Mathematics

Jensen Polynomials, the Riemann-zeta Function, and SYK

A new paper by Michael Griffin, Ken Ono, Larry Rolen, and Don Zagier appears to have a made some intriguing steps when it comes to the Riemann Hypothesis (RH). The paper is titled, ‘Jensen polynomials for the Riemann zeta function and other sequences’. The preprint originally appeared in arXiv [arXiv:1902.07321 [math.NT]] in February 2019. It was one of a long list of papers that I wanted to read over the summer. And with the final version now in the Proceedings of the National Academy of Sciences (PNAS), I would like to discuss a bit about the author’s work and one way in which it relates to my own research.

First, the regular reader will recall that in a past post on string mass and the Riemann-zeta function, we discussed the RH very briefly, including the late Sir Michael Atiyah’s claim to have solved it, and finally the separate idea of a stringy proof. The status of Atiyah’s claim still seems unclear, though I mentioned previously that it doesn’t look like it will hold. The idea of a stringy proof also remains a distant dream. But we may at least recall from this earlier post some basic properties of the RH.

What is very interesting about the Griffin et al paper is that it returns to a rather old approach to the RH, based on George Pólya’s research in 1927. The authors also build on the work of Johan Jensen. The connection is as follows. It was the former, Pólya, a Hungarian mathematician, who proved that, for the Riemann-zeta function \zeta{s} at its point of symmetry, the RH is equivalent to the hyperbolicity of Jensen polynomials. For the inquisitive reader, as an entry I recommend this 1990 article in the Journal of Mathematical Analysis and Applications by George Csordas, Rirchard S. Varga, and Istvan Vincze titled, ‘Jensen polynomials with applications to the Riemann zeta-function’.

Pólya’s work is generally very interesting, something I have been familiarising myself with in relation to the Sachdev-Ye-Kitaev model (more on this later) and quantum gravity. When it comes to the RH, his approach was left mostly abandoned for decades. But Griffin et al formulate what is basically a new general framework, leveraging Pólya’s insights, and in the process proving a few new theorems and even proving criterion pertaining to the RH.

1. Hyperbolicity of Polynomials

I won’t discuss their paper in length, instead focusing on a particular section of the work. But as a short entry to their study, Griffin et al pick up from the work of Pólya, summarising his result about how the RH is equivalent to the hyperbolicity of all Jensen polynomials associated with a particular sequence of Taylor coefficients,

\displaystyle (-1 + 4z^{2}) \Lambda(\frac{1}{2} + z) = \sum_{n=0}^{\infty} \frac{\gamma (n)}{n!} \cdot z^{2n} \ \ (1)

Where {\Lambda(s) = \pi^{-s/2} \Gamma (s/2)\zeta{s} = \Lambda (1 - s)}, as stated in the paper. Now, if I am not mistaken, the sequence of Taylor coefficients belongs to what is called the Laguerre-Pólya class, in which case if there is some function {f(x)} that belongs to this class, the function satisfies the Laguerre inequalities.

Additionally,  the Jensen polynomial can be seen in (1). Written generally, a Jensen polynomial is of the form {g_{n}(t) := \sum_{k = 0}^{n} {n \choose k} \gamma_{k}t^{k}}, where {\gamma_{k}}‘s are positive and they satisfy the Turán inequalities {\gamma_{k}^{2} - \gamma_{k - 1} \gamma_{k + 1} \geq 0}.

Now, given that a polynomial with real coefficients is hyperbolic if all of its zeros are real, where read in Griffin et al how the Jensen polynomial of degree {d} and shift {n} in the arbitrary sequence of real numbers {\{ \alpha (0), \alpha (1), ... \}} is the following polynomial,

\displaystyle J_{\alpha}^{d,n} (X) := \sum_{j = 0}^{d} {d \choose j} \alpha (n + j)X^{j} \ \ (2)

Where {n} and {d} are the non-negative integers and where, I think, {J_{\alpha}^{d,n} (X)} is the hyperbolicity of polynomials. Now, recall that we have our previous Taylor coefficients {\gamma}. From the above result, the following statement is given that the RH is equivalent to {J_{\gamma}^{d,n}(X)} – the hyperbolicity of polynomials – for all non-negative integers. What is very curious, and what I would like to look into a bit more, is how this conditions holds under differentiation. In any case, as the authors point out, one can prove the RH by showing hyperbolicity for {J_{\alpha}^{d,n} (X)}; but proving the RH is of course notoriously difficult!

Alternatively, another path may be chosen. My understanding is that Griffin-Ono-Rolen-Zagier use shifts in {n} for small {d}, because, from what I understand about hyperbolic polynomials, one wants to limit the hyperbolicity in the {d} direction. Then the idea, should I not be corrected, is to study the asymptotic behaviour of {\gamma(n)}.

This is the general entry, from which the authors then go on to consider a number of theorems. I won’t go through all of the theorems. One can just as well read the paper and the proofs. What I want to do is focus particularly on Theorem 3.

2. Theorem 3

Aside from the more general considerations and potential breakthroughs with respect to the RH, one of my interests triggered in the Griffin-Ono-Rolen-Zagier paper has to do with my ongoing studies concerning Gaussian Unitary Ensembles (GUE) and Random Matrix Theory (RMT) in the context of the Sachdev-Ye-Kitaev (SYK) model (plus similar models) and quantum gravity. Moreover, RMT has become an interest in relation to chaos and complexity, not least because in SYK and similar models we consider late-time behaviour of quantum black holes in relation to theories of quantum chaos and random matrices.

But for now, one thing that is quite fascinating about Jensen polynomials for the Riemann-zeta function is the proof in Griffin et al of the GUE random matrix model prediction. That is, the derivative aspect GUE random matrix model prediction for the zeros of Jensen polynomials. One of the claims here is that the GUE and the RH are satisfied by the symmetric version of the zeta function. To quote in length,

‘To make this precise, recall that Dyson, Montgomery, and Odlyzko [9, 10, 11] conjecture that the nontrivial zeros of the Riemann zeta function are distributed like the eigenvalues of random Hermitian matrices. These eigenvalues satisfy Wigner’s Semicircular Law, as do the roots of the Hermite polynomials {H_{d}(X)}, when suitably normalized, as {d \rightarrow +\infty} (see Chapter 3 of [12]). The roots of {J){\gamma}^{d,0} (X)}, as {d \rightarrow +\infty} approximate the zeros of {\Lambda (\frac{1}{2} + z)} (see [1] or Lemma 2.2 of [13]), and so GUE predicts that these roots also obey the Semicircular Law. Since the derivatives of {\Lambda (\frac{1}{2} + z)} are also predicted to satisfy GUE, it is natural to consider the limiting behavior of {J_{\gamma}^{d,n}(X)} as {n \rightarrow +\infty}. The work here proves that these derivative aspect limits are the Hermite polynomials {H_{d}(X)}, which, as mentioned above, satisfy GUE in degree aspect.’

I think Theorem 3 raises some very interesting, albeit searching questions. I also think it possibly raises or inspires (even if naively) some course of thought about the connection of insights being made in SYK and SYK-like models, RMT more generally, and even studies of the zeros of the Riemann-zeta function in relation to quantum black holes. In my own mind, I also immediately think of the Hilbert-Polya hypothesis and the Jensen polynomials in this context, as well as ideas pertaining to the eigenvalues of Hamiltonians in different random matrix models of quantum chaos. There is connection and certainly also an interesting analogy here. To what degree? It is not entirely clear, from my current vantage. There are also some differences that need to be considered in all of these areas. But it may not be naive to ask, in relation to some developing inclinations in SYK and other tensor models, about how GUE random matrices and local Riemann zeros are or may be connected.

Perhaps I should save such considerations for a separate article.

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