Quantum Field Theories and Prime Numbers Spectrum

Abstract

The Riemann hypothesis states that all nontrivial zeros of the zeta function lie on the critical line $\Re (s)=1/2$. Hilbert and Pólya suggested a possible approach to prove it, based on spectral theory. Within this context, some authors formulated the question: is there a quantum mechanical system related to the sequence of prime numbers? In this Letter we show that such a sequence is not zeta regularizable. Therefore, there are no physical systems described by self-adjoint operators with countably infinite number of degrees of freedom with spectra given by the sequence of primes numbers.

10 pages; revised version

Cited by in corpus (3)

• Riemann zeta zeros and prime number spectra in quantum field theory
• Directed Polymers and Interfaces in Disordered Media
• Riemann zeta zeros and zero-point energy