1
On various equations concerning “High-Dimensional non-linear Schrodinger
Equations”. Possible mathematical connections with several Ramanujan’s
formulas, the κ formula regarding the Zeros of Riemann Zeta Function and
some topics of String Theory
Michele Nardelli
1
, Antonio Nardelli
2
Abstract
In this research thesis, we analyze some equations concerning “High-Dimensional
non-linear Schrodinger Equations”. We describe the possible mathematical
connections with several Ramanujan’s formulas, the κ formula regarding the Zeros
of Riemann Zeta Function and some 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:
A (CONCENTRATION-)COMPACT ATTRACTOR FOR HIGH-
DIMENSIONAL NON LINEAR SCHRODINGER EQUATIONS - TERENCE
TAO - arXiv:math/0611402v6 [math.AP] 27 Jan 2014
15
16
17
18
19
Now:
20
x + 1/24 ≤ 1
Input:
Inequality plot:
Alternate forms:
21
Solution:
Interval notation:
N ≈ -10^(-8) (9.58333×10^7)
Input interpretation:
Result:
-0.958333 = N
From:
For η
3
= 23/24 = 0.9583333…..
(-0.958333^0.958333)+1/24
Input interpretation:
22
Result:
-0.918367297….
(-0.958333^0.958333)+1/24 ≤ 1
Input interpretation:
Result:
From (37) and (40) we have that:
(-0.958333^-0.958333)
Input interpretation:
Result:
-1.041629815….
23
-0.958333^[5*(2/(20/19)-1)]
Input interpretation:
Result:
-0.82570325….
From which:
-2*(((-0.958333^[5*(2/(20/19)-1)])))
Input interpretation:
Result:
1.6514065…. result very near to the 14th root of the following Ramanujan’s class
invariant =
505
/
101/5
3
= 1164.2696 i.e. 1.65578...
24
From the algebraic sum of three previous results, inverting the sign, we obtain:
0.918367294 + 1.041629815 + 0.82570325116424 = 2.78570036016424
From which:
(0.918367294 + 1.041629815 + 0.82570325116424) - 4((((√(10-2√5) -2))⁄((√5-1))))
Input interpretation:
Result:
1.64938418…. ≈ ζ(2) =
2
6
= 1.644934
From:
25
(-0.958333^-0.958333)
Input interpretation:
Result:
-1.041629815….
We note that:
is equal to:
i.e. -1.041629815….
-1/2*1/(-0.958333^0.958333)
Input interpretation:
26
Result:
0.5208149075018…..
From which:
2(((-1/2*1/(-0.958333^0.958333))+((((√(10-2√5) -2))⁄((√5-1))))))
Input interpretation:
Result:
1.60978790268…. result that is a very good approximation to the value of the golden
ratio 1.618033988749...
(((-1/2*1/(-0.958333^0.958333))+4((((√(10-2√5) -2))⁄((√5-1))))))
Input interpretation:
Result:
1.65713108286….. result very near to the 14th root of the following Ramanujan’s
class invariant =
505
/
101/5
3
= 1164.2696 i.e. 1.65578...
27
1/3([(((-1/2*1/(-0.958333^0.958333))+4((((√(10-2√5) -2))⁄((√5-1))))))]+[2(((-1/2*1/(-
0.958333^0.958333))+((((√(10-2√5) -2))⁄((√5-
1))))))]+[(0.918367+1.041629+0.8257)-4((((√(10-2√5) -2))⁄((√5-1))))])
Input interpretation:
Result:
1.63876627006…. result very near to the mean between ζ(2) =
2
6
= 1.644934 , the
value of golden ratio 1.61803398… and the 14th root of the Ramanujan’s class
invariant =
505
/
101/5
3
= 1164.2696 i.e. 1.65578..., i.e. 1.63958266
(1.6387662700621655)-((((√(10-2√5) -2))⁄((√5-1))))^Pi
Input interpretation:
Result:
1.61958273685…. result that is a very good approximation to the value of the golden
ratio 1.618033988749...
28
Series representations:
29
(1.6387662700621655)+((((√(10-2√5) -2))⁄((√5-1))))^4
Input interpretation:
Result:
1.64527890583…. ≈ ζ(2) =
2
6
= 1.644934
From:
30
From:
We have:
That is equal to:
(-0.958333^0.958333)
Input interpretation:
Result:
-0.9600339636…. 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:
31
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)
-(((((√(10-2√5) -2))⁄((√5-1))))^1/(5+0.918367294))((-0.958333^0.958333)+(-
0.958333^-0.958333))
Input interpretation:
Result:
1.618233549909…. result that is a very good approximation to the value of the
golden ratio 1.618033988749...
32
Now, we have that:
From which, if we consider:
=
22
From the following Ramanujan expression:
22
24 + 4372
22
= 64
1 +
2
12
+
1
2
12
22
= 64
1 +
2
12
+
1
2
12
+ 24 4372
22
33
64[(1+√2)^12+(1-√2)^12]+24-4372e^(-π√22)
Input:
Exact result:
Decimal approximation:
2508951.9982…
Property:
Alternate forms:
Expanded form:
34
Series representations:
35
36
e^(π√22)
Input:
37
Decimal approximation:
2508951.9982….
Property:
Series representations:
Integral representation:
Thence, from the above expression, we can obtain easily 4096, 1728 and a value very
near to ζ(2)
38
-(((2508951.9982 - 64[(1+√2)^12+(1-√2)^12]-24 ))/(e^(-π√22))+24*6+276)
Input interpretation:
Result:
4096.11… ≈ 4096 = 64
2
Series representations:
39
40
27sqrt((-(((2508951.9982 - 64[(1+√2)^12+(1-√2)^12]-24 ))/(e^(-π√22))+24*6+276)))
Input interpretation:
Result:
1728.02…
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)
Series representations:
41
[27sqrt((-(((2508951.9982 - 64[(1+√2)^12+(1-√2)^12]-24 ))/(e^(-
π√22))+24*6+276)))]^1/15
Input interpretation:
42
Result:
1.64375334906922…. ≈ ζ(2) =
2
6
= 1.644934
Or:
64[(1+√2)^12+(1-√2)^12]+24
Input:
Result:
2508952
equalizing and dividing, we obtain:
(((2508952-24)/([(1+√2)^12+(1-√2)^12] )))^2
Input:
Result:
4096 = 64
2
43
27sqrt(((((2508952-24)/([(1+√2)^12+(1-√2)^12] )))^2))+1
Input:
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. The number 1728 is one less than the Hardy–Ramanujan number 1729
(taxicab number)
((27sqrt(((((2508952-24)/([(1+√2)^12+(1-√2)^12] )))^2))+1))^1/15
Input:
Result:
Decimal approximation:
1.6438152287…. ≈ ζ(2) =
2
6
= 1.644934
44
Alternate form:
All 15th roots of 1 + 67741056/((1 - sqrt(2))^12 + (1 + sqrt(2))^12):
We note that:
2508952 / 8 = 313619
i.e.
1/8((64[(1+√2)^12+(1-√2)^12]+24))
45
Input:
Result:
313619 that is a Prime number
Input:
Prime factorization:
Properties:
313632 – 13
Where 313632:
313632/16 = 19602
From 19602:
19602-8192-4096-2048-1024-512-256-128-64 = 3282
313632/24 = 13068
46
From 13068:
13068-8192-4096-512-256 = 12
313632/48 = 6534
From 6534:
6534-4096-2048-256 = 134
……………...
313632/144 = 2178
……………...
313632/288 = 1089
Where 1089:
From 1089:
1089-1024 = 65
3282 + 12 + 134 + 65 = 3493 ;
4096 – 3493 – 576 = 27
27 * 65 – 27 = 1728
47
From:
We consider the first inequality equal to 1/8 and the second equal to 1/2. Dividing by
κ
1/7
, where κ = ((((√(10-2√5) -2))⁄((√5-1)))) and adding 1, we obtain:
(((((1/8)^2+1/2))*1/((((√(10-2√5) -2))⁄((√5-1))))^1/7))+1
Input:
48
Exact result:
Decimal approximation:
1.61718317821…. result that is a very good approximation to the value of the golden
ratio 1.618033988749...
Alternate forms:
49
And:
1/[(((((1/8)^2+1/2))*1/((((√(10-2√5) -2))⁄((√5-1))))^1/7))+1]
Input:
Exact result:
Decimal approximation:
0.61835914…. result that is a very good approximation to the value of the golden
ratio conjugate 0.618033988749...
Alternate forms:
50
From:
Manuscript Book 2 of Srinivasa Ramanujan
For x = 2:
(2Pi)^2 (((cosh(2Pi*sqrt2) + cos(2Pi*sqrt2)))) / (((cosh(2Pi*sqrt2) - cos(2Pi*sqrt2))))
Input:
Exact result:
51
Decimal approximation:
39.4596730889…..
Alternate forms:
Expanded form:
Alternative representations:
52
Series representations:
53
Integral representations:
Multiple-argument formulas:
54
1+64Pi[(coth(Pi))/(17)+(2coth(2Pi))/(32)+(3coth(3Pi))/(97)+(4coth(4Pi))/(272)+(5cot
h(5Pi))/(641)+(6coth(6Pi))/(1312)+(7coth(7Pi))/(2417)+(8coth(8Pi))/(4112)+(9coth(9
Pi))/(6577)+(10coth(10Pi))/(10016)]
Input:
Exact result:
Decimal approximation:
38.55027800574….
We previously have calculated:
55
0.918367294 + 1.041629815 + 0.82570325116424 = 2.78570036016424
Now, we note that:
1/2((((1/14[(2Pi)^2 (((cosh(2Pi*sqrt2) + cos(2Pi*sqrt2)))) / (((cosh(2Pi*sqrt2) -
cos(2Pi*sqrt2))))] + 1/15[(2Pi)^2 (((cosh(2Pi*sqrt2) + cos(2Pi*sqrt2)))) /
(((cosh(2Pi*sqrt2) - cos(2Pi*sqrt2))))]))))
Input:
Exact result:
Decimal approximation:
2.724596475186……result very near to the above value 2.78570036016424
Alternate forms:
56
Expanded form:
Alternative representations:
57
Series representations:
58
Integral representations:
Multiple-argument formulas:
59
Furthermore, we have also:
1/15[(2Pi)^2 (((cosh(2Pi*sqrt2) + cos(2Pi*sqrt2)))) / (((cosh(2Pi*sqrt2) -
cos(2Pi*sqrt2))))]
Input:
Exact result:
Decimal approximation:
2.6306448725….
60
Alternate forms:
And again:
sqrt(((1/15[(2Pi)^2 (((cosh(2Pi*sqrt2) + cos(2Pi*sqrt2)))) / (((cosh(2Pi*sqrt2) -
cos(2Pi*sqrt2))))])))
Input:
Exact result:
61
Decimal approximation:
1.6219262845…. result that is a good approximation to the value of the golden ratio
1.618033988749...
Alternate form:
All 2nd roots of (4 π^2 (cos(2 sqrt(2) π) + cosh(2 sqrt(2) π)))/(15 (cosh(2 sqrt(2)
π) - cos(2 sqrt(2) π))):
Alternative representations:
62
Series representations:
63
Integral representations:
64
Multiple-argument formulas:
65
We have also:
(47+7+2)/
(((1+64Pi[(coth(Pi))/(17)+(2coth(2Pi))/(32)+(3coth(3Pi))/(97)+(4coth(4Pi))/(272)])))
Input:
Exact result:
Decimal approximation:
1.617885037218…. result that is a very good approximation to the value of the
golden ratio 1.618033988749...
Alternate forms:
66
Alternative representations:
67
Series representations:
And:
(55+2)/
(((1+64Pi[(coth(Pi))/(17)+(2coth(2Pi))/(32)+(3coth(3Pi))/(97)+(4coth(4Pi))/(272)])))
Input:
68
Exact result:
Decimal approximation:
1.64677584145…. ≈ ζ(2) =
2
6
= 1.644934
Alternate forms:
Alternative representations:
69
Series representations:
70
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
71
“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
72
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:
73
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.
74
For
exp((-Pi*sqrt(18)) we obtain:
Input:
Exact result:
Decimal approximation:
1.6272016… * 10
-6
Property:
Series representations:
75
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
Now:
ln
18
= 13.328648814475 =
18
And:
(1.6272016* 10^-6) *1/ (0.000244140625)
Input interpretation:
Result:
0.006665017...
76
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:
Result:
0.00666501785…
Series representations:
77
Now:
6+
= 0.0066650177536
=
= 0.00666501785…
From:
ln(0.00666501784619)
Input interpretation:
Result:
-5.010882647757…
Alternative representations:
78
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:
79
(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:
80
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:
81
(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...
Series representations:
82
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…..
83
Series representations:
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
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:
104
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
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
110
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.
A (CONCENTRATION-)COMPACT ATTRACTOR FOR HIGH-
DIMENSIONAL NON LINEAR SCHRODINGER EQUATIONS - TERENCE
TAO - arXiv:math/0611402v6 [math.AP] 27 Jan 2014
Manuscript Book 2 of Srinivasa Ramanujan
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