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. IV
Michele Nardelli
1
, Antonio Nardelli
2
Abstract
In this research thesis (part IV), 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.
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. 2 e 3, Jacopo Riccati, In Lucca, presso Iacopo Giusti, 1762-1764.
We have:
15
From:
Sqrt(2abz) / sqrt(ac+2/3bg+bz)
Input:
Exact result:
Alternate forms:
Alternate forms assuming a, b, c, g, and z are positive:
16
Real roots:
Root for the variable z:
17
Series expansion at z = 0:
Series expansion at z = ∞:
Derivative:
Indefinite integral:
18
Sqrt(2abz) / sqrt(ac+2/3bg+bz) = u
Input:
Exact result:
Alternate forms:
Alternate forms assuming a, b, c, g, u, and z are positive:
19
Real solutions:
Derivative:
Limit:
20
for:
a = ((sqrt(3 π) sqrt(r))/3)) b = 2; c = 3; x = 1; v = 16 ; z = 11.9 or 92.1
From:
Sqrt(2abz) / sqrt(ac+2/3bg+bz) = u
Sqrt(((2(((sqrt(3 π) sqrt(4/3))/3)))*2*11.9))) / sqrt(((3((sqrt(3 π)
sqrt(4/3))/3))+2/3*2*16+2*11.9)))
Input:
Result:
1.07492…
21
Series representations:
22
From which:
((Sqrt(((2(((sqrt(3 π) sqrt(4/3))/3)))*2*11.9)) / sqrt(((3((sqrt(3 π)
sqrt(4/3))/3))+2/3*2*16+2*11.9))))^7
Input:
Result:
1.65823140595…. result very near to the 14th root of the following Ramanujan’s
class invariant =
505
/
101/5
3
= 1164.2696 i.e. 1.65578...
23
We have:
24
From:
25
From:
26
27
From Corollary I:
For a = ((sqrt(3 π) sqrt(r))/3)) b = 2; c = 3; x = 1; v = 16 ; z = 11.9 or 92.1
Integrate(1+((2sqrt(2*11.9))) / ((sqrt(3(((sqrt(3 π) sqrt(r))/3))))) = ((2sqrt(2*11.9))) /
((sqrt(3(((sqrt(3 π) sqrt(r))/3)))))
28
Indefinite integral:
6.18222x
((2sqrt(2*11.9))) / ((sqrt(3(((sqrt(3 π) sqrt(4/3))/3)))))
Input:
Result:
5.18222….
Series representations:
29
From Corollary II:
For a = (((sqrt(3 π) sqrt(4/3))/3)) b = 2; c = 3; x = 1; v = 16 ; z = 11.9 or 92.1
[3(((sqrt(3 π) sqrt(4/3))/3))] / [2 sqrt((2*4*((sqrt(3 π) sqrt(4/3))/3)))]
Input:
Exact result:
30
Decimal approximation:
0.576485123….
Property:
Series representations:
31
32
Integrate(1+2/(((sqrt(((3(((((sqrt(3 π) sqrt(4/3))/3)))))+2*11.9)/(2*11.9)))-1))) *
(((1/4 sqrt(3) π^(1/4))))
Indefinite integral:
16.6147x
Plot of the integral:
Or, for z = 92.1 :
Integrate(1+2/(((sqrt(((3(((((sqrt(3 π) sqrt(4/3))/3)))))+2*92.1)/(2*92.1)))-1))) *
(((1/4 sqrt(3) π^(1/4))))
Indefinite integral:
120.971x
33
Plot of the integral:
From the algebraic sum, we obtain:
Integrate(1+2/(((sqrt(((3(((((sqrt(3 π) sqrt(4/3))/3)))))+2*11.9)/(2*11.9)))-1))) *
(((1/4 sqrt(3) π^(1/4)))) + 120.971
Indefinite integral:
137.586x
Plot of the integral:
34
(137.586)^1/10
Input interpretation:
Result:
1.636279…. 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
Integrate(1+2/(((sqrt(((3(((((sqrt(3 π) sqrt(4/3))/3)))))+2*11.9)/(2*11.9)))-1))) *
(((1/4 sqrt(3) π^(1/4)))) - 120.971
Indefinite integral:
-104.356x
Plot of the integral:
35
Integrate((((1+2/(((sqrt(((3(((((sqrt(3 π) sqrt(4/3))/3)))))+2*11.9)/(2*11.9)))-1)))
(((1/4 sqrt(3) π^(1/4)))) - 120.971) *1/5))^-1 * -34
Indefinite integral:
1.62903x
For x = 1 we have 1.62903, 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
Plot of the integral:
36
Now, we have that:
37
38
39
From:
After the integration of
= ;
=
We have:
If multiply both the sides by dt and put v = ds/dt , integrating
40
=
2
2
;
=
= , we obtain:
From which:
For P = 8, p = 4, m = 2 , M = 16, AC = 32 , AE = 64, s = 1/2
From:
(8 – (4*64/32))*1/2*t^2 = (16+(2*64^2/32^2)) *1/2
41
(8 – (4*64/32))*1/2*t^2 - (16+(2*64^2/32^2)) * 1/2
Input:
Exact result:
-12
Indeed for t = 0:
(8 – (4*64/32))*1/2*0 - (16+(2*64^2/32^2)) * 1/2
Input:
Exact result:
-12
Or:
- (16+(2*64^2/32^2)) * 1/2
Input:
Exact result:
-12
42
From:
From which:
we obtain:
sqrt((16*32^2+2*64^2) / (8*32+4*64))
Input:
Result:
Decimal approximation:
6.92820323….
43
Now, from the previous expression:
(8 – (4*64/32))*1/2*(4sqrt3)^2 = (16+(2*64^2/32^2)) *1/2
(8 – (4*64/32))*1/2*(4sqrt3)^2
Input:
Exact result:
And again:
(8 – (4*64/32))*1/2*(4sqrt3)^2-(16+(2*64^2/32^2)) *1/2
Input:
Exact result:
-12
233/((((8 – (4*64/32))*1/2*(4sqrt3)^2-(16+(2*64^2/32^2)) *1/2)))^2
Input:
44
Exact result:
Decimal approximation:
1.61805555…. result that is a very good approximation to the value of the golden
ratio 1.618033988749...
12((((8 – (4*64/32))*1/2*(4sqrt3)^2-(16+(2*64^2/32^2)) *1/2)))^2+1
Input:
Exact result:
1729
This result is very near to the mass of candidate glueball f
0
(1710) scalar meson.
Furthermore, 1728 occurs in the algebraic formula for the j-invariant of an elliptic
curve. The number 1728 is one less than the Hardy–Ramanujan number 1729
(taxicab number)
[12((((8 – (4*64/32))*1/2*(4sqrt3)^2-(16+(2*64^2/32^2)) *1/2)))^2+1]^1/15
Input:
45
Exact result:
Decimal approximation:
1.6438152287…. ≈ ζ(2) =
2
6
= 1.644934
((([12((((8 – (4*64/32))*1/2*(4sqrt3)^2-(16+(2*64^2/32^2)) *1/2)))^2+1]^1/15-
1)))^1/32
Input:
Exact result:
Decimal approximation:
0.986333511553…. result very near to the value of the following Rogers-Ramanujan
continued fraction:
46
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
= 0.9863870313564812915… (π
2
/6 – 1)^1/32
16[((-2*((((8 – (4*64/32))*1/2*(4sqrt3)^2-(16+(2*64^2/32^2)) *1/2)))))^2-64]
Input:
Exact result:
8192
The total amplitude vanishes for gauge group SO(8192), while the vacuum energy is
negative and independent of the gauge group.
47
The vacuum energy and dilaton tadpole to lowest non-trivial order for the open
bosonic string. While the vacuum energy is non-zero and independent of the gauge
group, the dilaton tadpole is zero for a unique choice of gauge group, SO(2
13
) i.e.
SO(8192). (From: “Dilaton Tadpole for the Open Bosonic String “ Michael R.
Douglas and Benjamin Grinstein - September 2,1986)
8[((-2*((((8 – (4*64/32))*1/2*(4sqrt3)^2-(16+(2*64^2/32^2)) *1/2)))))^2-64]+276
Input:
Exact result:
4372
Where 4372 is a value indicated in the fundamental Ramanujan paper “Modular
equations and Approximations to π”
48
From:
Broken Scale Invariance, Gravity Mass, and Dark Energy in Modified Einstein
Gravity with Two Measure Finsler Like Variables - Panayiotis Stavrinos, Sergiu
I. Vacaru - arXiv:2106.00462v1 [physics.gen-ph] 28 May 2021
We have that:
Thence:
If we consider:
For:
= P
49
= (p AE)/AC 1/2 t
2
= - (M + (m AE
2
)/AC
2
) s
We obtain
P - ((p AE)/(AC) 1/2 t^2) - (M + (m (AE)^2)/((AC)^2)) s
Input:
Result:
Alternate forms:
50
Expanded form:
Roots:
Derivative:
Indefinite integral:
51
Alternative representation:
Series representations:
From:
-s ((e^2 m)/C^2 + M) - (e p t^2)/(2 C) + P
For P = 8, p = 4, m = 2 , M = 16, AC =8*4 = 32 , AE = 8*8 = 64, s = 1/2
52
-1/2 ((8^2 * 2)/4^2 + 16) - (8*4 (4sqrt3)^2)/(2*4) + 8
Input:
Exact result:
-196
From the indefinite integral, we obtain:
-(e^2 m s^2)/(2 C^2) - (e p s t^2)/(2 C) - (M s^2)/2 + P s
-(8^2*2*1/4) / (2*4^2) – (8*4*1/2*(4sqrt3)^2) / (2*4) – (16*1/4)*1/2 + 8*1/2
Input:
Exact result:
-95
From the derivative, we obtain:
-(8^2 *2)/(4^2) – 16
Input:
53
Result:
-24 value that is linked to the "Ramanujan function" (an elliptic modular function
that satisfies the need for "conformal symmetry") that has 24 "modes" corresponding
to the physical vibrations of a bosonic string.
From the root t, we obtain:
(sqrt(2/8) sqrt(4^2 *8 - 8^2*16*1/2 - 4^2*16*1/2))/(sqrt(4) sqrt(4))
Input:
Result:
Decimal approximation:
Polar coordinates:
2√2
Polar forms:
54
Approximate form
(((sqrt(2/8) sqrt(4^2 *8 - 8^2*16*1/2 - 4^2*16*1/2))/(sqrt(4) sqrt(4))))^2
Input:
Exact result:
-8
[(((sqrt(2/8) sqrt(4^2 *8 - 8^2*16*1/2 - 4^2*16*1/2))/(sqrt(4) sqrt(4))))^2]^4+276
Input:
Exact result:
4372
Where 4372 is a value indicated in the fundamental Ramanujan paper “Modular
equations and Approximations to π”
55
From the algebraic sum:
we obtain:
-((-1/2 ((128)/16 + 16) - (32 (4sqrt3)^2)/(8) + 8))+(-(128*1/4)/(32) –
(16*(4sqrt3)^2)/(8) – (4)*1/2 + 8*1/2)-((-(128)/(16) – 16))
Input:
Exact result:
125 result very near to the Higgs boson mass 125.18 GeV
56
1/2((sqrt(([-((-1/2 ((128)/16 + 16) - (32 (4sqrt3)^2)/(8) + 8))+(-(128*1/4)/(32) –
(16*(4sqrt3)^2)/(8) – (4)*1/2 + 8*1/2)-((-(128)/(16) – 16))]^1/3))+1))
Input:
Result:
Decimal approximation:
1.6180339887…. = golden ratio
Minimal polynomial:
Expanded form:
We have that:
Thence:
57
i.e.
From:
= P
= (p AE)/AC 1/2 t
2
= - κ
2
(M + (m AE
2
)/AC
2
) s
P - ((36*p AE)/(AC) 1/2 t^2) – κ^2(M + (m (AE)^2)/((AC)^2)) s
Input:
Result:
Alternate forms:
58
Expanded form:
Roots:
59
Roots for the variable κ:
Derivative:
Indefinite integral:
Alternative representation:
Series representations:
60
For P = 8, p = 4, m = 2 , M = 16, AC =8*4 = 32 , AE = 8*8 = 64, s = 1/2
From:
-(((√(10-2√5) -2))⁄((√5-1)))^2 * 1/2 (64*2/16 + 16) – 1/4(18*8*4*t^2)+8
where (((√(10-2√5) -2))⁄((√5-1))) = κ
Input:
61
Exact result:
Plots:
Polynomial discriminant:
62
For t = 0.9:
8 - (12 (-2 + sqrt(10 - 2 sqrt(5)))^2)/(-1 + sqrt(5))^2 - 144 *0.9^2
Input:
Result:
-109.60841083….
For t = 4.3:
8 - (12 (-2 + sqrt(10 - 2 sqrt(5)))^2)/(-1 + sqrt(5))^2 - 144 *4.3^2
Input:
Result:
-2655.52841083…
From the two expressions, after some calculations, we obtain:
-2*[(8 - (12 (-2 + sqrt(10 - 2 sqrt(5)))^2)/(-1 + sqrt(5))^2 - 144 *0.9^2)+ 8 - (12 (-2 +
sqrt(10 - 2 sqrt(5)))^2)/(-1 + sqrt(5))^2 - 144 *4.3^2]-128*11-26-(((√(10-2√5) -
2))⁄((√5-1)))
63
Input:
Result:
4095.99…. ≈ 4096 = 64
2
-2*[(8 - (12 (-2 + sqrt(10 - 2 sqrt(5)))^2)/(-1 + sqrt(5))^2 - 144 *0.9^2)+ 8 - (12 (-2 +
sqrt(10 - 2 sqrt(5)))^2)/(-1 + sqrt(5))^2 - 144 *4.3^2]-128*11-26+276-(((√(10-2√5) -
2))⁄((√5-1)))
Input:
Result:
4371.99…. ≈ 4372
Where 4372 is a value indicated in the fundamental Ramanujan paper “Modular
equations and Approximations to π”
65
From:
66
For Planck mass = 2,176434(24)×10
−8
; Planck density =
5,154 849 × 10
96
;
H = 2.2*10^-18
And:
M = 32.5 σv = 3*10^-26 (from WIMP - Wikipedia)
From:
(2Pi^2)/45*4.55*((32.5^3)*(3*10^-26))/((2.2e-18)*32.5)
Input interpretation:
Result:
0.0000287471….
67
For:
1/(((4.55)*(32^3))) = N
i
Input:
Result:
6.7071600274…*10
-6
Repeating decimal:
Rational approximation:
From:
68
d/dx*6.7071600274725e-6 = -0.0000287471/x^2*[(6.7071600274725e-6)^2-1]
Input interpretation:
d/dx(6.7071600274725e-6)
Derivative:
-0.0000287471/x^2*[(6.7071600274725e-6)^2-1]
Input interpretation:
Result:
Plots:
69
Alternate form assuming x is real:
Roots:
Properties as a real function:
Domain
Range
Parity
Derivative:
70
Indefinite integral:
Limit:
For x = -1:
(0.0000287471)/(-1)^2
Input interpretation:
Result:
0.0000287471
For x = 5.1
(0.0000287471)/(5.1)^2
Input interpretation:
Result:
71
Repeating decimal:
1.105232602845…*10
-6
From the two above expressions, we obtain:
1/((0.0000287471)/(5.1)^2) * ((0.0000287471)/(-1)^2)
Input interpretation:
Result:
26.01 ≈ 26
Rational form:
From which:
8((([1/((0.0000287471)/(5.1)^2) * ((0.0000287471)/(-1)^2) - 2-((((√(10-2√5) -
2))⁄((√5-1))))^3]^2-64)))+5
Input interpretation:
72
Result:
4096.04… ≈ 4096 = 64
2
8((([1/((0.0000287471)/(5.1)^2) * ((0.0000287471)/(-1)^2) - 2-((((√(10-2√5) -
2))⁄((√5-1))))^3]^2-64)))+5+276
Input interpretation:
Result:
4372.04…. ≈ 4372
Where 4372 is a value indicated in the fundamental Ramanujan paper “Modular
equations and Approximations to π”
73
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
74
“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
75
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:
76
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.
77
For
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
78
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...
Thence:
0.000244140625
6+
=
18
Dividing both sides by 0.000244140625, we obtain:
0.000244140625
0.000244140625
6+
=
1
0.000244140625
18
79
6+
= 0.0066650177536
((((exp((-Pi*sqrt(18)))))))*1/0.000244140625
Input interpretation:
Result:
0.00666501785…
Series representations:
Now:
80
6+
= 0.0066650177536
=
= 0.00666501785…
From:
ln(0.00666501784619)
Input interpretation:
Result:
-5.010882647757…
Alternative representations:
Series representations:
81
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:
82
(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:
83
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)))))
Input interpretation:
84
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:
From
85
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…..
Series representations:
86
87
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:
88
1.610609533…. result that is a good approximation to the value of the golden ratio
1.618033988749...
Series representations:
89
Now, we have:
For:
ξ = 1
= 0.989117352243
90
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:
91
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:
92
113+5
505
8
+
105+5
505
8
3
14
= 1,65578
Series representations:
93
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:
94
Result:
-0.034547055658…
Series representations:
95
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))]))))))))
96
Input interpretation:
Result:
1.6237116159…. result that is an approximation to the value of the golden ratio
1.618033988749...
Series representations:
97
98
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
99
Series representations:
100
101
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...
102
Series representations:
103
104
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
105
Series representations:
106
107
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:
108
Series representations:
109
110
From the previous expression
= -0.034547055658…
we have also:
111
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:
112
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
113
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. 2 e 3, Jacopo Riccati, In Lucca, presso Iacopo Giusti, 1762-1764.
Broken Scale Invariance, Gravity Mass, and Dark Energy in Modified Einstein
Gravity with Two Measure Finsler Like Variables - Panayiotis Stavrinos, Sergiu
I. Vacaru - arXiv:2106.00462v1 [physics.gen-ph] 28 May 2021
Cosmology - Part III Mathematical Tripos - Daniel Baumann
dbaumann@damtp.cam.ac.uk - https://pdfslide.net/reader/f/part-iii-mathematical-
tripos-particle-liantaowmy-teachingdark-matter-2016-06-27
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