Physical realization for Riemann zeros from black hole physics

Abstract

According to a conjecture attributed to Polya and Hilbert, there is a self-adjoint operator whose eigenvalues are the the nontrivial zeros of the Riemann zeta function. We show that the near-horizon dynamics of a massive scalar field in the Schwarzscild black hole spacetime, under a reasonable boundary condition, gives rise to normal mode frequencies that coincide with the nontrivial Riemann zeros. In achieving this result, we exploit the Bekenstein conjecture of black hole area quantization, and argue that it is responsible for the breaking of the continuous scale symmetry of the near horizon dynamics into a discrete one.