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?

1. June 30, 2019 2:57 am

What is the connection to P=NP?

June 30, 2019 6:54 am

Probably, Jensen polynomials and their use to decompose structures of combinatorial problems. I’m personaly working on my own L-function that describes MAXCLIQUE problem and I’m using very similar polynomials(my own) with special series(also my own, they are very convinient in a work with symmetry) to explain all non-trivial zeroes on a critical line of my L-function. Then I’m using my own new logic that I previously used to deal with undecidability and gently excluded it from any computational model and also excluded such TMs that never halt, but their tape will always be different with almost no repititions. I advanced that logic to start to describe “perfect asymmetry” and based new set theory on that, combining all together I made a proof that MAXCLIQUE cannot be solved by ANY kind of deterministic search algorithm that takes ANY polynomial boundary. I simply can reconstruct any search to L-Function and then show that it has zeros at different places and it is OBSTRUCTION, means that on the inputs related to actual zeros algorithm will never stop or will be wrong. I was happy at first, but then realized that you can try to solve this problem without searching at all, so it’s not a P!=NP proof, I hoped for ETH for 3SAT, but proof doesn’t imply it, alas. So L-Functions + correct Polynomials are very strong couple, IMHO. Now I’m trying to get into representation theory and vector analysis, but it is teethbreaking for me, I never learned math in appropriate way(I was too poor to get education in university), but I need RT+VA for making everything perfectly rigorous. I have no idea what books could help me.

June 30, 2019 8:55 am

Dear Ivan Gusev:

Wow. A lot going on here. I would humbly suggest to try to do less. It may be more. That is focus on your new ideas and not change everything.

Best

Dick

July 12, 2019 10:46 pm

I was trying to invent logic that I would be able to use to verify intellectual behavior of AI and to allow AI to use the logic as the primal source of any its conclusion. I started it in 16 yo and got the concept done in 21, after that I wasted 2-3 years formalizing the logic and writing a new set theory of locally-nondeterministic sets(AD w/o “perfect information” available), super-continuums(way more powerful and uncountable for uncountable), asymmetry as superposition. So, I can say that I’m always on focus on my ideas, whole life wasted on them ;D. In 16 I stared my quest to a “perfect AI” and very quickly after diving in theory, I found out about PvNP problem, I realized, that if I could prove them equal, I could never reveal it to the world and make my AI unbeatable for a long time. PvNP was always on a second place for me, especially in the country that calls CS by the name “informatika”. Now I have very bad health issues, this is why I tried to verify my ideas “in public” for a bit and without giving up its structure. I feel that I have no more powers to struggle against chronical diseases and I see that after everything that Putin has done to Russia outside of Moscow I have no way of getting any kind of medical help, also I can’t find a job because of health problems and have no money to get education in university. You said that I shouldn’t try to change everything, but 3 days ago I was thinking about the application of my “theory of asymmetrical cascades” to a physics of electron and by an accident realized how to solve “Inscribed square problem”. Unfortunately, about the topology, I do know only its name and nothing more of its consistency. here is the “sketch” of the proof that I cannot finish because of zero-knowledge in maths.

Can you give me some suggestions that I would be able to use to learn math required for such proofs?

June 30, 2019 9:00 am

Dear domotorp:

Thanks for keeping us on track. We lost the point a bit. The approximation property also applies to P=NP question. Many try to show, for example, that P=NP or that P not = NP. I would like a better AP type of questions. As we have said many times: even showing SAT is not very easy is an open question. Linear time anyone?

Best

Dick

July 2, 2019 12:23 pm

What?

You try to suggest that we make you believe “everything is connected” is wrong? 🙂

— RH

2. June 30, 2019 5:18 am

Right now the central authority is not Bombieri on the topic.

June 30, 2019 8:52 am

Dear RH:

Interesting name. Stands for Robin Holmes? Anyway. He is one of the greats, but others may be the current experts. I believe that when it comes to predictions on conjectures we all may be experts.

Best

Dick

June 30, 2019 10:57 pm

Neeeeerds!!!

July 1, 2019 2:52 pm

Dear Olih Yurchenko:

Not sure what you mean. We are nerds indeed.

Best

Dick

July 2, 2019 12:02 pm

Oh, no! 🙂

“Neeeeerds!!!” is Quaylian? 🙂

Larry may know why, perhaps? 🙂 🙂 🙂

— RH

July 3, 2019 9:13 am

I am always curious how some mathematicians are able to discern importance of results when I can’t decipher anything beyond what is obviously apparent. For instance, why is this result more important than this similar result for Permanent vs Determinant by Briquel and Burgisser – https://arxiv.org/pdf/1806.00417.pdf? Here they show that Koiran’s Real-Tau conjecture is true in expectation for “random” polynomials (read the paper for details on the distribution – the distribution has algebro-geometric significance). I don’t mean to ask this rhetorically, I really want to understand what heuristics one uses to gauge importance of an advancement. I’d be grateful if anyone can answer this.

July 5, 2019 10:22 am

No question is stupid. Why is a result important is a great question. The result you cite seems to me to not be too surprising. Random objects behavior very differently from “natural” objects. We know that random objects usually are hard for just about any measure. So while the result is not easy, it is not too surprising.

On the RH result. We have some experts that think it can be a breakthrough. So that is one point that helps make us think it could be an approach too RH. They also solve some technical but neat results. And they extend some known results on the Jensen polynomials.

But I am still unclear why one result is better.

Best

Dick