FRACTAL SUPERSYMMETRIC QM, GEOMETRIC PROBABILITY AND THE RIEMANN HYPOTHESIS

Abstract

The Riemann's hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form sn=1/2+iλn. Earlier work on the RH based on supersymmetric QM, whose potential was related to the Gauss–Jacobi theta series, allows us to provide the proper framework to construct the well-defined algorithm to compute the density of zeros in the critical line, which would complement the existing formulas in the literature for the density of zeros in the critical strip. Geometric probability theory furnishes the answer to the difficult question whether the probability that the RH is true is indeed equal to unity or not. To test the validity of this geometric probabilistic framework to compute the probability if the RH is true, we apply it directly to the the hyperbolic sine function sinh (s) case which obeys a trivial analog of the RH (the HSRH). Its zeros are equally spaced in the imaginary axis sn=0+inπ. The geometric probability to find a zero (and an infinity of zeros) in the imaginary axis is exactly unity. We proceed with a fractal supersymmetric quantum mechanical (SUSY-QM) model implementing the Hilbert–Polya proposal to prove the RH by postulating a Hermitian operator that reproduces all the λn for its spectrum. Quantum inverse scattering methods related to a fractal potential given by a Weierstrass function (continuous but nowhere differentiable) are applied to the fractal analog of the Comtet–Bandrauk–Campbell (CBC) formula in SUSY QM. It requires using suitable fractal derivatives and integrals of irrational order whose parameter β is one-half the fractal dimension (D=1.5) of the Weierstrass function. An ordinary SUSY-QM oscillator is also constructed whose eigenvalues are of the form λn=nπ and which coincide with the imaginary parts of the zeros of the function sinh (s). Finally, we discuss the relationship to the theory of 1/f noise.