Comments on the Riemann conjecture and index theory on Cantorian fractal space-time

Abstract

ABSTRACT: An heuristic proof of the Riemman conjecture is proposed. It is based on the old idea of Polya-Hilbert. A discrete/fractal derivative self adjoint operator whose spectrum may contain the nontrivial zeroes of the zeta function is presented. To substantiate this heuristic proposal we show using generalized index-theory arguments, corresponding to the (fractal) spectral dimensions of fractal branes living in Cantorian-fractal space- time, how the required negative traces associated with those derivative operators naturally agree with the zeta function evaluated at the spectral dimensions. The ζ(0) = −1/2 plays a fundamental role. Final remarks on the recent developments in the proof of the Riemann conjecture are made.