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 spacetime, 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.

One-loop effective action and the Riemann zeros

“…Let us present one situation where the basic quantity (5) shows up. Consider a simple model of two Euclidean self-interacting scalar fields φ 1 (x) and φ 2 (x) with masses m 1 and m 2 , respectively.…”

“…The renormalized zero-point energy of this system was presented using an analytic regularization procedure. 5 Another route to exploit connections between number theory and quantum theory is known as arithmetic quantum field theory. 6 In these theories, the partition function of hypothetical systems are related to the Riemann zeta function or to more general Dirichlet series.…”

Thermodynamics of the bosonic randomized Riemann gas

“…For other approaches discussing the physics of Riemann zeros see [26][27][28][29] . Also, in the literature we can find other recent results connecting number theory and quantum field theory [30][31][32][33][34] .…”

Functional equations for regularized zeta-functions and diffusion processes

“…Introducing randomness in the bosonic Riemann gas and studying the thermodynamic variables of this arithmetic gas in the complex β-plane, the connection between the zeros of the Riemann zeta-function and physics was established. For other papers discussing formal aspects of quantum field theory and number theory see the references [21][22][23][24][25][26][27][28][29][30][31].…”