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

TL;DR

This paper heuristically approaches the Riemann conjecture by proposing a fractal spectral operator whose spectrum may include the nontrivial zeros of the zeta function, supported by index theory on fractal space-time.

Contribution

It introduces a novel fractal derivative operator and applies generalized index theory to connect spectral dimensions with the zeros of the zeta function, offering a new heuristic perspective.

Findings

Spectral operator's negative traces align with zeta function values at spectral dimensions.

The operator's spectrum potentially contains the nontrivial zeros of the zeta function.

The value ζ(0) = -1/2 is fundamental in this framework.

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 $\zeta (0) = - 1/2$ plays a fundamental role. Final remarks on the recent developments in the proof of the Riemann conjecture are made.