Number Theory

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

Authors: Carlos Castro
Comments: 20 Pages. This article appeared in the Int. Jour. of Geom. Methods of Modern Physics, 4, no. 5 (2007) 881-895.

The Riemann's hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form s_n = 1/2 + iλ_n. 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 P a weighted superposition of Weierstrass functions ΣW(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 completion relation ( "trace formula") related to a superposition of eigenfunctions of these scaling-like operators is defined. If the completion relation is satisfied this could be another test of the Riemann Hypothesis. In an appendix we briefly describe our recent findings showing why the Riemann Hypothesis is a consequence of CT -invariant Quantum Mechanics, because < Ψ_s | CT | Ψ_s > ≠ 0 where s are the complex eigenvalues of the scaling-like operators.
Category: Number Theory