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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.

(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

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