1
On the possible mathematical connections between the κ formula regarding the
Riemann Zeta Function, some Ramanujan equations, the study of a Riccati
equation and some topics of String Theory
Michele Nardelli
1
, Antonio Nardelli
2
Abstract
In this research thesis, we describe the possible mathematical connections between
the κ formula regarding the Riemann Zeta Function, some Ramanujan equations, the
study of a Riccati equation 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
2
Conte Jacopo Riccati
Mathematician
(1676 – 1754)
Vesuvius landscape with gorse – Naples
4
5
From:
On the Zeros of the Davenport Heilbronn Function
S. A. Gritsenko - Received May 15, 2016 - ISSN 0081-5438, Proceedings of the
Steklov Institute of Mathematics, 2017, Vol. 296, pp. 65–87.
We have:
(
10 2
5 2) (
5 1 ) =
6
Input:
Decimal approximation:
0.28407904384…. = κ
Alternate forms:
Minimal polynomial:
Expanded forms:
7
For ((((√(10-2√5) -2))⁄((√5-1)))) = 8πG; G = 0.011303146014
Indeed:
((((√(10-2√5) -2))⁄((√5-1))))/(8π)
Input:
Result:
Decimal approximation:
0.01130314…. = g (gravitational coupling constant)
Property:
8
Alternate forms:
Expanded forms:
Series representations:
9
We note that:
((2 i (sqrt(5) - 1) t + sqrt(5) - 1)/(2 (sqrt(2 (5 - sqrt(5))) - 2)))
Input:
Exact result:
Plot:
10
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:
Exact result:
12
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.
We have the following equation: (in Italian)
Now:
=
2
+
3
×
2
=
2
+
2
×
+ =
2
+
2
×
+
2
+
2
= ((
2
+
2
) ×
)
1
2
+
2
+
2
+
2
=
We consider various calculations.
15
a)
integrate ([(x + y)] /(x^2+y^2))dx - integrate ( sqrty) dx
Input:
Result:
3D plot:
(figure that can be related to a D-brane)
16
Contour plot:
Alternate forms:
Series expansion of the integral at x = 0:
Series expansion of the integral at x = ∞:
17
Indefinite integral:
From
1/2 log(x^2 + y^2) + x (-sqrt(y)) + tan^(-1)(x/y)
For x = y = 2 :
-2 sqrt(2) + tan^(-1)(2/2) + 1/2 log(2^2 + 2^2)
Input:
Exact Result:
Decimal approximation:
-1.0033081905….
18
Alternate forms:
Alternative representations:
19
Series representations:
Integral representations:
20
Continued fraction representations:
For x = y = 1 :
-1 sqrt(1) + tan^(-1) + 1/2 log(1^2 + 1^2)
Input:
21
Exact Result:
Decimal approximation:
-0.727801738….
Alternate form:
Alternative representations:
22
Series representations:
Integral representations:
23
Continued fraction representations:
b)
integrate[(x + y)/((x^2+y^2))] - integrate (sqrty)
Input:
24
Result:
3D plot:
(figure that can be related to a D-brane)
Contour plot:
25
Alternate forms:
Series expansion of the integral at x = 0:
Series expansion of the integral at x = ∞:
Indefinite integral:
26
c)
integrate((x + y)dxdy) *(1/(x^2+y^2))= integrate (sqrt(y))
Input:
Result:
Implicit plot:
Alternate form assuming x and y are real:
Alternate form:
27
Alternate form assuming x and y are positive:
Expanded form:
Real solutions:
Solutions for the variable y:
28
Implicit derivatives:
d)
integrate((x + y)dxdy) *(1/(x^2+y^2)) - integrate (sqrt(y))
Input:
Result:
29
3D plot:
(figure that can be related to a D-brane)
Contour plot:
Alternate forms:
30
Expanded form:
Series expansion of the integral at x = 0:
Series expansion of the integral at x = ∞:
Indefinite integral:
We have also:
integrate ([(x + y)]dxdy) *(1/(x^2+y^2))= integrate (sqrtx)
Input:
31
Result:
Implicit plot:
Alternate form:
Alternate form assuming x and y are positive:
Expanded form:
Solutions:
32
Solution:
Solutions for the variable y:
Implicit derivatives:
From:
33
For x = 9/16 :
(((48*(9/16)^4.5 – 64*(9/16)^5 + 9(9/16)^4)^0.5 – 3(9/16)^2)) / ((2(3*9/16-
4*(9/16)^1.5))
From:
(-3 x^2 + sqrt(9 x^4 + 48 x^(9/2) - 64 x^5))/(2 (3 x - 4 x^(3/2)))
Input:
Result:
Input:
Result:
34
Plots:
Alternate form assuming x is real:
Alternate forms:
35
Expanded form:
Derivative:
36
Indefinite integral:
From this integral, we obtain:
(1024 x^(3/2) - 1296 ln((3 - 4 sqrt(x)) x^2) - 81 tan^(-1)(((8 sqrt(x) - 3) sqrt((-64 x +
48 sqrt(x) + 9) x^4))/(x^2 (64 x - 48 sqrt(x) - 9))))/4096
Input:
37
Result:
Plots:
Alternate form assuming x is real:
Alternate forms:
38
Expanded forms:
Alternate form assuming x>0:
39
Derivative:
Indefinite integral:
40
((1296 ln(-8 x^(5/2) + sqrt((-64 x + 48 sqrt(x) + 9) x^4) + 9 x^2) - (sqrt((-64 x + 48
sqrt(x) + 9) x^4) (256 x^(3/2) + 224x +192sqrt(x)+171))/x^2 + 1152x + 1728 sqrt(x)
+ 1296 ln(3-4 sqrt(x))))/4096
Input:
Result:
41
Plots:
Alternate form assuming x is real:
Alternate forms:
42
Alternate form assuming x>0:
43
Expanded forms:
44
Derivative:
45
From:
x = 2
(1024 2^(3/2) - 1296 ln((3 - 4 sqrt(2)) 2^2) - 81 tan^(-1)(((8 sqrt(2) - 3) sqrt((-64*2 +
48 sqrt(2) + 9) 2^4))/(2^2 (64*2 - 48 sqrt(2) - 9))))/4096
Input:
Exact Result:
Decimal approximation:
46
Polar coordinates:
1.02211716….
Polar forms:
Approximate form
47
Alternate forms:
48
Expanded forms:
Alternative representations:
49
Series representations:
50
51
Integral representations:
52
(1296 log((sqrt(-64*2 + 48 sqrt(2) + 9) - 8 sqrt(2) + 9) 2^2) + 1152*2 + 1728 sqrt(2) -
sqrt(-64*2 + 48 sqrt(2) + 9) (32 sqrt(2) (8*2 + 7 sqrt(2) + 6) + 171) + 1296 log(3 - 4
sqrt(2)))/4096
Input:
53
Exact result:
Decimal approximation:
Polar coordinates:
2.825965074….
54
Polar forms:
Approximate form
55
Alternate forms:
56
Expanded forms:
Alternative representations:
57
Series representations:
58
59
Integral representations:
We have also:
(1296 log((sqrt(-64*2 + 48 sqrt(2) + 9) - 8 sqrt(2) + 9) 2^2) + 1152*2 + 1728 sqrt(2) -
sqrt(-64*2 + 48 sqrt(2) + 9) (32 sqrt(2) (8*2 + 7 sqrt(2) + 6) + 171) + 1296 log(3 - 4
sqrt(2)))/2.825965074
60
Input interpretation:
Result:
Polar coordinates:
4096 = 64
2
Polar forms:
61
Alternative representations:
62
Series representations:
63
64
65
Integral representation:
27sqrt(((1296 ln((sqrt(-128+48 sqrt(2) + 9)-8sqrt(2)+9)2^2)+1152*2+1728 sqrt(2)-
sqrt(-128+48 sqrt(2)+9)(32 sqrt(2) (16 + 7 sqrt(2) + 6) + 171) + 1296 ln(3 - 4
sqrt(2)))/2.825965074))
Input interpretation:
Result:
Polar coordinates:
1728
66
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)
Polar forms:
Alternative representations:
67
Series representations:
68
69
Integral representation:
[27sqrt(((1296 ln((sqrt(-128+48 sqrt(2) + 9)-8sqrt(2)+9)2^2)+1152*2+1728 sqrt(2)-
sqrt(-128+48 sqrt(2)+9)(32 sqrt(2) (16 + 7 sqrt(2) + 6) + 171) + 1296 ln(3 - 4
sqrt(2)))/2.825965074))]^1/15
Input interpretation:
Result:
70
Polar coordinates:
1.64375 ≈ ζ(2) =
2
6
= 1.644934
Polar forms:
(1296 log((sqrt(-128+48sqrt(2)+ 9)-8 sqrt(2) + 9) 2^2) + 1152*2 + x* sqrt(2) - sqrt(-
128 + 48 sqrt(2) + 9) (32 sqrt(2) (16 + 7 sqrt(2) + 6) + 171) + 1296 log(3 - 4
sqrt(2)))/4096 = 2.825965074
Input interpretation:
Result:
71
Alternate forms:
Expanded form:
Alternate forms assuming x>0:
72
Alternate form assuming x is real:
Complex solution:
Result:
Polar coordinates:
4372.07 ≈ 4096 + 276
where 4372 is a value indicated in the fundamental Ramanujan paper “Modular
equations and Approximations to π”
73
We have:
(-0.0717626221871- 1.0195948363578i) + (2.5450758814147 - 1.2282782098824i)
Input interpretation:
Result:
Polar coordinates:
3.3421866659844
Polar forms:
74
(2sqrt3)/(7) (((-0.0717626221871- 1.0195948363578i) + (2.5450758814147 -
1.2282782098824i)))
Input interpretation:
Result:
Polar coordinates:
1.6539534611041 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:
113+5
505
8
+
105+5
505
8
3
14
= 1,65578
75
Polar forms:
And, for ((((√(10-2√5) -2))⁄((√5-1)))) = κ :
1/2 (((-0.0717626221871- 1.0195948363578i) + (2.5450758814147 -
1.2282782098824i))) - ((((√(10-2√5) -2))⁄((√5-1))))^2
Input interpretation:
Result:
Polar coordinates:
1.6122862492589 result that is a very good approximation to the value of the golden
ratio 1.618033988749...
76
Polar forms:
Observations
We note that, from the number 8, we obtain as follows:
77
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
“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
78
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:
79
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
80
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
81
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
82
6+
= 0.0066650177536
((((exp((-Pi*sqrt(18)))))))*1/0.000244140625
Input interpretation:
Result:
0.00666501785…
Series representations:
Now:
83
6+
= 0.0066650177536
=
= 0.00666501785…
From:
ln(0.00666501784619)
Input interpretation:
Result:
-5.010882647757…
Alternative representations:
Series representations:
84
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:
85
(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:
86
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:
87
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
88
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:
89
90
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:
91
Polar coordinates:
1.610609533…. result that is a good approximation to the value of the golden ratio
1.618033988749...
Series representations:
92
Now, we have:
93
For:
ξ = 1
= 0.989117352243
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….
94
Series representations:
From which:
1+1/(((4((2*e^(-0.989117352243/2))) /
((((1+sqrt(((1+1/3*(4Pi^2)/25*e^(2*0.989117352243)))))))))))
Input interpretation:
95
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:
113+5
505
8
+
105+5
505
8
3
14
= 1,65578
Series representations:
96
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)]
97
Input interpretation:
Result:
-0.034547055658…
Series representations:
98
99
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:
Result:
1.6237116159…. result that is an approximation to the value of the golden ratio
1.618033988749...
100
Series representations:
101
102
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
103
Series representations:
104
105
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...
106
Series representations:
107
108
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
109
Series representations:
110
111
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:
112
Series representations:
113
114
From the previous expression
= -0.034547055658…
115
we have also:
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:
116
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
117
References
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
March 27, 2018
AdS Vacua from Dilaton Tadpoles and Form Fluxes
J. Mourad and A. Sagnotti - arXiv:1612.08566v2 [hep-th] 22 Feb 2017