A fractal SUSY-QM model and the Riemann hypothesis

The Riemann’s hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form s = 1/2+iλn. Hilbert-Polya argued that if a Hermitian operator exists whose eigenvalues are the imaginary parts of the zeta zeros, λn’s, then the RH is true. In this paper a fractal supersymmetric quantum mechanical (SUSY-QM) model is proposed to prove the RH. It is based on a quantum inverse scattering method related to a fractal potential given by a Weierstrass function (continuous but nowhere differentiable) that is present in the fractal analog of the CBC (Comtet, Bandrauk, Campbell) formula in SUSY QM. It requires using suitable fractal derivatives and integrals of irrational order whose parameter β is one-half the fractal dimension of the Weierstrass function. An ordinary SUSY-QM oscillator is constructed whose eigenvalues are of the form λn = nπ, and which coincide with the imaginary parts of the zeros of the function sin(iz). This sine function obeys a trivial analog of the RH. A review of our earlier proof of the RH based on a SUSY QM model whose potential is related to the Gauss-Jacobi theta series is also included. The spectrum is given by s(1 − s) which is real in the critical line (location of the nontrivial zeros) and in the real axis (location of the trivial zeros).