  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.
