A recent paper by Hartnoll and Yang study a BKL dynamics of gravity close to a spacelike singularity, where one has a mapping onto the motion of a particle within half the fundamental domain of PSL(2,Z), the modular group. The semiclassical quantization of this motion gives a conformal quantum mechanics where states are modular invariant. Each state defines an odd automorphic L-function, where in particular, for a basis of dilatation eigenstates the wavefunction is proportional to the L-function along the critical axis, hence vanishing at the critical zeros.
It is shown that the L-function along the positive real axis is equal to the partition function of a gas of non-interacting charged oscillators labeled by prime numbers. This generalizes Bernard Julia’s notion of a primon gas. The author’s close the paper with comments on the adeles and p-adic holography.
Why is this relevant to the Riemann hypothesis? Certainly, the zeta is an L-function, however, unlike the local zeta functions of the Weil conjectures one lacks a compact geometry in which to define a self-adjoint operator. The Hilbert-Polya dream is to find a self-adjoint operator in which the non-trivial zeros can be recovered from its real spectrum. Yet, the Riemann zeta function most likely will be proven by invoking new mathematical structures that generalize the geometries of the Weil conjectures. Such geometries will, of course, be related to the adeles and generalize p-adic holography. I give this paper 10/10 for being inspirational.
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