We present a remarkable connection between the asymptotic behavior of the Riemann zeros and one-loop effective action in Euclidean scalar field theory. We show that in a two-dimensional space, the asymptotic behavior of the Fourier transform of two-p
oint correlation functions fits the asymptotic distribution of the non-trivial zeros of the Riemann zeta function. We work out an explicit example, namely the non-linear sigma model in the leading order in $1/N$ expansion.
The Riemann hypothesis states that all nontrivial zeros of the zeta function lie in the critical line $Re(s)=1/2$. Hilbert and Polya suggested that one possible way to prove the Riemann hypothesis is to interpret the nontrivial zeros in the light of
spectral theory. Following this approach, we discuss a necessary condition that such a sequence of numbers should obey in order to be associated with the spectrum of a linear differential operator of a system with countably infinite number of degrees of freedom described by quantum field theory. The sequence of nontrivial zeros is zeta regularizable. Then, functional integrals associated with hypothetical systems described by self-adjoint operators whose spectra is given by this sequence can be constructed. However, if one considers the same situation with primes numbers, the associated functional integral cannot be constructed, due to the fact that the sequence of prime numbers is not zeta regularizable. Finally, we extend this result to sequences whose asymptotic distributions are not far away from the asymptotic distribution of prime numbers.
We consider an Unruh-DeWitt detector interacting with a massless Dirac field. Assuming that the detector is moving along an hyperbolic trajectory, we modeled the effects of fluctuations in the event horizon using a Dirac equation with random coeffici
ents. First, we develop the perturbation theory for the fermionic field in a random media. Further we evaluate corrections due to the randomness in the response function associated to different model detectors.
