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Gaussian Hamiltonian ZetaZeros:

If the appearance of ZetaZeros is “Normal’ :random and the randomness

is a space-filling Gaussian normal, the the energy spectrum can be calculated

in an Hamiltonian, using a Gaussian radial wave function on the complex plane.

It appears the radius increase as a function of ZetaZero[n] when this Hamiltonian is solved

for zero potential energy.

(

Clear[phi, phi0, r, n, m]

(* physics :so called natural units*)

hbar = 1;

c = 1;

(* s as deviation of Gaussian normal wave function*)

phi0 = (Sqrt[2/π] Sqrt[1/

s[n]])

(* quantum mass*)

m[n_] = hbar/(s[n]*c)

phi[r_, n_] = phi0*Exp[-r^2/(2*s[n])]

1/Integrate[phi[r, n], {r, 0, Infinity}]

ConditionalExpression[1, Re[1/s[n]] > 0]

(* kinetic energy*)

T = (hbar^2/(2*m[n]))*D[phi[r, n], {r, 2}]

(* Hamiltonian*)

H[n_] = T + V[n]*phi[r, n]

Solve[H[n] - ZetaZero[n]*phi[r, n] == 0, V[n]]

(* solving for zero potential energy*)

phi[r, n] /.

Solve[(-c hbar r^2 + c hbar s[n] + 2 s[n] ZetaZero[n])/(2 s[n]) == 0,

s[n]][[1]]

FullSimplify[

ExpandAll[

T /. Solve[(-c hbar r^2 + c hbar s[n] + 2 s[n] ZetaZero[n])/(2 s[n]) == 0,

s[n]][[1]]]]

(* setting Kinetic energy equal to the ZetaZero spectrum in n*)

(* solving \

for Gaussian radius on the complex plane*)

Solve[(E^(-(1/2) - ZetaZero[n]) ZetaZero[n] Sqrt[(2 + 4 ZetaZero[n])/r^2])/

Sqrt[π] - ZetaZero[n] == 0, r]

(* Plotting for ZetaZeros: the Gaussian radius on the complex plane*)

rr = Table[{Re[

N[E^(1/2 (-1 - 2 ZetaZero[n])) Sqrt[2/π] Sqrt[1 + 2 ZetaZero[n]]]],

Im[N[E^(1/2 (-1 - 2 ZetaZero[n])) Sqrt[2/π] Sqrt[

1 + 2 ZetaZero[n]]]]}, {n, 1, 300}];

g0 = ListPlot[rr, PlotStyle -> Red, ImageSize -> 1000, PlotRange -> All]

Max[Abs[rr]]

Min[Abs[rr]]

rr1 = Table[{Re[

N[E^(1/2 (-1 - 2 ZetaZero[n])) Sqrt[2/π] Sqrt[1 + 2 ZetaZero[n]]]],

Im[N[E^(1/2 (-1 - 2 ZetaZero[n])) Sqrt[2/π] Sqrt[

1 + 2 ZetaZero[n]]]]}, {n, 301, 600}];

g1 = ListPlot[rr1, PlotStyle -> Red, ImageSize -> 1000, PlotRange -> All]

Show[{g0, g1}]

(* end*)

If the appearance of ZetaZeros is “Normal’ :random and the randomness

is a space-filling Gaussian normal, the the energy spectrum can be calculated

in an Hamiltonian, using a Gaussian radial wave function on the complex plane.

It appears the radius increase as a function of ZetaZero[n] when this Hamiltonian is solved

for zero potential energy.

(

**mathematica**)Clear[phi, phi0, r, n, m]

(* physics :so called natural units*)

hbar = 1;

c = 1;

(* s as deviation of Gaussian normal wave function*)

phi0 = (Sqrt[2/π] Sqrt[1/

s[n]])

(* quantum mass*)

m[n_] = hbar/(s[n]*c)

phi[r_, n_] = phi0*Exp[-r^2/(2*s[n])]

1/Integrate[phi[r, n], {r, 0, Infinity}]

ConditionalExpression[1, Re[1/s[n]] > 0]

(* kinetic energy*)

T = (hbar^2/(2*m[n]))*D[phi[r, n], {r, 2}]

(* Hamiltonian*)

H[n_] = T + V[n]*phi[r, n]

Solve[H[n] - ZetaZero[n]*phi[r, n] == 0, V[n]]

(* solving for zero potential energy*)

phi[r, n] /.

Solve[(-c hbar r^2 + c hbar s[n] + 2 s[n] ZetaZero[n])/(2 s[n]) == 0,

s[n]][[1]]

FullSimplify[

ExpandAll[

T /. Solve[(-c hbar r^2 + c hbar s[n] + 2 s[n] ZetaZero[n])/(2 s[n]) == 0,

s[n]][[1]]]]

(* setting Kinetic energy equal to the ZetaZero spectrum in n*)

(* solving \

for Gaussian radius on the complex plane*)

Solve[(E^(-(1/2) - ZetaZero[n]) ZetaZero[n] Sqrt[(2 + 4 ZetaZero[n])/r^2])/

Sqrt[π] - ZetaZero[n] == 0, r]

(* Plotting for ZetaZeros: the Gaussian radius on the complex plane*)

rr = Table[{Re[

N[E^(1/2 (-1 - 2 ZetaZero[n])) Sqrt[2/π] Sqrt[1 + 2 ZetaZero[n]]]],

Im[N[E^(1/2 (-1 - 2 ZetaZero[n])) Sqrt[2/π] Sqrt[

1 + 2 ZetaZero[n]]]]}, {n, 1, 300}];

g0 = ListPlot[rr, PlotStyle -> Red, ImageSize -> 1000, PlotRange -> All]

Max[Abs[rr]]

Min[Abs[rr]]

rr1 = Table[{Re[

N[E^(1/2 (-1 - 2 ZetaZero[n])) Sqrt[2/π] Sqrt[1 + 2 ZetaZero[n]]]],

Im[N[E^(1/2 (-1 - 2 ZetaZero[n])) Sqrt[2/π] Sqrt[

1 + 2 ZetaZero[n]]]]}, {n, 301, 600}];

g1 = ListPlot[rr1, PlotStyle -> Red, ImageSize -> 1000, PlotRange -> All]

Show[{g0, g1}]

(* end*)

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News Link:

Brunel University News (press release)

Physicists make major breakthrough towards proof of Riemann hypothesis

Brunel University News (press release) - Mar 24, 2017

The function is useful in number theory, such as for investigating properties of prime numbers. Yet in the century and a half since, and in spite of hard efforts by many mathematicians, no one has been able to prove that all of the (nontrivial) zeros ...

http://news.google.com/news/url?sr=1&ct2=us%2F1_0_s_3_1_a&sa=t&usg=AFQjCNHp7SFo375mHYQ74uzdn8q7v8kGcQ&cid=null&url=http%3A%2F%2Fwww.brunel.ac.uk%2Fnews-and-events%2Fnews%2Farticles%2FPhysicists-make-major-breakthrough-towards-proof-of-Riemann-hypothesis&ei=n4zWWMDJJtDEqQK5g6xo&rt=SECTION&vm=STANDARD&bvm=section&did=-7587097314474524690&sid=-5323590909069302167&ssid=cstm&st=2&at=dt0

Brunel University News (press release)

Physicists make major breakthrough towards proof of Riemann hypothesis

Brunel University News (press release) - Mar 24, 2017

The function is useful in number theory, such as for investigating properties of prime numbers. Yet in the century and a half since, and in spite of hard efforts by many mathematicians, no one has been able to prove that all of the (nontrivial) zeros ...

http://news.google.com/news/url?sr=1&ct2=us%2F1_0_s_3_1_a&sa=t&usg=AFQjCNHp7SFo375mHYQ74uzdn8q7v8kGcQ&cid=null&url=http%3A%2F%2Fwww.brunel.ac.uk%2Fnews-and-events%2Fnews%2Farticles%2FPhysicists-make-major-breakthrough-towards-proof-of-Riemann-hypothesis&ei=n4zWWMDJJtDEqQK5g6xo&rt=SECTION&vm=STANDARD&bvm=section&did=-7587097314474524690&sid=-5323590909069302167&ssid=cstm&st=2&at=dt0

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Four Proof A Theorem Called Piyush Therorem

https://edupediapublications.org/journals/index.php/IJR/article/view/3743/3589

https://edupediapublications.org/journals/index.php/IJR/article/view/3743/3589

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General news link:

https://blog.oup.com/2017/03/defense-mathematics-excerpt/

https://blog.oup.com/2017/03/defense-mathematics-excerpt/

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