nLab Riemann hypothesis and physics

Idea

There are several ways in which the Riemann hypothesis has or might have an incarnation in or leave an imprint on physics, specifically in quantum physics.

General

• Guilherme França, André LeClair: A theory for the zeros of Riemann $\zeta$ and other $L$-functions (updated) [arXiv:1407.4358]

Via string theory

By the general discussion at zeta function and at functional determinant zeta functions are closely related to 1-loop vacuum amplitudes and to vacuum energy in quantum field theory in general and in string theory in particular.

Concretely, the Rankin-Selberg-Zagier method implies that the partition function of the superstring asymptotes for small proper time to a constant times a converging oscillatory term (reminiscent of the explicit formulae for L-functions) whose frequencies are proportional to the imaginary values of the zeros of the Riemann zeta function (ACER 11).

Similarly there the Veneziano amplitude of the string has an expression in terms of the Riemann zeta function (HJM 15).

Via noncommutative geometry

In other parts of the literature there is the desire to interpret the Riemann zeta function instead as a partition function of a quantum mechanical system. (Notice that the 1-loop vacuum amplitude mentioned before is instead a Mellin transform of the partition function.) The main example here is maybe the Bost-Connes system.

Related entries

• number theory and physics

• Selberg trace formula

• random matrix theory

References

Via string theory

via string theory:

• Sergio Cacciatori, Matteo Cardella, Equidistribution Rates, Closed String Amplitudes, and the Riemann Hypothesis (arXiv:1007.3717)

  “the Riemann hypothesis can be rephrased in terms of ultraviolet relations occurring in perturbative closed string theory”

• Carlo Angelantonj, Matteo Cardella, Shmuel Elitzur, Eliezer Rabinovici, Vacuum stability, string density of states and the Riemann zeta function, JHEP 1102 024 (2011) [arXiv:1012.5091]

• Sergio L. Cacciatori, Matteo A. Cardella, Uniformization, Unipotent Flows and the Riemann Hypothesis (arXiv:1102.1201)

• Yang-Hui He, Vishnu Jejjala, Djordje Minic, From Veneziano to Riemann: A String Theory Statement of the Riemann Hypothesis (arXiv:1501.01975)

Via scattering amplitudes

via scattering amplitudes in perturbative quantum field theory:

• Grant N. Remmen, Amplitudes and the Riemann Zeta Function (arXiv:2108.07820)

Via noncommutative geometry

via noncommutative geometry:

• Matthew Watkins, the nontrivial Riemann zeta zeros interpreted as a spectrum of energy levels

• Alain Connes, Noncommutative geometry and the Riemann zeta-function (pdf)

• Masoud Khalkhali, What is new with Connes’ approach to the Riemann hypothesis? pdf

via random matrix theory:

• Nina Snaith, Random matrix theory and zeta functions, 2000 (pdf)