The Riemann hypothesis and the Hamiltonian of a quantum mechanical system. (English) Zbl 0693.10030

Armitage, J. V.

Number theory and dynamical systems, Lond. Math. Soc. Lect. Note Ser. 134, 153-172 (1989).

The article under review is an extended version of the author’s lecture on an approach to the Riemann hypothesis in terms of diffusion processes. The author starts from the observation that the non-trivial zeros of \(\zeta\) give rise to eigenvalues of a family of operators, and this suggests the application of methods of functional analysis and in particular of perturbation theory to characterize the critical numbers \(+i\tau\).
Montgomery’s pair correlation conjecture led M. V. Berry [Lect. Notes Phys. 263, 1–17 (1986; Zbl 0664.10021)] to study certain quantum-mechanical systems whose energy levels reveal numerically a close connection with the non-trivial zeros of \(\zeta\). The author suggests two possible candidates for the corresponding Hamiltonian. These are related with the Fokker-Planck equation and the harmonic oscillator. Moreover, the author uses a random walk approach to the Ornstein-Uhlenbeck process to exhibit a polynomial whose zeros should yield the zeros of \(\zeta\) under a limiting process. The author also suggests a formal argument which leads to a generalization of the approximate functional equation and he presents a possible strategy for a proof of the Riemann hypothesis, “but we do not claim satisfactorily to have settled those questions, let alone to have proved the Riemann Hypothesis.”
[For the entire collection see Zbl 0678.00019.]

Reviewer: Jürgen Elstrodt (Münster)

MSC:

11M26	Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
81Q10	Selfadjoint operator theory in quantum theory, including spectral analysis

Keywords:

Riemann zeta-function; Riemann hypothesis; diffusion processes; non-trivial zeros; Hamiltonian; Fokker-Planck equation; harmonic oscillator; random walk; Ornstein-Uhlenbeck

Citations:

Zbl 0678.00019; Zbl 0664.10021