On Strategies Towards the Riemann Hypothesis: Fractal Supersymmetric Qm and a Trace Formula

Abstract

The Riemann's hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form sn = 1/2+iλn. An improvement of our previous construction to prove the RH is presented by implementing the Hilbert-Polya proposal and furnishing the Fractal Supersymmetric Quantum Mechanical (SUSY-QM) model whose spectrum reproduces the imaginary parts of the zeta zeros. We model the fractal fluctuations of the smooth Wu-Sprung potential ( that capture the average level density of zeros ) by recurring to a weighted superposition of Weierstrass functions p W (x, p, D) and where the summation has to be performed over all primes p in order to recapture the connection between the distribution of zeta zeros and prime numbers. We proceed next with the construction of a smooth version of the fractal QM wave equation by writing an ordinary Schroedinger 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. The second approach to prove the RH relies on the existence of a continuous family of scaling-like operators involving the Gauss-Jacobi theta series. An explicit trace formula related to a superposition of eigenfunctions of these scaling-like operators is defined. If the trace relation is satisfied this could be another test of the Riemann Hypothesis

THE RIEMANN HYPOTHESIS IS A CONSEQUENCE OF $\mathcal{CT}$-INVARIANT QUANTUM MECHANICS

“…For example, when l = −2; k = 1 4 , one has Ψ so |Ψ sn = −Z[s n ] = 0; Ψ so |Ψ sm = −Z[s m ] = 0 but Ψ sm |Ψ sn = −Z[s * m + s n − 1 2 ] = 0. The procedure on how to construct a discrete ortho-normal basis of states was outlined in [39].…”

“…In [39] we generalized our previous strategy [3] to prove the RH based on extending the Wu and Sprung QM problem by invoking a judicious superposition of an infinite family of fractal Weierstrass functions parametrized by the prime numbers p in order to improve the expression for the fractal potential. A fractal SUSY QM model whose spectrum furnished the imaginary parts of the zeta zeros λ n was studied in [3,39] based on 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.…”

“…A fractal SUSY QM model whose spectrum furnished the imaginary parts of the zeta zeros λ n was studied in [3,39] based on 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. A model of fractal spin has been constructed by Wellington da Cruz [22] in connection with the fractional quantum Hall effect based on the filling factors associated with the Farey fractions.…”

On the Riemann Hypothesis, Area Quantization, Dirac Operators, Modularity, and Renormalization Group

“…For these reasons, before entering into the next two sections we deem it very important to review the results [10,16] based on a family of scaling-like operators in one dimension involving the Gauss-Jacobi theta series and an infinite parameter family of theta series where the inner product of their eigenfunctions Ψ s (t; l) is given by ( There is a one-to-one correspondence among the zeta zeros s n (Z[s n ] = 0 ⇒ ζ(s n ) = 0) with the eigenfunctions Ψ sn (t; l) (of the latter scaling-like operators) when the latter are orthogonal to the "ground" reference state Ψ so (t; l); where s o = 1 2 + i0 is the center of symmetry of the location of the nontrivial zeta zeros. We shall present a concise review [10] and show why the RH follows from a CT invariance when the pseudo-norm of the eigenfunctions Ψ s |CT |Ψ s is not null.…”

“…The Scaling Operators related to the Gauss-Jacobi Theta series and the Riemann zeros [16] are given by also real-valued. The eigenvalues of D 1 are complex-valued numbers s. The charge conjugation operation C acting on the eigenfunctions is defined as ψ s (t) = t −s+k e V (t) → ψ s * (t) = t −s * +k e V (t) = t −s * +s ψ s (t), (1.2b) which is related to scalings transformations of ψ s (t) by t-dependent (local) scaling factors t −s * +s = e (−s * +s) ln t = e 2iIm(s) ln t ⇒ a phase rotation, (1.2c) where Im(s) is the imaginary part of s. Since local t-dependent (ln t dependent to be precise) phase rotations resemble U (1) gauge transformations one can then interpret the (dV /d ln t) term in D 1 as a gauge field (potential) in one dimension that gauges the scalings transformations.…”