1
Further mathematical connections between some Ramanujan equations
concerning p(n) and τ(n), several equations concerning Mock Modularity in M-
theory duality, various parameters concerning Particle Physics, 𝝓 and ζ(2). II
Michele Nardelli
1
, Antonio Nardelli
2
Abstract
In this paper we describe and analyze further new mathematical connections between
some Ramanujan formulas concerning p(n) and τ(n), several equations concerning
Mock Modularity in M-theory duality, various parameters concerning Particle
Physics, 𝜙 and ζ(2).
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
https://blog.vedicmathsindia.org/srinivasa-ramanujan/
From:
Duality and Mock Modularity
Atish Dabholkar, Pavel Putrov, Edward Witten - arXiv:2004.14387v1 [hep-th] 29
Apr 2020
3
Thence, we have:
That is:
4
Now, we have that:
For: n
1
=9/2 ; n
2
= 1/2 ; -3.05774*10
16
= η(τ); n = 2; z = 3/2
From
we obtain:
1/(-3.05774e+16)^2 * (exp(2Pi))^(((81/4+1/4)/2)+(9/2+1/2)3/2)
Input interpretation:
Result:
2.91447…*10
15
From which:
1/(1364-29)(((1/(-3.05774e+16)^2 * (exp(2Pi))^(((81/4+1/4)/2)+(9/2+1/2)3/2))))^3
where 1364 and 29 are Lucas numbers
Input interpretation:
5
Result:
1.85437…*10
43
result practically equal to the Planck frequency 1.8549×10
43
Hz
We have also:
1/(2.718281828459)[1/(((1/(-3.05774e+16)^2 *
(exp(2Pi))^(((81/4+1/4)/2)+(9/2+1/2)3/2))))]^1/3
Input interpretation:
Result:
2.57545…*10
-6
Now, from:
For eq. (3.46)
and q = 0.5 , we obtain:
sum (((0.5^(5n+1)(1-0.5^2)(1+0.5^(5n+2))(1-0.5^(5n+2)))))/((((1-0.5^(5n+1))(1-
0.5^(5n+3))^2))), n=-20..infinity
6
Input interpretation:
Approximated sum:
63.5858
Convergence tests:
Partial sum formula:
For q = exp(2Pi) = 535.49165 , we obtain:
sum (((535.49165^(5n+1)(1-0.5^2)(1+535.49165^(5n+2))(1-
535.49165^(5n+2)))))/((((1-535.49165^(5n+1))(1-535.49165^(5n+3))^2))), n=-
2..infinity
Input interpretation:
Result:
7
Convergence tests:
Partial sum formula:
that is:
8.32545×10^-8 ψ_0^(0)(m+1.2) - 2.65006×10^-9 ψ_0^(1)(m+1.6)+2.61551×10^-6
(m+1)+4.9026×10^-9
We note that:
(((535.49165^(5*2+1)(1-0.5^2)(1+535.49165^(5*2+2))(1-
535.49165^(5*2+2)))))/((((1-535.49165^(5*2+1))(1-535.49165^(5*2+3))^2)))
Input interpretation:
Result:
2.615506821126….*10
-6
result very near to the value previously obtained
2.57545…*10
-6
From the above expression,
we obtain also:
48log(((1/(-3.05774e+16)^2 * (exp(2Pi))^(((81/4+1/4)/2)+(9/2+1/2)3/2))))+18+2
8
Input interpretation:
Result:
1729.206…
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)
From which:
(((48log(((1/(-3.05774e+16)^2 *
(exp(2Pi))^(((81/4+1/4)/2)+(9/2+1/2)3/2))))+18+2)))^1/15
Input interpretation:
Result:
1.6438283….≈ ζ(2) =
=1.644934…
And again:
4log(((1/(-3.05774e+16)^2 * (exp(2Pi))^(((81/4+1/4)/2)+(9/2+1/2)3/2))))-Pi
Input interpretation:
9
Result:
139.2923…. result practically equal to the rest mass of Pion meson 139.57 MeV
4log(((1/(-3.05774e+16)^2 * (exp(2Pi))^(((81/4+1/4)/2)+(9/2+1/2)3/2))))-18+1
Input interpretation:
Result:
125.4339…. result very near to the Higgs boson mass 125.18 GeV
Now, from
For: : n
1
=9/2 ; n
2
= 1/2 ; -3.05774*10
16
= η(τ); -3.3805503749….*10
34
= τ
2
(n);
z = 3/2; we obtain:
1/(-3.05774e+16)^2 * (-1)^(9/2+1/2+1) *
(exp(2Pi))^[((((((9/2+1/2)^2+(1/2+1/2)^2))/2+(9/2+1/2+1)3/2)))]
Input interpretation:
10
Result:
1.15281…*10
27
From which, we obtain also:
2ln(((1/(-3.05774e+16)^2 * (-1)^(9/2+1/2+1) *
(exp(2Pi))^[((((((9/2+1/2)^2+(1/2+1/2)^2))/2+(9/2+1/2+1)3/2)))])))+1
Input interpretation:
Result:
125.6240…. result very near to the Higgs boson mass 125.18 GeV
2ln(((1/(-3.05774e+16)^2 * (-1)^(9/2+1/2+1) *
(exp(2Pi))^[((((((9/2+1/2)^2+(1/2+1/2)^2))/2+(9/2+1/2+1)3/2)))])))+18-3
Input interpretation:
Result:
139.6240…. result practically equal to the rest mass of Pion meson 139.57 MeV
11
27*1/2(((2ln(((1/(-3.05774e+16)^2 (-
1)^(9/2+1/2+1)(exp(2Pi))^[((((9/2+1/2)^2+(1/2+1/2)^2))/2+(9/2+1/2+1)3/2)])))+Pi)))
+4
Input interpretation:
Result:
1728.836….
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)
With regard 27 (From Wikipedia):
“The fundamental group of the complex form, compact real form, or any algebraic
version of E
6
is the cyclic group Z/3Z, and its outer automorphism group is the cyclic
group Z/2Z. Its fundamental representation is 27-dimensional (complex), and a basis
is given by the 27 lines on a cubic surface. The dual representation, which is
inequivalent, is also 27-dimensional. In particle physics, E
6
plays a role in
some grand unified theories”.
(((((27*1/2(((2ln(((1/(-3.05774e+16)^2 (-
1)^(9/2+1/2+1)(exp(2Pi))^[((((9/2+1/2)^2+(1/2+1/2)^2))/2+(9/2+1/2+1)3/2)])))+Pi)))
+4)))))^1/15
12
Input interpretation:
Result:
1.6438048….≈ ζ(2) =
=1.644934…
Dividing with the previous result, i.e.
2.91447…*10
15
we obtain:
((((1/(-3.05774e+16)^2 * (-1)^(9/2+1/2+1) *
(exp(2Pi))^[((((((9/2+1/2)^2+(1/2+1/2)^2))/2+(9/2+1/2+1)3/2)))]))))*1/(2.91447*10^
15)
Input interpretation:
Result:
3.95548…*10
11
From which, we obtain:
13
1/(64Pi)*((((1/(-3.05774e+16)^2 * (-1)^(9/2+1/2+1) *
(exp(2Pi))^[((((((9/2+1/2)^2+(1/2+1/2)^2))/2+(9/2+1/2+1)3/2)))]))))*1/(2.91447*10^
15)
Input interpretation:
Result:
1.96729…*10
9
result very near to the value of the Planck energy 1.9561×10
9
J
From:
Mock modularity and a secondary elliptic genus
Davide Gaiotto and Theo Johnson-Freyd - arXiv:1904.05788v1 [hep-th] 11 Apr 2019
We have that:
For q = exp(2Pi) = 535.49165; x = exp(-2Pi) = 0.001867442731708; n = 2 we
obtain:
1/(((0.001867442731708^0.5-0.001867442731708^-0.5)))*((1-535.49165^2)^2) /
(((1-0.001867442731708*535.49165^2)(1-0.001867442731708^-1*535.49165^2)))
14
Input interpretation:
Result:
-0.0433755…
From which:
-5/((((1/(((0.0018674427^0.5-0.0018674427^-0.5)))*((1-535.49165^2)^2) / (((1-
0.0018674427*535.49165^2)(1-0.0018674427^-1*535.49165^2)))))))+21+Pi
Input interpretation:
Result:
139.414…. result practically equal to the rest mass of Pion meson 139.57 MeV
Series representations:
15
Integral representations:
-5/((((1/(((0.0018674427^0.5-0.0018674427^-0.5)))*((1-535.49165^2)^2) / (((1-
0.0018674427*535.49165^2)(1-0.0018674427^-1*535.49165^2)))))))+11-1
Input interpretation:
Result:
125.273…. result very near to the Higgs boson mass 125.18 GeV
16
27*1/2(((-5/((((1/(((0.0018674427^0.5-0.0018674427^-0.5)))*((1-535.49165^2)^2) /
(((1-0.0018674427*535.49165^2)(1-0.0018674427^-1*535.49165^2)))))))+13)))-e
Input interpretation:
Result:
1728.96…
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:
17
[27*1/2(((-5/((((1/(((0.0018674427^0.5-0.0018674427^-0.5)))*((1-535.49165^2)^2) /
(((1-0.0018674427*535.49165^2)(1-0.0018674427^-1*535.49165^2)))))))+13)))-
e]^1/15
Input interpretation:
Result:
1.643813…≈ ζ(2) =
=1.644934…
18
Now, we have that:
From
https://mss-
cat.trin.cam.ac.uk/manuscripts/uv/view.php?n=Add.Ms.a.94.13#?c=0&m=0&s=0&c
v=26&xywh=-269%2C1068%2C4667%2C2707
For x = 2 , Q = -312.388; 11J = 3071.13; J = 279.1936; R = 9.83114i
we obtain:
p(4)*2 + p(23)*2^2 + p(42)*2^3 + p(61)*2^4 (1-2^19)(1-2^38)(1-2^57) = 5*
2*9.83114i((1-2)(1-2^2)(1-2^3))^24 +19*279.1936
(((5*2 + 1255*2^2 + 53174*2^3 + 1121505*2^4 (1-2^19)(1-2^38)(1-2^57))))x = 5*
2*9.83114*i*((1-2)(1-2^2)(1-2^3))^24 +19*279.1936
Input interpretation:
Result:
19
Alternate form:
Complex solution:
Input interpretation:
Result:
Polar coordinates:
1.42734*10
-8
Indeed, we have that:
(((5*2 + 1255*2^2 + 53174*2^3 + 1121505*2^4 (1-2^19)(1-2^38)(1-
2^57))))*(1.42734e-8)
Input interpretation:
Result:
Repeating decimal:
-5.3194571…*10
33
5* 2*9.83114*i*((1-2)(1-2^2)(1-2^3))^24 +19*279.1936
20
Input interpretation:
Result:
Polar coordinates:
5.31945*10
33
From
we obtain:
-1/((((((5*2 + 1255*2^2 + 53174*2^3 + 1121505*2^4 (1-2^19)(1-2^38)(1-
2^57))))*(1.42734e-8)))) * 10^16
Input interpretation:
Result:
1.8798910886…*10
-18
result very near to the value of the Planck charge
1.875545956(41)×10
−18
C
And:
1/(2207+322-18)((((((5*2 + 1255*2^2 + 53174*2^3 + 1121505*2^4 (1-2^19)(1-
2^38)(1-2^57))))*(1.42734e-8))))^4 * 1.616255e-35
where 2207, 322 and 18 are Lucas numbers, while 1.616255e-35 is the Planck length
21
Input interpretation:
Result:
5.15385756…*10
96
result very near to the value of Planck density
5.1550×10
96
kg/m
3
We have also:
11*2log(((((5*2 + 1255*2^2 + 53174*2^3 + 1121505*2^4 (1-2^19)(1-2^38)(1-
2^57))))*(1.42734e-8)))+18
Input interpretation:
Result:
Polar coordinates:
1727.83….
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)
(((11*2log(((((5*2 + 1255*2^2 + 53174*2^3 + 1121505*2^4 (1-2^19)(1-2^38)(1-
2^57))))*(1.42734e-8)))+18)))^1/15
22
Input interpretation:
Result:
Polar coordinates:
1.64374 ≈ ζ(2) =
=1.644934…
And again:
2log(((((5*2 + 1255*2^2 + 53174*2^3 + 1121505*2^4 (1-2^19)(1-2^38)(1-
2^57))))*(1.42734e-8)))-18+2
Input interpretation:
Result:
Polar coordinates:
139.455 result practically equal to the rest mass of Pion meson 139.57 MeV
2log(((((5*2 + 1255*2^2 + 53174*2^3 + 1121505*2^4 (1-2^19)(1-2^38)(1-
2^57))))*(1.42734e-8)))-29-1
23
Input interpretation:
Result:
Polar coordinates:
125.471 result very near to the Higgs boson mass 125.18 GeV
Now, we have:
For x = 2 , Q = -312.388; 11J = 3071.13; J = 279.1936; R = 9.83114i
we obtain:
(1*2+1575*2^2+124754*2^3+4087968*2^4)*(1-2^23)(1-2^46)(1-2^69)
Input:
24
Result:
Result:
-2.314113975….*10
49
(-312.388*9.831141i)*2(((((1-2)(1-2^2)(1-2^3)))))^24
Input interpretation:
Result:
Polar coordinates:
3.32347*10
35
We have that:
(-6.96295×10^13 i)*((((-312.388*9.831141i)*2(((((1-2)(1-2^2)(1-2^3)))))^24)))
Input interpretation:
Result:
Result:
-2.31411327328….*10
49
From the previous expression:
25
we obtain, performing the ratio between the two results:
sqrt55/((((((((5*2 + 1255*2^2 + 53174*2^3 + 1121505*2^4 (1-2^19)(1-2^38)(1-
2^57))))*(1.42734e-8) / (-
23141132732825011171211765990445654368965130832732))))))^2
Input interpretation:
Result:
1.40351…*10
32
result very near to the value of Planck temperature
1.416785(16)×10
32
K
Now, we have that:
For x = 2 , Q = -312.388; R = 9.83114i , we obtain:
(-312.388)^2*2((1-2)(1-2^2)(1-2^3))^24
Input interpretation:
26
Result:
1.0560433705….*10
37
(-312.388^2*9.83114i)*2((1-2)(1-2^2)(1-2^3))^24
Input interpretation:
Result:
Polar coordinates:
1.03821*10
38
Multiplying the two results and performing the following calculations, we obtain:
(138^3)/(2(sqrt2+1)) * sqrt[((((-312.388^2*9.83114i)*2((1-2)(1-2^2)(1-
2^3))^24)))*((((-312.388)^2*2((1-2)(1-2^2)(1-2^3))^24)))]
where 138
3
is a Ramanujan cube (see Ramanujan’s taxicab numbers)
Input interpretation:
Result:
Polar coordinates:
1.80225 * 10
43
result very near to the value of the Planck frequency 1.8549×10
43
Hz
27
Furthermore, we have also:
10*ln[((((-312.388^2*9.83114i)*2((1-2)(1-2^2)(1-2^3))^24)))*((((-312.388)^2*2((1-
2)(1-2^2)(1-2^3))^24)))]
Input interpretation:
Result:
Polar coordinates:
1727.93
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)
Alternative representations:
28
Series representations:
Integral representations:
((((10*ln[((((-312.388^2*9.83114i)*2((1-2)(1-2^2)(1-2^3))^24)))*((((-
312.388)^2*2((1-2)(1-2^2)(1-2^3))^24)))]))))^1/15
Input interpretation:
29
Result:
Polar coordinates:
1.64375….≈ ζ(2) =
=1.644934…
From:
Topics in Analytic Number Theory - Authors: Rademacher, Hans - © 1973
We have the following equation:
(pag.274-276)
30
From:
For:
= 16
=
We know that (sequence OEIS A000041):
p(576)= 134003339931725153597473
from the above equation
31
we obtain:
sqrt2/Pi * 16^(9/4) / (sqrt(576-1/24))^3*[((10/99((((Pi*sqrt(2/3)*(sqrt(576-
1/24))/16))))^3+8/11((((((sinh(((Pi*sqrt(2/3))(sqrt(576-1/24))/16)))-
((Pi*sqrt(2/3))sqrt(576-1/24))/16)))))))]
Input:
Exact result:
Decimal approximation:
0.3333712569717784………..
Alternate forms:
Alternative representations:
32
Alternative representations:
Series representations:
Integral representations:
33
34
From
We obtain:
16^(-3/4)*x =
(0.333371256971778445271446116998556334523759081116569981866)
Input interpretation:
Result:
Plot:
35
Alternate form:
Solution:
2.66697005577…..
and 0.3333712569717… < 1/2
Thence:
(pag. 278)
we obtain:
exp((((Pi*sqrt(2/3)*(sqrt(576-1/24))*1/((sqrt(576-1/24))
partial derivative [exp((((Pi*sqrt(2/3)*(sqrt(n-1/24))))))*1/((sqrt(n-1/24)))]
Derivative:
Plots:
36
Alternate forms:
Root:
Series expansion at n = 0:
37
Big‐O notation »
Series expansion at n = ∞:
Big‐O notation »
Indefinite integral:
From
For n = 576, we obtain:
(4 sqrt(6) e^(1/6 π sqrt(24 *576 - 1)) (π (24 *576 - 1)^(3/2) - 144 *576 + 6))/(24 *576
- 1)^(5/2)
Input:
38
Exact result:
Decimal approximation:
1.19072230527….*10
24
Alternate forms:
Series representations:
39
In conclusion, we obtain:
1/(2Pi*sqrt2)*( 1.1907223052777592320795791961112576317405153985450700 ×
10^24) + 0.333371256971778445271446116998556334523759081116569981866
Input interpretation:
Result:
134003339931721156234346.64929 = 1.3400333993*10
23
And
p(576)= 134003339931725153597473 = 1.34003339931725153597473 × 10
23
From:
40
From the eq.(16.2.42), for n = 576, we obtain:
1/(sqrt2) * 576^(-27/4) * exp(4Pi*sqrt576)
Input:
Exact result:
Decimal approximation:
1.5731249093917….*10
112
41
Property:
Series representations:
The ratio between the two results is:
(((1/(sqrt2) * 576^(-27/4) * exp(4Pi*sqrt576))))/134003339931725153597473
Input:
42
Exact result:
Decimal approximation:
1.173944552571….*10
89
Property:
Series representations:
43
From which:
((2(pi - sqrt(3))/2))^2 / ((3+golden ratio^2)) * sqrt[(((1/(sqrt2) * 576^(-27/4) *
exp(4Pi*sqrt576))))/134003339931725153597473]
Input:
Exact result:
Decimal approximation:
1.2117007517…*10
44
result very near to the value of Planck force 1.2103×10
44
N
Alternate forms:
44
Series representations:
45
We have also:
ln((((((1/(sqrt2) * 576^(-27/4) *
exp(4Pi*sqrt576))))/134003339931725153597473)))-76-4
where 76 and 4 are Lucas numbers
Input:
Exact result:
Decimal approximation:
125.09044276…. result very near to the Higgs boson mass 125.18 GeV
Alternate forms:
46
Alternative representations:
Series representations:
47
Integral representations:
48
ln((((((1/(sqrt2) * 576^(-27/4) *
exp(4Pi*sqrt576))))/134003339931725153597473)))-47-7-11-1/2
where 47, 7 and 11 are Lucas numbers
Input:
Exact result:
Decimal approximation:
139.59044276…. result practically equal to the rest mass of Pion meson 139.57 MeV
Alternate forms:
Alternative representations:
49
Series representations:
50
Integral representations:
27*1/2*(((ln((((((1/(sqrt2) * 576^(-27/4) *
exp(4Pi*sqrt576))))/134003339931725153597473)))-76-1)))
Input:
Exact result:
Decimal approximation:
1729.220977….
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
51
curve. The number 1728 is one less than the Hardy–Ramanujan number 1729
(taxicab number)
Alternate forms:
Alternative representations:
52
Series representations:
Integral representations:
53
(((27*1/2*(((ln((((((1/(sqrt2) * 576^(-27/4) *
exp(4Pi*sqrt576))))/134003339931725153597473)))-76-1))))))^1/15
Input:
Exact result:
Decimal approximation:
1.6438292339….≈ ζ(2) =
=1.644934…
Alternate forms:
54
All 15th roots of 27/2 (log(e^(96
π)/(469809459062265264115215027256239228715008 sqrt(3))) - 77):
55
Alternative representations:
Series representations:
56
Integral representations:
57
58
Observations
From:
https://www.scientificamerican.com/article/mathematics-
ramanujan/?fbclid=IwAR2caRXrn_RpOSvJ1QxWsVLBcJ6KVgd_Af_hrmDYBNyU8m
pSjRs1BDeremA
Ramanujan's statement concerned the deceptively simple concept of partitions—the
different ways in which a whole number can be subdivided into smaller numbers.
Ramanujan's original statement, in fact, stemmed from the observation of patterns,
such as the fact that p(9) = 30, p(9 + 5) = 135, p(9 + 10) = 490, p(9 + 15) = 1,575
and so on are all divisible by 5. Note that here the n's come at intervals of five units.
Ramanujan posited that this pattern should go on forever, and that similar patterns
exist when 5 is replaced by 7 or 11—there are infinite sequences of p(n) that are all
divisible by 7 or 11, or, as mathematicians say, in which the "moduli" are 7 or 11.
Then, in nearly oracular tone Ramanujan went on: "There appear to be
corresponding properties," he wrote in his 1919 paper, "in which the moduli are
powers of 5, 7 or 11...and no simple properties for any moduli involving primes other
than these three." (Primes are whole numbers that are only divisible by themselves or
by 1.) Thus, for instance, there should be formulas for an infinity of n's separated by
5^3 = 125 units, saying that the corresponding p(n)'s should all be divisible by 125.
In the past methods developed to understand partitions have later been applied to
physics problems such as the theory of the strong nuclear force or the entropy of
black holes.
From Wikipedia
In particle physics, Yukawa's interaction or Yukawa coupling, named after Hideki
Yukawa, is an interaction between a scalar field ϕ and a Dirac field ψ. The Yukawa
interaction can be used to describe the nuclear force between nucleons (which
are fermions), mediated by pions (which are pseudoscalar mesons). The Yukawa
interaction is also used in the Standard Model to describe the coupling between
the Higgs field and massless quark and lepton fields (i.e., the fundamental fermion
particles). Through spontaneous symmetry breaking, these fermions acquire a mass
proportional to the vacuum expectation value of the Higgs field.
Can be this the motivation that from the development of the Ramanujan’s equations
we obtain results very near to the dilaton mass calculated as a type of Higgs boson:
59
125 GeV for T = 0 and to the Higgs boson mass 125.18 GeV and practically equal to
the rest mass of Pion meson 139.57 MeV
Note that:
Thence:
And
That are connected with 64, 128, 256, 512, 1024 and 4096 = 64
2
(Modular equations and approximations to π - S. Ramanujan - Quarterly Journal of
Mathematics, XLV, 1914, 350 – 372)
All the results of the most important connections are signed in blue throughout the
drafting of the paper. We highlight as in the development of the various equations we
use always the constants π, ϕ, 1/ϕ, the Fibonacci and Lucas numbers, linked to the
golden ratio, that play a fundamental role in the development, and therefore, in the
final results of the analyzed expressions.
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In mathematics, the Fibonacci numbers, commonly denoted F
n
, form a sequence,
called the Fibonacci sequence, such that each number is the sum of the two preceding
ones, starting from 0 and 1. Fibonacci numbers are strongly related to the golden
ratio: Binet's formula expresses the nth Fibonacci number in terms of n and the
golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends
to the golden ratio as n increases.
Fibonacci numbers are also closely related to Lucas numbers ,in that the Fibonacci
and Lucas numbers form a complementary pair of Lucas sequences
The beginning of the sequence is thus:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765,
10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040,
1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169,
63245986, 102334155...
The Lucas numbers or Lucas series are an integer sequence named after the
mathematician François Édouard Anatole Lucas (1842–91), who studied both that
sequence and the closely related Fibonacci numbers. Lucas numbers and Fibonacci
numbers form complementary instances of Lucas sequences.
The Lucas sequence has the same recursive relationship as the Fibonacci sequence,
where each term is the sum of the two previous terms, but with different starting
values. This produces a sequence where the ratios of successive terms approach
the golden ratio, and in fact the terms themselves are roundings of integer powers of
the golden ratio.
[1]
The sequence also has a variety of relationships with the
Fibonacci numbers, like the fact that adding any two Fibonacci numbers two terms
apart in the Fibonacci sequence results in the Lucas number in between.
The sequence of Lucas numbers is:
2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778,
9349, 15127, 24476, 39603, 64079, 103682, 167761, 271443, 439204, 710647,
1149851, 1860498, 3010349, 4870847, 7881196, 12752043, 20633239, 33385282,
54018521, 87403803……
All Fibonacci-like integer sequences appear in shifted form as a row of the Wythoff
array; the Fibonacci sequence itself is the first row and the Lucas sequence is the
second row. Also like all Fibonacci-like integer sequences, the ratio between two
consecutive Lucas numbers converges to the golden ratio.
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A Lucas prime is a Lucas number that is prime. The first few Lucas primes are:
2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451,
6643838879, ... (sequence A005479 in the OEIS).
In geometry, a golden spiral is a logarithmic spiral whose growth factor is φ,
the golden ratio.
[1]
That is, a golden spiral gets wider (or further from its origin) by a
factor of φ for every quarter turn it makes. Approximate logarithmic spirals can
occur in nature, for example the arms of spiral galaxies
[3]
- golden spirals are one
special case of these logarithmic spirals
We observe that 1728 and 1729 are results 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. As a consequence, it is sometimes called a Zagier as a
pun on the Gross–Zagier theorem. The number 1728 is one less than the Hardy–
Ramanujan number 1729 (taxicab number).
Furthermore, we obtain as results of our computations, always values very near to the
Higgs boson mass 125.18 GeV and practically equals to the rest mass of Pion meson
139.57 MeV. In conclusion we obtain also many results that are very good
approximations to the value of the golden ratio 1.618033988749... and to ζ(2) =
=1.644934…
We note how the following three values: 137.508 (golden angle), 139.57 (mass of
the Pion - meson Pi) and 125.18 (mass of the Higgs boson), are connected to each
other. In fact, just add 2 to 137.508 to obtain a result very close to the mass of
the Pion and subtract 12 to 137.508 to obtain a result that is also very close to
the mass of the Higgs boson. We can therefore hypothesize that it is the golden
angle (and the related golden ratio inherent in it) to be a fundamental ingredient
both in the structures of the microcosm and in those of the macrocosm.
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References
Duality and Mock Modularity
Atish Dabholkar, Pavel Putrov, Edward Witten - arXiv:2004.14387v1 [hep-th] 29
Apr 2020
Mock modularity and a secondary elliptic genus
Davide Gaiotto and Theo Johnson-Freyd - arXiv:1904.05788v1 [hep-th] 11 Apr 2019
https://mss-
cat.trin.cam.ac.uk/manuscripts/uv/view.php?n=Add.Ms.a.94.13#?c=0&m=0&s=0&c
v=26&xywh=-269%2C1068%2C4667%2C2707
Topics in Analytic Number Theory - Authors: Rademacher, Hans - © 1973