129

The Riemann Hypothesis is a mathematical hypothesis that describes the distribution of prime numbers. It is one of the seven Millennium Problems put forth by the Clay Mathematics Institute, notorious for its difficulty and has a $1,000,000 prize for the person that is able to prove it. Several attempts at a proof have been made in the past (including that of a Fields medallist), but so far, they have all failed and it is considered an unsolved problem.

Since yesterday, I have been seeing news that an Indian mathematical physicist named Kumar Easwaran has claimed to solve the Riemann Hypothesis. This article, for example, describes the peer-review process:

Over five years ago, Eswaran had placed his research on the internet. The research paper was titled 'The final and exhaustive proof of the Riemann Hypothesis from first principles'. Despite all this, the editors of international journals were reluctant to put the paper through a detailed peer review, according to a report in The Times of India.

“The hypothesis was important to prove as it would enable mathematicians to exactly count the prime numbers,” said Eswaran.

With over thousands of downloads, in 2020, an expert committee was constituted to look into the proof developed by Eswaran. The committee comprised of eight mathematicians and theoretical physicists.

More than 1,200 mathematicians were invited by the committee to take part in an open review in which – the referees would willingly have their names and institutional affiliations revealed in an open manner. This is necessary so that nothing is done anonymously. Also, nothing should remain hidden from the other experts to see. Soon after, seven international scholars responded to the call.

After scrutinising the comments of the seven reviewers and the responses of the author, the committee concluded that Eswaran's proof of the RH is correct.

So, has the committee actually come to the conclusion that the Riemann Hypothesis has been solved by Kumar Eswaran?

Note:

  • As of writing this question, the Clay Mathematical Institute's webpage on the RH states that the problem is unsolved.

  • I have not yet come across any international news outlet covering this. All the articles I have read are by Indian news outlets. If this is actually true, wouldn't international news and SciComm also publicize this?

15
  • 13
    While this is a notable question, it is not one that site members can answer. We'll have to wait for the Clay Mathematics Institute to say yea or nay. Jun 29 at 15:11
  • 19
    Closed version of this question at math.stackexchange.com/questions/4185354/…
    – Henry
    Jun 29 at 15:48
  • 18
    I note that the closed Math.SE question now has links in comments to a Reddit thread discussing the proof's flaws, and from there to a published rebuttal from several years ago.
    – IMSoP
    Jun 29 at 17:03
  • 6
    While this site cannot evaluate the details of an arcane math proof, it can and should evaluate whether the evidence presented makes a convincing case that the proof has been evaluated by people capable of assessing it. Perhaps a reason to subtly reword the question but not a reason to close it.
    – matt_black
    Jun 30 at 10:29
  • 8
    I retracted my close vote. We have a new member who does have the chops to be able to answer this question. Jun 30 at 12:26
247

Summary

The current version of the question asks two things:

Have mathematicians concluded that an Indian mathematical physicist has solved the Riemann Hypothesis?

No. There isn't a single professional mathematician who publicly stated that they have read Kumar's proof and determined that it is correct.

So, has the committee actually come to the conclusion that the Riemann Hypothesis has been solved by Kumar Eswaran?

Yes, the committee stated that they believe Kumar has solved the problem. But only one member of the committee is a mathematician, and the committee stated that it arrived at its conclusion by relying on the opinions of reviewers, all of whom either explicitly disagree with the committee's conclusions or are themselves not mathematicians.

Furthermore, the reasoning provided by the committee for how they arrived at their conclusion is plagued by many misleading statements and an apparent lack of understanding of mathematical reasoning and the meaning of mathematical proof, reducing any credibility the committee might have had otherwise to zero.

The detailed analysis below requires no mathematical background to understand. For what it's worth, I'm a professional mathematician with expertise in the areas related to the Riemann hypothesis, but the validity of my answer does not depend on that fact in any way.

Detailed analysis

The main source for the claim from the news media saying Eswaran's proof has been validated appears to be this 208-page report titled "Open reviews of the proof of the Riemann Hypothesis of Kumar Eswaran: An Expert Committee's Report". The effort that the makers of this report put in is impressive, and they are academics with (presumably) some credibility in their respective fields, so I can see why this would catch the attention of the media. In the Foreword of this document, Dr. K.T. Mahhe, who has the also-impressive title "Chairman Sreenidhi Group of Institutions", writes (page i of the report, emphasis in the original):

I can vouch for the fact that, under the circumstances, this expert committee has done due diligence; it has done its utmost to conduct an intellectually rigorous, honest, and fair assessment of the proposed proof. On the basis of the assessment, this expert committee has concluded that Dr. Kumar Eswaran’s proof of the Riemann Hypothesis is correct.

Moreover, the expert committee writes towards the end of its summary opinion (pages 1-2 of the report):

After careful perusal of all the arguments in the proof and the reviews from the experts who have responded, the Committee felt that there are no negative arguments that could technically invalidate the proof and therefore have arrived at the firm conclusion that the proof by Dr. K. Eswaran is both credible and acceptable and that the RH can be considered as proven.

Sounds great, right? But wait, let's dig deeper. The names and titles of the 8 members of the expert committee are listed on page 3 of the report PDF. Only one of them, K. Srinivasa Rao, is a mathematician. Only four of them, listed as "Members of Technical Committee", signed the summary opinion. Those four include K. Srinivasa Rao, two physicists, and Vinayak Eswaran, the brother of Kumar Eswayan who is apparently a professor of mechanical and aerospace engineering.

The experts' summary opinion is itself troubling. First, the statement "there are no negative arguments that could technically invalidate the proof" is concerning, and not something I'd ever expect a mathematician to say. Mathematical proofs aren't considered valid by default until someone can point out a "negative argument" that "technically invalidates it"; actually it is the exact opposite, the burden of convincing people that the proof is correct lies fully with the person claiming to have a proof. So the criterion the experts seem, by their own admission, to be applying in determining that the proof is correct is prima facie invalid.

Second, the experts who signed the opinion don't claim to have read the proof and understood it. They only state that they reached their conclusion "after careful perusal of all the arguments in the proof" and after reading six reviews of the proof written by seven reviewers who read Eswayan's paper. (To be fair, two of the experts — both physicists — were themselves reviewers.) So apart from those two physicists, the other experts' conclusion (including that of Rao, the only mathematician in the group) is actually reached through a second-hand reliance on other sources of authority.

Now, the summary opinion provides quotes from each of the six reviews from the seven reviewers, as a way of supporting the conclusion that "the RH can be considered as proven". Let's examine those quotes, what they contain, and, tellingly, what they omit, which can be found by looking at the full reviews appended later in the report.

Review 1: "By Prof. Ken Roberts and Prof. SR Valluri. Univ. of Western Ontario, Canada."

We found Dr. Eswaran's work quite stimulating of mathematical ideas, and believe that his work should be brought to the attention of a wider scholarly audience, That is, the proof (or selected portions of the methodology) should be published. The Riemann Hypothesis has resisted the efforts of many of the best mathematicians for over 120 years,...” (in email to convener page 58).

Hmm. Well, do Profs. Roberts and SR Valluri actually say they think the proof is correct? Nope. In fact, in their detailed report, included later in the document, they write (page 66) "We are at present undecided on whether the proof is accurate in all respects. There are some aspects which we believe require clarification in order to construct a fully justified proof." In the email on page 58, they wrote: "Our report is incomplete, in that we did not examine all aspects of the proposed proof. There are some aspects of the Riemann zeta function with which we are not sufficiently familiar in order to speak authoritatively." This is not mentioned in the summary opinion.

Review 2: "By Prof. WladislawNarkiewicz (sic), University of Worclaw (sic), Poland, A well-known Polish Number Theorist."

This review is in the form of very detailed technical discussions over emails conducted by Professor WladislawNarkiewicz who had worked through many parts of the paper and asked queries and examined the replies, the discussions extending nearly 60 pages (page 86 -140). The committee commends the painstaking review and discussion which was conducted in the spirit of an open and sincere investigation revealing the many subtleties of RH - we thank him. His entire discussion is given in this report for the benefit of readers and posterity.

In his penultimate email Professor Narkiewicz said that arguments were “heuristic”, though he says “I agree that the similarity of the considered sequence of values of the lambda-function with a random walk gives some reasons to believe in the truth of the conjecture”(page 129)

Prof Narkiewicz’s final reply ended with this sentence: “I want also to stress that the word "heuristic" has no negative meaning. A lot of work of really great mathematicians has been performed in a heuristic way. This applies not only to old times (Euler, Laplace, the Bernoulli’s, ...) but also to recent times”(page 139).

Hmm. The experts certainly seem to think Prof. Narkiewicz is an authority on the subject and that his opinion counts for a lot, since they thought to include in the report a printout of his Wikipedia page (pages 86-87). But, does he actually say he thinks the proof is correct? Nope. Actually, in the 60 page email correspondence he states very clearly and multiple times that he believes the proof is incorrect. (See pages 88, 94, 100, 110, 129.) But the experts don't mention this in their summary, instead taking one of Narkiewicz's quotes out of context, and a weak quote at that that says nothing about the proof's correctness or lack of correctness.

Review 3: "By Professor German Sierra, Dept. of Physics University of Madrid, Spain"

This is the only negative review (page 141-143) The Reviewer seems to have believed (erroneously) that Eswaran was trying to prove the randomness of primes and also imputed that he (Eswaran) felt Equal Probabilities is sufficient for the proof. Eswaran, in his reply,(page 144-154) protested that he does not hold to either of these views. Since there was no reference by the Reviewer of more than 3/4th of the paper, Eswaran requested that the Reviewer kindly read the rest of the paper for the details of the actual proof. Since the Reviewer did not respond, this Review has necessarily to be treated as incomplete and infructuous.

Again, there is a conceptual error in this dismissal of Prof. Sierra's negative review. The experts seem to be assuming the premise that in order to determine that the proof is incorrect, one has to read all of it. This is not true, and is the opposite of how mathematical proofs work: to decide that it's true you need to read all of it, but to decide that it's not a valid proof it's enough to find one incorrect (or vague, or meaningless "not even wrong"-style) step.

(Indeed, the history of claimed proofs of the Riemann hypothesis and other famous conjectures shows that historically they have almost always been able to be dismissed fairly easily by finding one obviously wrong or nonsensical statement, and the criticism "how can you dismiss it? You haven't even read all of it!" is a perennial complaint from crackpot proof-claimers with a poor grasp of mathematical reasoning.)

Review 4: "By Prof. M. Seetharaman, Formerly Dept. of Theoretical Physics Univ of Madras" (one of the expert committee members)

This Reviewer after studying the papers of Kumar Eswaran, was very definitive and said the following; “The author’s analysis is exhaustive, unambiguous, and every step in the analysis is explained in great detail and established. The conclusions of the author and his result must therefore be considered proven.” (Page 7)

This reviewer is not a mathematician. But okay.

Review 5: "By Professor V. Srinivasan, Formerly Professor of Physics and Dean Univ of Hyderabad" (also one of the expert committee members)

Professor Srinivasan reviewed the various steps of the proof saying that “by Judiciously using the properties of the random walk problem”, it was shown “that Riemann’s Hypothesis is true. There is also a numerical proof given.” “I compliment the author for solving the Riemann’s Hypothesis.”(Page 38-39)

This reviewer is also not a mathematician. And his use of the phrase "numerical proof" indicates yet another misunderstanding of how mathematical proofs work, and that he attributes a completely different meaning to the word "proof" than what mathematicians mean when they use that word.

Review 6: "By Professor Vinayak Eswaran, Dept of Mechanical and Aerospace Engineering, IIT Hyderabad."

Professor Vinayak, who is Kumar Eswaran’s younger brother had taken the trouble to spend the best part of two years to understand and study the background material and understand the proof of RH. Therefore he was invited by the Committee to write a review of the proof. He has submitted a very detailed review that summarized all the arguments of the proof and says that there is no doubt that the Riemann Hypothesis is proved. He also submitted an essay which discusses why the RH was so difficult to prove, as there is perhaps only one way it could have been done (pages 8-35)

This man is not a mathematician. Even if he were not a close family member of the person whose proof he is evaluating, his opinion about the proof is mostly irrelevant in the context of the current discussion.

Conclusion

The 208-page report contains many claims that are misleading, either intentionally or unintentionally. Since I am a professional mathematician I can state unequivocally that it adds zero credibility among professional mathematicians to Eswaran's claims of having found a proof of the Riemann hypothesis. And if professional mathematicians won't accept the claim, there is no meaningful sense in which the Riemann hypothesis can be considered as "solved" by Kumar Eswaran at this point in time.

4
  • Comments are not for extended discussion; this conversation has been moved to chat.
    – fredsbend
    Jun 30 at 21:23
  • Take it to chat. And be polite.
    – Oddthinking
    Jul 2 at 16:58
  • 2
    @fredsbend Why were so many of the comments deleted before moving it to chat? That chat discussion doesn't resemble the comment thread that was here, and this is to so much of an extent that I got confused and thought I had clicked on a chat link for a completely different answer. Jul 2 at 17:05
  • 17
    Honestly, the fact that the man's brother didn't immediately recuse himself from reviewing the paper due to a conflict of interest feels pretty damning in its own right.
    – nick012000
    Jul 2 at 19:40
63

Following up @DanRomik's full, accurate answer: I was one of the thousands of people asked to study and comment upon the original document. After a brief look, I realized that it was of a fairly typical sort of not-really-mathematical approaches to many things. Numerical stuff, heuristics, etc. I did not read it carefully, but just far enough to be quite confident that I didn't want to spend time on it.

I do have considerable respect for the marvelous mathematical ideas that have been generated by physics and physicists... but/and also an awareness of the pitfalls that a too-optimistic/naive perception of some of those ideas can lead to pseudo/non-proofs of many things. Especially when otherwise-wonderful ideas are taken far out of their tested operating range.

And, yes, just because someone sounds unorthodox/crank-y doesn't mean they're wrong. So I do try to get beyond my initial recoil-reaction to bombast and such, to avoid missing potentially wonderful things. But, still, in this case, I think I can't find so much that would interest me.

12
  • 17
    You should check out the correspondence between the physicist and Prof. Wladislaw Narkiewicz. Prof. Narkiewicz is very polite but his final email conceals some extremely dry humor about why he kept up the correspondence.
    – Avery
    Jun 30 at 22:50
  • 6
    Here is a direct link sreenidhi.edu.in/pdffls/…
    – Avery
    Jun 30 at 23:11
  • 13
    @paulgarrett After checking out the professor's interactions with Dr. Narkiewicz, in particular, his fallcious understanding of probability spaces, I'm deeply concerned by his insistence that the work of others, who properly apply notions of probability theory, are "definitely wrong". Jul 1 at 19:06
  • 13
    "... want you to know that I enjoyed our discussion which showed that the notion of a proof may have different interpretations". Ouch, Wladyslaw is not holding back! He's basically saying that Eswaran does not even know what constitutes a mathematical proof.... Jul 3 at 6:22
  • 7
    @PerAlexandersson: Sadly, Narkiewicz's irony, while clear to any mathematician, will be lost when read by a casual reader, including members of the committee. Jul 5 at 6:17
13

"So, has the committee actually come to the conclusion that the Riemann Hypothesis has been solved by Kumar Eswaran?"

Yes they did: "On the basis of the assessment, this expert committee has concluded that Dr. Kumar Eswaran’s proof of the Riemann Hypothesis is correct." However this was not an independent committee (it was formed by the university where Kumar Eswaran works), and would not meet the standards for the Clay Institute to mark the Riemann Hypothesis as "solved". This alone doesn't mean the proof is wrong, so now I'll summarize the actual reviews in the report (which you can access by clicking the link at the beginning of this paragraph).

Summary:

  • Review 1 was done by two people who said that they didn't have sufficient expertise.
  • Reviews 2 and 3 were done by experts, but they both dismissed the proof as "heuristic".
  • Reviews 4-6 were done by a more "local" committee including Eswaran's own brother.





Appendix (explaining the summary in more detail):


Review 1: Ken Roberts & SR Valluri, University of Western Ontario, Canada

  • "Our report is incomplete, in that we did not examine all aspects of the proposed proof. There are some aspects of the Riemann zeta function with which we are not sufficiently familiar in order to speak authoritatively."

Review 2: Władysław Narkiewicz, University of Wrocław, Poland

This is presented as a series of emails between reviewer and reviewee.

  • 22 Feb 2020 (Narkiewicz): The review contains 4 sections. In section 1-2 the reviewer agrees with some of the author's assertions, but gives nearly 1-line proofs of them. In sections 3-4 he claims there's flaws in Eswaran's proof, for example he says "This is simply untrue".
  • 24 Feb 2020 (Eswaran): Gives a 2-page response.
  • 2 Mar 2020 (Narkiewicz): The review can be split into parts (a) and (b). He agrees with (a) and says it was proven in 1898 by H. von Mangoldt, but says "assertion b) has no correct proof."
  • 5 Mar 2020 (Eswaran): Contains a direct response about (b).
  • 9 Mar 2020 (Narkiewicz): He says "The main problem with your approach to RH lies in the proof of (1)" and "nowhere in your paper you mention how one can formally deduce your assertion about (1) from this well-known theorem. I wonder whether such a proof is possible, as the elements of the sequence λ(n) depend on n, and elements in a random sequence do not have that property". He begins to discourage Eswaran, by saying "Do not worry about this situation. Several excellent mathematicians tried without success to prove Riemann Hypothesis."
  • 13 Mar 2020 (Eswaran): Response is "taking me some time."
  • 17 Mar 2020 (Eswaran): Gives a 4-page response.
  • 30 Mar 2020 (Narkiewicz): He says "Your proof consists of the following steps" and gives 5 steps. he says step 4 is incorrect.
  • 2 Apr 2020 (Eswaran): Gives a 4+ page response.
  • 3 Apr 2020 (Narkiewicz): Things are becoming more focused. He says "This thime I send you a short message with only one question" then "It seems that the problem of your proof lies in the fact that you disregard the difference of these two notions of probability. If you really have a proof of your assertion, then I would like to be able to see it."
  • 5 Apr 2020 (Eswaran), 14 Apr (Narkiewicz), 16 Apr (Eswaran): More back-and-forth.
  • 17 Apr 2020 (Narkiewicz): This is the last email from Narkiewicz in the report. There's no more specific attacks on Ewaran's work, but he calls Eswaran's argument "heuristic" and indicates that he personally wouldn't call it a "proof".

Reviewer 3: Germán Sierra, Instituto de Fisica Teorica, Spain

  • 9 Sep 2020 (Sierra): Summarizes the proof and says "This idea has been used heuristically to conjecture the moments of the Riemann zeta function, but again there are not rigorous proofs except for a few cases."
  • 6 Oct 2020 (Eswaran): Gives a response with a 10-page PDF, but unlike in the case of Narkiewicz, the report contains no response from the reviewer.
11
  • 4
    Can you clarify or highlight where you answer adds incremental information or sheds additional light over Dan Romik's earlier answer, which takes the same tack (analyzing the reviewers, their credentials, and their conclusions), but in depth?
    – Dan Bron
    Jul 2 at 17:10
  • 2
    @user1271772 Ah, I see. I do think Dan’s answer highlights, early on and throughout, that the reviewers did not approve of the proof. But the other element - evaluating their credentials - I do see as a clear difference between the two answers.
    – Dan Bron
    Jul 2 at 18:01
  • 2
    @DanBron Thanks! I think our analyses of the reviews are also different. For example, immediately below "Review 1" in Dan's answer, is a quote from the two reviewers "We found Dr. Eswaran's work quite stimulating of mathematical ideas, and believe that his work should be brought to the attention of a wider scholarly audience; that is, the proof (or selected portions of the methodology) should be published" which at first made me think the proof was approved. The quote I showed, from the same reviewers, was the part that says "we are not sufficiently familiar in order to speak authoritatively." Jul 2 at 18:12
  • 4
    @user1271772: The thing is, in his April 3 email Narkiewicz (item 1) pointed out at the specific issues that E. has to address. In his reply, instead of addressing this issue, E. directs him to a 1943 paper in a physics journal. Instead of specifying, as requested, a probability space, sigma-algebra and measure, and explaining why almost-every behavior that ergodic arguments typically establish, would apply to every element of the probability space, he simply refers to the 1943 paper. No wonder that N. simply finished the correspondence in his next email. Jul 2 at 19:08
  • 3
    Narkiewicz considers Eswaran's argument to be a heuristic one, which is not what Narkiewicz says he would consider a "proof". Jul 2 at 19:18

This site is temporarily in read only mode and not accepting new answers.

Not the answer you're looking for? Browse other questions tagged .