### Will RH be Proved by a Physicist?

The Riemann Hypothesis (RH) states that all the non-trivial (non-real) zeros of the zeta function lie on a line, the critical line, ${\Re(s) = 1/2}$. By a simple change of variable, we can have them lying on the real axis. But the eigenvalues of any hermitian matrix are real. This led to the Hilbert-Polya Conjecture:

The non-trivial zeros of ${\zeta(s)}$ are the
eigenvalues of a hermitian operator.

Is there a Riemann operator? What could this operater be? What dynamical system would it represent? Are prime numbers and quantum mechanics linked? Will RH be proved by a physicist?

This last question might make a purest-of-the-pure number theorist squirm. But it is salutary to recall that, of the nine papers that Riemann published during his lifetime, four were on physics!

Random Hermitian Matrices

The spectra of random hermitian matrices are not random: the eigenvalues are spread out as if there is “repulsion” between closely spaced values.

Left panel: Eigenvalues of a random ${256\times256}$ hermitian matrix. Right panel: Random numbers from a uniform distribution.

The figure  above (left panel) shows a subset of the eigenvalues of a random ${256\times256}$ hermitian matrix (only the central 52 values are shown, and they are normalized to span the range ${[0,100]}$). The right panel shows ${52}$ randomly generated numbers from a uniform distribution over the same range. We see that the two distributions appear to be quite distinct in character. The eigenvalues are, broadly speaking, fairly evenly spaced out. The random numbers are clustered in places and quite sparse in others (a vertical gap corresponds to wide spacing, a quasi-horizontal group means a cluster).

In contrast, the figure below (left panel) shows the eigenvalues (as before), while the right panel shows the imaginary parts of ${52}$ consecutive non-trivial zeros of ${\zeta(s)}$. We see that the two distributions appear to be similar in character. Neither resembles the very uneven pattern seen with the random numbers.

Left panel: Eigenvalues of a random ${256\times256}$ hermitian matrix. Right panel: Imaginary parts of ${52}$ consecutive zeros of ${\zeta(s)}$.

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An Explicit Formula

The Riemann zeta-function provides information about the distribution of the prime numbers. Much more precisely, there is an explicit formula for the number of primes in the interval ${[0,x]}$ that involves the zeros of the Riemann zeta function on the critical line ${\Re(s) =1/2}$. Of course, RH states that all the non-trivial zeros are on this line. Quoting Granville (2008) pg. 337, “It is astonishing that there can be such a formula, an exact expression for the number of primes up to ${x}$, in terms of the zeros of a complicated function: you can see why Riemann’s work stretched people’s imagination and had such an impact.”

In order to understand the distribution of primes in short intervals, we need to understand the distribution of zeros of the zeta function in short intervals. This question was investigated by Hugh Montgomery in the 1970s. He found that the proportion of pairs of zeros whose difference is less than ${\alpha}$ times the average gap between consecutive zeros is given by the following integral:

$\displaystyle \int_0^\alpha \left[ 1 - \left(\frac{\sin\pi\theta }{\pi\theta}\right)^2 \right]\, \mathrm{d}\theta \,. \ \ \ \ \ (1)$

For a random sequence, the proportion would be ${\alpha}$, but the value of the integral is far smaller than this. The implication of this is that there are far fewer pairs of zeros close together than would be expected if they were distributed in random fashion. We saw this in the above figures. It is is often expressed in an informal way by saying that the zeros repel one another.

A Serendipitous Encounter

The story of the meeting between Hugh Montgomery and Freeman Dyson has been told many times. Briefly, when Montgomery was visiting the Institute for Advanced Study in Princeton in 1973, he was introduced to Dyson, the renowned physicist. Dyson asked him about his research and was astounded when Montgomery told him about the distribution of the zeros of the Riemann zeta function. Dyson had been studying the statistics of the energy levels of large nuclei, using random matrix theory, and he immediately recognised the integrand of the integral (1) as the mean spectrum of the eigenvalues of random hermitian matrices. This indicated a likely link between the Riemann Hypothesis (a problem in pure number theory) and the energy spectrum of heavy nuclei (a question in quantum mechanics).

Much deeper investigations showed beyond doubt the close correspondence between the two problems. This raised the possibility of a proof of Riemann’s hypothesis through the study of physical systems. It even suggested the prospect of a proof of the central problem of number theory by a physicist, which might be regarded as an “appaling vista” by some mathematicians.

The GUE Hypothesis

The Gaussian Unitary Ensemble (GUE) models random Hamiltonians without time reversal symmetry. The GUE of degree ${N}$ consists of the set of all ${N\times N}$ hermitian matrices, together with a certain probability measure, the unique probability measure that is invariant under conjugation by unitary matrices.

As seen above, the eigenvalues of random hermitian matrices do not behave like independent random variables: the probability of finding eigenvalues close together is far less than a random distribution would suggest. In simple terms, the eigenvalues repel each other. This is shown by the pair correlation function. Montgomery found the pair correlation for the non-trivial zeros, and Dyson recognized it as the pair correlation function for random hermitian matrices in the GUE.

The Montgomery-Odlyzko Law. Histogram: Distribution of spacings for the 90,001st to 100,000th zeta-function zeros. Graph: curve from GUE theory. [From Derbyshire (2003)].

Further numerical evidence confirmed that the zeros obey imply the GUE statistics. The connection found by Montgomery is usually expressed as the Montgomery-Odlyzko Law:

The Riemann spectrum is statistically identical to the distribution
of eigenvalues of an ensemble of Gaussian hermitian matrices.

Michael Berry proposed that there exists a classical dynamical system with periodic orbits having lengths corresponding to the rational primes. The search for a dynamical system of this sort is one approach to proving the Riemann hypothesis.

Bender, Brody and Mueller (2017) constructed a Hamiltonian operator ${\hat H}$ such that, if the eigenfunctions obey a suitable boundary condition, the eigenvalues correspond to the nontrivial zeros of the Riemann zeta function. However, while they presented heuristic arguments that ${\hat H}$ is self-adjoint, they did not prove rigorously that it is Hermitian. If this could be done, then RH would be proved. It is not clear whether this approach will be fruitful.

Sources

${\bullet}$ Bender, Carl M., Dorje C. Brody and Markus P. Mueller, 2017: Hamiltonian for the Zeros of the Riemann Zeta Function. Phys. Rev. Lett., 118, 130201.

${\bullet}$ Derbyshire, John, 2003: Prime Obsession. Plume Books, 422pp. ISBN: 978-0-4522-8525-5.

${\bullet}$ Mehta, M.L., 2004: Random Matrices. Elsevier/Academic Press. Amsterdam. ISBN: 0-12-088409-7.

${\bullet}$ Granville, Andrew: Analytic Number Theory. § IV.2 in The Princeton Companion to Mathematics, Ed. Timothy Gowers. Princeton University Press (pp. 332-348). ISBN: 978-0-6911-1880-2.

${\bullet}$ Pomerance, Carl: Computational Number Theory. § IV.3 in The Princeton Companion to Mathematics, Ed. Timothy Gowers. Princeton University Press (pp. 348-362).