904… = κ
((√(10-2√5) -2))⁄((√5-1)) = κ
Input:
Decimal approximation:
0.28407904… = κ
Alternate forms:
Minimal polynomial:
50
Expanded forms:
We have that:
From:
(44Pi^2)/(225sqrt3sqrt21)+(Pi*sqrt2)/(150)*(21/720)^1/2*exp(((Pi*sqrt(2/3)*(sqrt72
1/21))))
Input:
51
Exact result:
Decimal approximation:
0.377574688….
Alternate forms:
52
Series representations:
53
54
From
that is equal to
we obtain the following final expression:
55
1/300 sqrt(7/30) e^(1/3 sqrt(206/21) π) π + (44 π^2)/(675 sqrt(7)) + 22^1.5/(2
sqrt(3)×720) exp(-π (sqrt(2/3)×sqrt(721)/21)) (1/3 + (sqrt(3)×22)/(5 π sqrt(2)
sqrt(720)))
Input:
Result:
0.37819323…
Alternative representations:
56
57
(44Pi^2)/(225sqrt3sqrt18)+(Pi*sqrt2)/(150)*(18/598)^1/2*exp(((Pi*sqrt(2/3)*(sqrt59
9/18))))+19^1.5/(2 sqrt(3)×598) exp(-π (sqrt(2/3)×sqrt(598)/18)) (1/3 +
(sqrt(3)×19)/(5 π sqrt(2) sqrt(598)))
Input:
Result:
0.43123074….
Now, we consider ((√(10-2√5) -2))⁄((√5-1)) = κ
[((1/(Pi*sqrt2) sum (e^-(2Pi*i*24)/k)*k^0.5 *((√(10-2√5) -2))⁄((√5-1))
(((sinh((Pi*sqrt(2/3)*sqrt(24-1/24))/k)*1/(sqrt(24-1/24))))), k =
1..infinity))]*0.848788
58
Input interpretation:
Result:
1575
((([((1/(Pi*sqrt2) sum (e^-(2Pi*i*24)/k)*k^0.5 *((√(10-2√5) -2))⁄((√5-1))
(((sinh((Pi*sqrt(2/3)*sqrt(24-1/24))/k)*1/(sqrt(24-1/24))))), k =
1..infinity))]*0.848788)))^1/15
Input interpretation:
Result:
1.63362 result very near to the mean between ζ(2) =
2
6
= 1.644934 and the value
of golden ratio 1.61803398…, i.e. 1.63148399
59
[((1/(Pi*sqrt2) sum (e^-(2Pi*i*48)/k)*k^0.5 *((√(10-2√5) -2))⁄((√5-1))
(((sinh((Pi*sqrt(2/3)*sqrt(48-1/24))/k)*1/(sqrt(48-1/24))))), k =
1..infinity))]*0.615231
Input interpretation:
Result:
147273
((([((1/(Pi*sqrt2) sum (e^-(2Pi*i*48)/k)*k^0.5 *((√(10-2√5) -2))⁄((√5-1))
(((sinh((Pi*sqrt(2/3)*sqrt(48-1/24))/k)*1/(sqrt(48-1/24))))), k =
1..infinity))]*0.615231)))^1/24
Input interpretation:
60
Result:
1.64187 ≈ ζ(2) =
2
6
= 1.644934
For n = 599:
[((1/(Pi*sqrt2) sum (e^-(2Pi*i*599)/k)*k^0.5 *((√(10-2√5) -2))⁄((√5-1))
(((sinh((Pi*sqrt(2/3)*sqrt(599-1/24))/k)*1/(sqrt(599-1/24))))), k =
1..infinity))]*0.181536
Input interpretation:
Result:
4.3535*10
23
((([((1/(Pi*sqrt2) sum (e^-(2Pi*i*599)/k)*k^0.5 *((√(10-2√5) -2))⁄((√5-1))
(((sinh((Pi*sqrt(2/3)*sqrt(599-1/24))/k)*1/(sqrt(599-1/24))))), k =
1..infinity))]*0.181536)))^1/110
Input interpretation:
61
Result:
1.64021 ≈ ζ(2) =
2
6
= 1.644934
For n = 721:
[((1/(Pi*sqrt2) sum (e^-(2Pi*i*721)/k)*k^0.5 *((√(10-2√5) -2))⁄((√5-1))
(((sinh((Pi*sqrt(2/3)*sqrt(721-1/24))/k)*1/(sqrt(721-1/24))))), k =
1..infinity))]*0.165702
Input interpretation:
Result:
62
1.61062*10
26
((([((1/(Pi*sqrt2) sum (e^-(2Pi*i*721)/k)*k^0.5 *((√(10-2√5) -2))⁄((√5-1))
(((sinh((Pi*sqrt(2/3)*sqrt(721-1/24))/k)*1/(sqrt(721-1/24))))), k =
1..infinity))]*0.165702)))^1/121
Input interpretation:
Result:
1.64659 ≈ ζ(2) =
2
6
= 1.644934
[((1/(Pi*sqrt2) sum (e^-(2Pi*i*72)/k)*k^0.5 *((√(10-2√5) -2))⁄((√5-1))
(((sinh((Pi*sqrt(2/3)*sqrt(72-1/24))/k)*1/(sqrt(72-1/24))))), k =
1..infinity))]*0.507764
Input interpretation:
63
Result:
5.39278*10
6
((([((1/(Pi*sqrt2) sum (e^-(2Pi*i*72)/k)*k^0.5 *((√(10-2√5) -2))⁄((√5-1))
(((sinh((Pi*sqrt(2/3)*sqrt(72-1/24))/k)*1/(sqrt(72-1/24))))), k =
1..infinity))]*0.507764)))^1/31
Input interpretation:
64
Result:
1.64875 ≈ ζ(2) =
2
6
= 1.644934
From the following terms
0.848788; 0.615231; 0.181536; 0.165702, we obtain:
(0.848788 / 0.615231) - (0.181536 / 0.165702)
Input interpretation:
Result:
0.284067804…. ≈ κ
From the exact value of κ, we obtain as follows:
(x / 0.615231) - (0.181536 / 0.165702) = ((√(10-2√5) -2))⁄((√5-1))
Input interpretation:
Result:
65
Alternate forms:
Expanded forms:
Solution:
0.848795
[((√(10-2√5) -2))⁄((√5-1))]x = 0.848795
x ≈ 2.98788
(0.848788/ x) - (0.181536 / 0.165702) = ((√(10-2√5) -2))⁄((√5-1))
Input interpretation:
66
Result:
Plot:
Alternate form assuming x is real:
Alternate forms:
67
Alternate form assuming x is positive:
Expanded forms:
Solution:
0.615226
[((√(10-2√5) -2))⁄((√5-1))]x = 0.615226
x ≈ 2.16569
(0.848788/ 0.615231) - (x / 0.165702) = ((√(10-2√5) -2))⁄((√5-1))
Input interpretation:
Result:
68
Alternate forms:
Expanded forms:
Solution:
0.181534
[((√(10-2√5) -2))⁄((√5-1))]x = 0.181534
x ≈ 0.639026
69
(0.848788/ 0.615231) - (0.181536/ x) = ((√(10-2√5) -2))⁄((√5-1))
Input interpretation:
Result:
Plot:
Alternate form assuming x is positive:
Expanded forms:
70
Solution:
0.165704
[((√(10-2√5) -2))⁄((√5-1))]x = 0.165704
x ≈ 0.583302
Thence, we have:
(2.98788 - 2.16569)
Input interpretation:
Result:
0.82219
And:
(0.639026 - 0.583302)
Input interpretation:
Result:
71
0.055724
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:
72
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:
73
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
exp((-Pi*sqrt(18)) we obtain:
Input:
Exact result:
74
Decimal approximation:
1.6272016… * 10
-6
Property:
Series representations:
Now, we have the following calculations:
6+
= 4096
18
18
= 1.6272016… * 10^-6
from which:
1
4096
6+
= 1.6272016… * 10^-6
0.000244140625
6+
=
18
= 1.6272016… * 10^-6
75
Now:
ln
18
= 13.328648814475 =
18
And:
(1.6272016* 10^-6) *1/ (0.000244140625)
Input interpretation:
Result:
0.006665017...
Thence:
0.000244140625
6+
=
18
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:
76
Result:
0.00666501785…
Series representations:
Now:
6+
= 0.0066650177536
=
= 0.00666501785…
From:
77
ln(0.00666501784619)
Input interpretation:
Result:
-5.010882647757…
Alternative representations:
Series representations:
78
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:
(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:
79
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:
From
80
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)))))
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...
81
Series representations:
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)]
82
Input interpretation:
Result:
Polar coordinates:
54.76072411…..
Series representations:
83
84
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...
85
Series representations:
86
Now, we have:
For:
ξ = 1
= 0.989117352243
87
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:
88
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:
89
113+5
505
8
+
105+5
505
8
3
14
= 1,65578
Series representations:
90
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:
91
Result:
-0.034547055658…
Series representations:
92
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))]))))))))
93
Input interpretation:
Result:
1.6237116159…. result that is an approximation to the value of the golden ratio
1.618033988749...
Series representations:
94
95
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
96
Series representations:
97
98
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...
99
Series representations:
100
101
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
102
Series representations:
103
104
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:
105
Series representations:
106
107
From the previous expression
= -0.034547055658…
we have also:
108
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:
109
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:
110
We have:
(M^2)/3*[1-(b/euler number * k/sqrt6) * (φ- sqrt6/k) * exp(-(k/sqrt6)(φ- sqrt6/k))]^2
i.e.
V = (M^2)/3*[1-(b/euler number * k/sqrt6) * (φ- sqrt6/k) * exp(-(k/sqrt6)(φ-
sqrt6/k))]^2
For k = 2 and φ = 0.9991104684, that is the value of the scalar field that is equal to
the value of the following Rogers-Ramanujan continued fraction:
we obtain:
V = (M^2)/3*[1-(b/euler number * 2/sqrt6) * (0.9991104684- sqrt6/2) * exp(-
(2/sqrt6)(0.9991104684- sqrt6/2))]^2
Input interpretation:
Result:
111
Solutions:
Alternate forms:
Expanded form:
Alternate form assuming b, M, and V are positive:
Alternate form assuming b, M, and V are real:
Derivative:
112
Implicit derivatives:
Global minimum:
113
Global minima:
From:
we obtain
(225.913 (-0.054323 M^2 + 6.58545×10^-10 sqrt(M^4)))/M^2
Input interpretation:
Result:
114
Plots:
Alternate form assuming M is real:
-12.2723 result very near to the black hole entropy value 12.1904 = ln(196884)
Alternate forms:
115
Expanded form:
Property as a function:
Parity
Series expansion at M = 0:
Series expansion at M = ∞:
Derivative:
Indefinite integral:
116
Global maximum:
Global minimum:
Limit:
Definite integral after subtraction of diverging parts:
From b that is equal to
117
from:
Result:
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:
Plots: (possible mathematical connection with an open string)
M = -0.5; M = 0.2
118
(possible mathematical connection with an open string)
M = 2 ; M = 3
Root:
Property as a function:
Parity
Series expansion at M = 0:
Series expansion at M = ∞:
119
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:
Result:
-4.38851344947*10
-16
120
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
For M = 3:
121
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
Input interpretation:
122
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:
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:
123
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...
From the Planck units:
Planck Length
5.729475 * 10
-35
Lorentz-Heaviside value
124
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
125
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
126
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
127
―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
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
128
References
Course of Field Theory and Gravity - Prof. Augusto Sagnotti (SNS Pisa-Italy)
ON THE PARTITION FUNCTION p(n).
By HANS RADEMACHER. - [Received 30 November, 1936.—Read 10 December,
1936.]
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
AdS Vacua from Dilaton Tadpoles and Form Fluxes - J. Mourad and A. Sagnotti
- arXiv:1612.08566v2 [hep-th] 22 Feb 2017 - March 27, 2018
Properties of Nilpotent Supergravity
E. Dudas, S. Ferrara, A. Kehagias and A. Sagnotti - arXiv:1507.07842v2 [hep-th] 14
Sep 2015