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On strategies towards the Riemann hypothesis: fractal supersymmetric QM and a trace formula. (English) Zbl 1204.11141
The author of this paper recently found a novel physical interpretation of the location of the nontrivial Riemann zeta zeros which corresponds to the presence of tachyonic-resonances (or condensates) in bosonic string theory. If there were zeros outside the critical line violating the RH these zeros would not correspond to poles of the string scattering amplitude [{\it C. Castro}, Int. J. Geom. Methods Mod. Phys. 3, No. 2, 187--199 (2006; Zbl 1090.81046)]. By writing, in Riemann’s hypothesis, the non-trivial complex zeros of the zeta-function $\zeta (s)$ in the form $s_n = 1/2\pm i\lambda_n$, the spectral properties of the $\lambda_n$’s are associated there with the random statistical fluctuations of the energy levels of a classical chaotic system (quantum chaos). {\it H. Montgomery}, [Proc. Int. Congr. Math., Vancouver 1974, Vol. 1, 379--381 (1975; Zbl 0342.10020)] showed that the two-level correlation function of the distribution of the $\lambda_n$’s coincides with the expression obtained by Dyson with the help of random matrices corresponding to a Gaussian unitary ensemble. {\it H. Wu} and {\it D. W. L. Sprung} [Phys. Rev. E 48, No. 4, 2595--2598 (1993), \url{http://link.aps.org/doi/10.1103/PhysRevE.48.2595}] numerically saw that the lower lying non-trivial zeros can be related to the eigenvalues of a Hamiltonian whose potential has a fractal shape and fractal dimension equal to $D = 1.5$, and made a remark concerning the construction of a one-dimensional integrable and time-reversal quantum Hamiltonian to model the imaginary parts of the zeros of zeta as an eigenvalue problem. This riddle of merging chaos with integrability is here solved by choosing a fractal local potential that captures the chaotic dynamics inherent with the zeta zeros. In the paper, the author generalizes his previous strategy to prove the RH based on extending the Wu and Sprung QM problem by invoking a convenient superposition of an infinite family of fractal Weierstrass functions parametrized by the prime numbers, in order to improve the expression for the fractal potential. The fractal SUSY QM model studied here is not a system with fractional supersymmetries, which are common in the string and M-theory literature, but only a Hamiltonian operator that admits a factorization into two factors involving fractional derivative operators whose fractional (irrational) order is one-half of the fractal dimension of the fractal potential. The author then proceeds with the construction of a smooth version of the fractal QM wave equation by writing an ordinary Schrödinger equation whose fluctuating potential (relative to the smooth Wu-Sprung potential) has the same functional form as the fluctuating part of the level density of zeros. A different approach to prove the RH, described in the paper, relies on the existence of a continuous family of scaling-like operators involving the Gauss-Jacobi theta series. An explicit completion relation (“trace formula”, as the author puts it) related to a superposition of eigenfunctions of these scaling-like operators is defined in the paper. If the completion relation is satisfied, this could presumably lead to another test of the Riemann Hypothesis.
##### MSC:
 11M26 Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses 11M06 $\zeta (s)$ and $L(s, \chi)$ 28A80 Fractals 46N50 Applications of functional analysis in quantum physics 11Z05 Miscellaneous applications of number theory 81Q60 Supersymmetry and quantum mechanics
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