1
On the possible mathematical connections between the κ formula regarding the
Riemann Zeta Function, some Ramanujan equations, the study of some Riccati
equations and various topics of String Theory. II
Michele Nardelli
1
, Antonio Nardelli
2
Abstract
In this research thesis (part II), we describe the possible mathematical connections
between the κ formula regarding the Riemann Zeta Function, some Ramanujan
equations, the study of some Riccati equations and various topics 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
2
A. Nardelli studies at the Università degli Studi di Napoli Federico II - Dipartimento di Studi Umanistici –
Sezione Filosofia - scholar of Theoretical Philosophy
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. 1, Jacopo Riccati, In Lucca, presso Iacopo Giusti, 1761.
Pg.483 - 484
15
16
From:
(x^3 dy + y^3 dx)/((x^2+y^2)*sqrt(x^2+y^2-x^2y^2))
a)
(x^3+ y^3)dy/((x^2+y^2)*sqrt(x^2+y^2-x^2y^2))
Input:
Result:
17
Indefinite integral:
For x = y = 2 :
1/2 (-(2 sqrt(4 (4) + 4 + 4))/(4 - 1) + ((sqrt(4) + 2) tanh^(-1)((4 - sqrt(4) (4 - 1) 2)/(4
sqrt(4 (4) + 4 + 4))))/2 + (2/sqrt(4) + 1) tanh^(-1)((sqrt(4) (4 - 1) 2 + 4)/(4 sqrt(4 - 4
(4 - 1)))))
Input:
Exact result:
18
Decimal approximation:
Polar coordinates:
2.2766
Polar forms:
Approximate form
19
Alternate forms:
Expanded form:
Alternative representations:
20
Series representations:
21
Integral representations:
22
From which:
1/2 (-(2 sqrt(4 (4) + 8))/(3) + ((sqrt(4) + 2) tanh^(-1)((4 - sqrt(4) (3) 2)/(4 sqrt(4 (4) +
8))))/2 + (2/sqrt(4) + 1) tanh^(-1)((sqrt(4) (3) 2 + 4)/(4 sqrt(4 - 4 (3)))))*(((√(10-2√5)
-2))⁄((√5-1)))
Where (((√(10-2√5) -2))⁄((√5-1))) = κ
Input:
23
Exact result:
Decimal approximation:
Polar coordinates:
0.64674 ≈ ζ(2) -1 =
2
6
1 = 1.644934 -1 = 0.644934…
Polar forms:
24
Approximate form
Alternate forms:
25
Expanded form:
Or, we calculate the above equation as follows:
(x^3+ y^3)dydx/((x^2+y^2)*sqrt(x^2+y^2-x^2y^2))
Indefinite integral:
26
2 + 2 + 1/2 (sqrt(4) + 2) tanh^(-1)((4 - 2 sqrt(4) (4 - 1))/(4 sqrt(4 (4) + 4 + 4))) + 1/2
(2 - sqrt(4)) tanh^(-1)((2 sqrt(4) (4 - 1) + 4)/(4 sqrt(4 (4) + 4 + 4)))
Input:
Exact result:
Decimal approximation:
3.1329852735….
Alternate forms:
27
Alternative representations:
Series representations:
28
Integral representations:
29
(1/2 - i/2) 2log((((((1 - i) (8- 2i - 2)))/(16 (2 + 2i)) - ((1 + i) sqrt(4 (4) + 4 + 4)/(2^3 (2
+ 2i ))))))
Input:
Exact result:
Decimal approximation:
Property:
Polar coordinates:
3.9785
30
Polar forms:
Approximate form
Alternate forms:
Alternative representations:
31
Series representations:
32
Integral representations:
(1/2 - i/2) 2log((((((1 - i) (8- 2i - 2)))/(16 (2 + 2i)) - ((1 + i) sqrt(4 (4) + 4 + 4)/(2^3 (2
+ 2i ))))))
Input:
33
Exact result:
Decimal approximation:
Property:
Polar coordinates:
3.9785
Polar forms:
34
Approximate form
Alternate forms:
Alternative representations:
35
Series representations:
36
Integral representations:
2 + 2 + 1/2 (sqrt(4) + 2) tanh^(-1)((4 - 2 sqrt(4) (4 - 1))/(4 sqrt(4 (4) + 4 + 4))) + 1/2
(2 - sqrt(4)) tanh^(-1)((2 sqrt(4) (4 - 1) + 4)/(4 sqrt(4 (4) + 4 + 4)))+(-7.1078272245 -
3.57664677049i)
Input interpretation:
Result:
37
Polar coordinates:
5.3471273275 final result
Polar forms:
Alternative representations:
38
Series representations:
39
40
41
Integral representations:
42
From which:
1/(((4 + 1/2 (sqrt(4) + 2) tanh^(-1)((4 - 2 sqrt(4) (4 - 1))/(4 sqrt(4 (4) + 8))) + 1/2 (2 -
sqrt(4)) tanh^(-1)((2 sqrt(4) (2) + 4)/(4 sqrt(4 (4) + 8)))+(-7.1078272245 -
3.57664677049i))))^1/4
Input interpretation:
Result:
Polar coordinates:
0.65761221050 result very near to the 14th root of the following Ramanujan’s class
invariant subtracting 1: =
505
/
101/5
3
= (1164.2696)
1/14
-1 i.e. 0.65578...
Polar forms:
43
Alternative representations:
44
Series representations:
45
46
Integral representations:
47
Pg. 503- 506
48
49
50
From:
i.e.
51
We have that:
2
=
2q*sqrt(b) = 2*sqrt(p*q)
b = (((m*sqrt(c) / 2q)))^2 = (((2*sqrt(p*q)*sqrt(c) / (2q))))^2
2q*sqrt(((((2*sqrt(p*q)*sqrt(c) / (2q))))^2)) = 2*sqrt(p*q)
Input:
Result:
Alternate form assuming c, p, and q are positive:
Real solutions:
52
Implicit derivatives:
53
For p = 2, q = 4
2*4*sqrt(((((2*sqrt(2*4)*sqrt(1) / (2*4))))^2)) = 2*sqrt(2*4)
Input:
Result:
Left hand side:
Right hand side:
2*4*sqrt(((((2*sqrt(2*4)*sqrt(1) / (2*4))))^2))
Input:
Result:
54
Decimal approximation:
5.656854249….
2*4*sqrt(((((2*sqrt(2*4)*sqrt(1) / (2*4))))^2)) – 4
Input:
Result:
Decimal approximation:
1.656854249…. 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:
55
113+5
505
8
+
105+5
505
8
3
14
= 1,65578
Alternate form:
Minimal polynomial:
56
From:
pg. 563
We have:
57
(-x dy^2 ddy – x dy ddy^2) / (2x^3 dy^3 dx)
ddy(-x dy^2 – x dy^2 d) / (2x^3 dy^3 dx)
i.e.
(((-d integrate dy^2 - d integrate dx^2)))1/(xdy^2) 1/(derivative 2x*x)
Input interpretation:
Result:
For x = -0.5 and y = 0.5 :
(-d 0.5^2 - d 0.5^2)/(4 d 0.5^2 0.5^2)
58
Input:
Result:
3D plot:
Contour plot:
59
Alternate form:
Expanded forms:
Derivative:
Indefinite integral:
60
Pg. 571
61
From:
We perform the integrate of:
Integrate(((-1/(x^2*sqrt(1+x^2))) + ((b/x^2))))dx
Indefinite integral:
62
3D plot:
Contour plot:
Alternate forms of the integral:
Expanded form of the integral:
63
Series expansion of the integral at x = 0:
Series expansion of the integral at x = ∞:
For x = 0.4 and b = 1.2 , from the result of the integral
(1 + x^2 - b sqrt(1 + x^2))/(x sqrt(1 + x^2))
we obtain:
(1 + 0.4^2 - 1.2 sqrt(1 + 0.4^2))/(0.4 sqrt(1 + 0.4^2))
Input:
Result:
-0.3074175964327….
From which:
1+(-d 0.5^2 - d 0.5^2)/(4 d 0.5^2 0.5^2)(1 + 0.4^2 - 1.2 sqrt(1 + 0.4^2))/(0.4 sqrt(1 +
0.4^2))
64
Input:
Result:
1.61484 result that is a very good approximation to the value of the golden ratio
1.618033988749...
Now, we have that:
i.e.
Indeed:
(1 + x^2 - b sqrt(1 + x^2))/(x sqrt(1 + x^2))
Input:
65
3D plot:
Contour plot:
Alternate forms:
66
Expanded form:
Real roots:
Properties as a real function:
Domain
Range
Parity
67
Roots for the variable x:
Series expansion at x = 0:
Series expansion at x = ∞:
Derivative:
Indefinite integral:
68
Limit:
From:
sqrt(1 + x^2) - 1/2 (-1 + b) log(1 - sqrt(1 + x^2)) - 1/2 (1 + b) log(1 + sqrt(1 + x^2))
we obtain:
1/(((sqrt(1 + 0.4^2) - 1/2 (-1 + 1.2) log(1 - sqrt(1 + 0.4^2)) - 1/2 (1 + 1.2) log(1 +
sqrt(1 + 0.4^2)))))
Input:
Result:
Polar coordinates:
1.62455 result quite near to the value of the golden ratio 1.618033988749...
69
Series representations:
70
Integral representation:
And:
sqrt(1 + 0.4^2) - 1/2 (-1 + 1.2) log(1 - sqrt(1 + 0.4^2)) - 1/2 (1 + 1.2) log(1 + sqrt(1 +
0.4^2))
Input:
Result:
Polar coordinates:
0.615555 result that is a very good approximation to the value of the golden ratio
conjugate 0.618033988749...
Polar forms:
71
Alternative representations:
Series representations:
72
Integral representation:
73
And again:
(((sqrt(1 + 0.4^2) - 1/2 (-1 + 1.2) log(1 - sqrt(1 + 0.4^2)) - 1/2 (1 + 1.2) log(1 + sqrt(1
+ 0.4^2)))))^1/32
Input:
Result:
Polar coordinates:
0.984951 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)
74
Note that
= 0.9863870313564812915… (π
2
/6 – 1)^1/32
Radians
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:
Series representations:
83
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:
Result:
0.99040073.... result very near to the dilaton value . = and to
the value of the following Rogers-Ramanujan continued fraction:
85
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:
Result:
-0.034547055658…
96
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))]))))))))
Input interpretation:
98
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
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:
115
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:
116
Solutions:
Alternate forms:
Expanded form:
Alternate form assuming b, M, and V are positive:
Alternate form assuming b, M, and V are real:
Derivative:
117
Implicit derivatives:
Global minimum:
118
Global minima:
From:
we obtain
(225.913 (-0.054323 M^2 + 6.58545×10^-10 sqrt(M^4)))/M^2
Input interpretation:
119
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:
120
Expanded form:
Property as a function:
Parity
Series expansion at M = 0:
Series expansion at M = ∞:
Derivative:
121
Indefinite integral:
Global maximum:
Global minimum:
Limit:
122
Definite integral after subtraction of diverging parts:
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:
123
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
124
Series expansion at M = 0:
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)
125
Input interpretation:
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:
126
Result:
7.021621519159*10
-17
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
127
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:
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)]
128
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:
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...
129
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
130
Planck’s Electric potential
1.042940*10
27
V Lorentz-Heaviside value
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
131
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.
Furthermore, Michele Nardelli wishes to thank the co-author Antonio Nardelli, for
his precious collaboration in the drafting of this work
132
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. 1, Jacopo Riccati, In Lucca, presso Iacopo Giusti, 1761.
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