Conceptually speaking, this is when " Number Theory Meets Quantum Mechanics", (DeJ). Or more specifically, this is when number theory meets plasma quantum dynamics providing and appropriate mathematical model for the (physical) Montgomery-Odlyzko law, (see below).Technically speaking, this is when the proposed Hermitian plasma quantum potential
operator meets a Gaussian Unitary Ensemple (GUE, i.e., the joint probability density function of Gaussian unitary matrices elements and the joint probability density function of the eigenvalues, (MeM)). "Their eigenvalues provide an excellent fit for the energy levels of the behavior of certain quantum dynamics
systems, where their spacing turned out to be not spaced at random", (DeJ). We note that in a certain sense Gaussian-random Hermitian matrices governing the behavior of quantum dynamics systems somehow also relates to the Landau damping phenomenon. The Montgomery conjecture never was proved, not even on the assumption that the RH is true. The claim is that interpreting the Gaussian-random Hermitian matrices as approximation operators to the proposed plasma quantum potential operator (interpreted as the Hermitian Berry-Keating operator) will prove the Montgomery conjecture and the RH. We also note that the distributional Hilbert space formulation of the Bagchi-Nyman-Beurling RH criterion, where the zeta function may be interpreted as a H(-1) distributional function (BaB), provides an appropriate framework to the proposed H(1/2) quantum potential energy Hilbert space equipped with an inner product induced by the Friedrichs extension of a symmetric and positive definite operator with a H(-1) domain. The Montgomery-Odlyzko law in a nutshell(DeJ) p. 292: "The distribution of the spacings between successive non-trivial zeros of the Riemann zeta function (suitable normalized) is statistically identical with the distribution of eigenvalue spacing in a Gaussian Unitary Ensemble (i.e. a collection of Gaussian unitary operators that share some common statistical properties)".(DeJ):p. 280 ff.: … The eigenvalues (of Gaussian-random Hermitian matrices)…
are struggling to keep their distance from each other. … The statistical
properties of spacings between long non-uniform string of numbers are encapsulated
in a creature called „pair correlations function“ and a certain ratio
associated with this function is called its „form factor“. … The form factor for the pair correlation of
random Hermitian matrices is the conjectured distribution function for the
differences between the non-trivial zeros of Riemann’s zeta function. … The following
points look pretty plausible on the basis of related comparing figures of „the
eigenvalues of a 269-by-269-random matrix“,(p. 285) & p. 289: „The first 269 values of „t“,
where ½+it is a non trivial zero of the zeta function“ (p. 289): repulsion 1. neither the zeta zeros nor the eigenvalues look much like randomly scattered points 2. they resemble each other 3. in particular, they both show the (energy
level) effect, trying to get as far as possible from each other, like a long
standing line of antisocial people".
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