properties of the hypothesised "Riemann dynamics"

This is an excerpt from the review article "The Riemann zeros and eigenvalue asymptotics" by M.V. Berry and J.P. Keating of Bristol University. It appeared in SIAM Review 41, no.2 (1999) 236-266.

6. Spectral Speculations. Although we do not know the conjectured Riemann operator H whose eigenvalues (all real) are the heights tn of the Riemann zeros, the analogies presented so far suggest a great deal about it. To summarize:

a. H has a classical counterpart (the "Riemann dynamics"), corresponding to a hamiltonian flow, or a symplectic transformation, in a phase space.

b. The Riemann dynamics is chaotic, that is, unstable and bounded.

c. The Riemann dynamics does not have time-reversal symmetry. In addition, we note the recent discovery [60,61] of modified statistics of the low zeros for the ensemble of Dirichlet L-fuctions, associated with a symplectic structure.

d. The Riemann dynamics is homogeneously unstable.

e. The classical periodic orbits of the Riemann dynamics have periods that are independent of "energy" t, and given by multiples of logarithms of prime numbers. In terms of symbolic dynamics, the Riemann dynamics is peculiar, and resembles Chinese: each primitive orbit is labelled by its own symbol (the prime p) in contrast to the usual situation where periodic orbits can be represented as words made of letters in a finite alphabet.

f. The Maslov phases associated with the orbits are also peculiar: they are all pi. The result appears paradoxical in view of the relation between these phases and the winding numbers of the stable and unstable manifolds associated with periodic orbits [22], but finds an explanation in a scheme of Connes [62].

g. The Riemann dynamics possesses complex periodic orbits (instantons) whose periods are multiples of i*pi.

h. For the Riemann operator, leading-order semiclassical mechanics is exact: as in the case of the Selberg trace formula [21], zeta(1/2 + it) is a product over classical periodic orbits, without corrections.

i. The Riemann dynamics is quasi-one-dimensional. There are two indications of this. First the number of zeros less than t increases as t log t; for a D-dimensional scaling system, with energy parameter alpha(E) proportional to 1/h-bar), the number of energy levels increases as alpha(E)D. Second, the presence of the factor p-m/2 in the counting function fluctuation formula (2.6) rather than the determinant in the more general Gutzwiller formula (2.9), suggests that there is a single expanding directon and no contracting direction.

j. The functional equation for zeta(s) resembles the corresponding relation - a consequence of hermiticity - for the quantum spectral determinant.

We have speculated [6] that the conjectured Riemann operator H might be some quantization of the following extraordinarily simple classical hamiltonian function H(X,P) of a single coordinate and its conjugate momentum P:

Hcl(X,P) = XP.

The authors then go on to outline the reasons for this tentative association on XP with zeta(s).


[6] M.V. Berry and J.P. Keating, H = xp and the Riemann zeros, in Supersymmetry and Trace Formulae: Chaos and Disorder, J. P. Keating, D.E. Khmelnitskii, and I.V. Lerner, eds., Plenum, New York, 1998, pp. 355-367.

[21] N.L. Balzas and A. Voros, Chaos on the pseudosphere, Phys. Rep., 143, (1986), pp. 109 - 240.

[22] J.M. Robbins , Maslov indices in the Gutzwiller trace formula, Nonlinearity, 4, (1991), pp. 343-363.

[60] N. Katz and P. Sarnak , Zeros of Zeta Functions, Their Spacings and Their Spectral Nature, preprint, 1997.

[61] P. Sarnak, Quantum Chaos, symmetry and zeta functions, Curr. Dev. Math. (1997), pp. 84 -115.

[62] A. Connes , Formule de trace en geometrie non-commutative et hypothese de Riemann, C.R. Acad. Sci. Paris, 323 (1996), pp. 1231-1236.

[(2.6) and (2.9) are internal references]

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