Riemann hypothesis and physics


Arithmetic geometry


physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics



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.

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.


Via string theory

via string theory:

Via scattering amplitudes

via scattering amplitudes in perturbative quantum field theory:

Via noncommutative geometry

via noncommutative geometry:

via random matrix theory: