Riemann zeta zeros and zero-point energy

Abstract

We postulate the existence of a self-adjoint operator associated to a system with countably infinite number of degrees of freedom whose spectrum is the sequence of the nontrivial zeros of the Riemann zeta function. We assume that it describes a massive scalar field coupled to a background field in a $(d+1)$-dimensional flat space-time. The scalar field is confined to the interval $[0,a]$ in one dimension and is not restricted in the other dimensions. The renormalized zero-point energy of this system is presented using techniques of dimensional and analytic regularization. In even dimensional space-time, the series that defines the regularized vacuum energy is finite. For the odd-dimensional case, to obtain a finite vacuum energy per unit area we are forced to introduce mass counterterms. A Riemann mass appears, which is the correction to the mass of the field generated by the nontrivial zeros of the Riemann zeta function.