A Prime Breakthrough

Gödel’s Lost Letter and P=NP 2019-08-20

A breakthrough on the Riemann Hypothesis

[ Composite of various sources ]

Michael Griffin, Ken Ono, Larry Rolen, and Don Zagier (GORZ) have recently published a paper on an old approach to the famous Riemann Hypothesis (RH).

Today we will discuss their work and its connection to P=NP.

Their paper is titled, “Jensen polynomials for the Riemann zeta function and other sequences.” The final version is in the Proceedings of the National Academy of Sciences (PNAS).

The RH is still open. We are not aware of any update on the status of Michael Atiyah’s claim to have solved it since this note on an AMS blog and our own discussion of his papers’ contents.

Recall the RH is a central conjecture in number theory, and it has the following properties:

If true, it would greatly enhance our understanding of the structure of the primes: ${2,3,5,7,\dots}$
It has resisted attacks for 160 years and counting.
There are an immense number of statements equivalent to the RH.

See, for example, the survey paper, “The Riemann Hypothesis,” by Brian Conrey.

Complexity theory has the P=NP question; number theory has the RH. Both seem to be beyond reach, and both are fundamental questions. The recent work of GORZ has created buzz among number theorists—perhaps they are on the verge of a breakthrough? Is there hope that we might see progress on P=NP? Or must we wait 160 years?

The buzz is reflected in a May 28 story in the online journal LiveScience. It quotes Enrico Bombieri, who wrote the official Clay Prize description of RH, as saying:

“Although this remains far away from proving the Riemann hypothesis, it is a big step forward. There is no doubt that this paper will inspire further fundamental work in other areas of number theory as well as in mathematical physics.”

Bombieri wrote an accompanying paper in PNAS, in which the above quoted sentences also appear. We will try to explain what the excitement is about.

The Approach and Results

RH is equivalent to the hyperbolicity of Jensen polynomials for the Riemann zeta function. Note: A real, polynomial ${p(x)}$ is hyperbolic if all its roots are real—a fancy name for a simple concept.

There is an analytic function ${E(z)}$ so that

$\displaystyle RH \iff E(z) \text{ has all real roots}.$

Almost a hundred years ago, Johan Jensen and George Pólya created an approach to the RH based on ${E(z)}$ . Rather than prove ${E(z)}$ has only real roots, they showed it is enough to show that a family of polynomials ${J(n, d)(x)}$ all are hyperbolic. The point is that polynomials are “simpler” than analytic functions—at least that is the hope.

What GORZ prove is this:

Theorem 1 For each ${d}$ and almost all ${n}$ , the polynomial ${J(n, d)(x)}$ is hyperbolic.

Of course, “almost all” means that there at most a finite number of polynomials that are not hyperbolic. They also show that for degree ${d}$ fixed,

$\displaystyle \lim_{n \rightarrow \infty} J(n, d)(x) = H_{d}(x).$

Where the polynomials ${H_{d}(x)}$ have only real roots.

This is explained further in a talk by Ono, which has this table showing how the polynomials converge:

Note: they use different notation for the polynomials.

The Approximation Property

There are many equivalent formulations of the RH. While all are equivalent, some are more equivalent than others. Some seem to be a more plausible path toward a resolution of the RH. Of course to date none have worked.

There is a way to distinguish equivalent formulations of the RH. Suppose that $S$ depends on some parameter $\beta$ . Suppose that as $\beta$ increases the statement $S(\beta)$ becomes a weaker version of the RH. Then let us informally say that the formulation $S(\beta)$ has the “Approximation Property” (AP). The point is that progress in proving caes of $S(\beta)$ —even as partial progress—is exciting. But if the equivalence only holds for $\beta=0$ , with no connections for higher $\beta$ , then the partial progress could be seen as morally useful—but it is not really mathematically useful.

There are equivalences of the RH that have AP and some that seem not to have it. The approximation for the RH is natural. The RH states that there is no zero $\sigma + it$ of the Riemann zeta function with $\sigma$ above $1/2$ . The weaker statement: If $\sigma + it$ is a zero, then $\sigma > \alpha$ is unknown for any $\alpha$ in the open interval $(1/2,1)$ . This can be used to get a property with the AP.

For example, this 2003 paper by Luis Baez-Duarte, titled “A new necessary and sufficient condition for the Riemann hypothesis,” has the AP. He notes, in our terminology, that his property is an AP.

Open Problems

Does the approach of GORZ have the AP? The issue is that while we get a universal statement in terms of $d$ , the “all but finitely many” condition on $n$ works against its being an approximation—what if the finite sets of exceptions grow in terms of $d$ in ways that offset the purpose behind the equivalence?