No, Riemann Hypothesis has NOT been solved, yet!

I woke up today (29th June 2021) with a piece of news that the Riemann Hypothesis has been solved. But looks like that was all just a fuss.

Indian newspapers, The Hindu, The Quint, Hindustan Times, etc., everyone reported that a Hyderabad-based mathematician has succeeded in solving this $1 million question.

Riemann Hypothesis is one of the unsolved problems in mathematics and it has a bounty of $1 million for anyone who can offer a solution, proof, or disproof.

For the last 161 years, thousands of mathematicians had tried to solve this problem but none has succeeded yet, obviously.

The news that it was solved by Kumar Easwaran, a mathematics physicist is nothing but a false statement.

[Update:] It is also reported that the $1 Million prize has been approved by Clay Mathematical Institute for K Easwaran. This fact is wrong as well.

But the truth is that Riemann Hypothesis is still unsolved.

I was surprised to know that following the news the Wikipedia page for Riemann Hypothesis was edited and it was erroneously declared that the problem is now solved. The edit has been reverted following a brief discussion.

The proof provided by Dr. K Easwaran (also: K. Eswaran) has already been declared flawed when he submitted it in 2018. Year after year he has been submitting the same paper with some modifications but as the foundation is flawed, the conclusion — no matter how hard you try, will always be flawed.

Riemann Hypothesis

Riemann Zeta Function in Riemann Hypothesis

Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. This is the conjecture.

All you have to do is to prove (or disprove) this.

Here Riemann zeta function $\zeta(s)$ is a function of a complex variable $s$, and is defined as:

$\zeta(s) = \sum_{n=1}^\infty n^{-s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots$ if $\textrm{Re}(s) \gt 1$

Also see: Poincare Conjecture

Why am I writing this?

I don’t write much about the latest mathematics developments anymore. But Riemann Hypothesis is possibly the most important unsolved problem in mathematics. Proof of this hypothesis changes everything and opens new space for further studies.

Progress till date

The Wikipedia page for Riemann Hypothesis has a list of all the important attempts made to solve the Riemann Hypothesis. These include solving the conjecture with:

  • Operator theory
  • Lee–Yang theorem
  • Turán’s result
  • Noncommutative geometry
  • Hilbert spaces of entire functions
  • Quasicrystals
  • Arithmetic zeta functions of models of elliptic curves over number fields
  • Multiple zeta functions

K Easwaran’s Proof

Here’s the excerpt from his paper’s abstract from https://www.researchgate.net/publication/325035649_The_Final_and_Exhaustive_Proof_of_the_Riemann_Hypothesis_from_First_Principles:

This proof of the Riemann Hypothesis (Riemann 1859) crucially depends on showing that the function F (s) ≡ ζ(2s)/ζ(s), has poles only on the critical line s = 1/2 + iy, which translates to having the non-trivial zeros of the ζ(s) function on the self-same critical line. It can be easily verified that all the non-trivial zeros of ζ(s) appear as poles in F (s), and all the trivial zeros cancel and so do not appear as poles in F (s).

He has even tried to explain this through videos on YouTube:

Why K Easwaran’s Proof is Flawed?

With every attempt K Easwaran made, attempts were made to crosscheck his claims. Garry Herrington showed the flaws in his proof in 2018 in this note (PDF). Garry’s claim was disputed by R. Raghwan in this reply just 20 days later. Raghwan supported Easwaran methods and followed his reply with one more letter to Garry Herrington. You can read the last copy here.

Where this proof goes wrong?

Dr. K Easwaran is thinking from the wrong perspective. This is formulated as a probability problem but by considering favored values (see this comment on reddit/math). Such proof claims are so regular — people start with a general theory pick a value of their choices and prove something.

[Update]: Reddit user The_Stutterer shares his review on this paper along with the flaws that he found.

Riemann Hypothesis, Image, Gaurav Tiwari
Click to enlarge image
Riemann Hypothesis, Image, Gaurav Tiwari
Click to enlarge image

The main idea of this proof hinges on the apparent similarity between the properties of a random walk and that of a deterministic sequence. This is what Garry Herrington claimed. Despite two or three attempts to save the proof, Garry’s claims override the others.

Easwaran is a physicist and his perspective is as such. In mathematics, there is no exception for a theorem nor a theorem is true for just a certain set of values.

This letter game may continue but there are more than enough reasons to say that K Easwaran’s claims are flawed but still worth having international attention. The author of the paper and the supporters have asked for favorable attention to gain positives from the paper.

To be honest there have been more than enough attempts to disapprove of the Riemann Hypothesis. See proposed (dis)proofs of the Riemann Hypothesis.

Suggested Readings

Frequently Asked Questions

Is Riemann Hypothesis Solved?

No. The Riemann Hypothesis is unsolved.

Who is K. Easwaran?

Dr. K. Easwaran or Dr. K. Eswaran is a Professor of Computer Science at SNIST, Ghatkesar Hyderabad.

What is Riemann Hypothesis?

Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2.”

Updates

  1. The Author of the paper responded. See the #comments. (Jun 29, 2021 at 16:32 EDT)
  2. Updated article to point out that the newspapers claim that $1M prize has been approved to K Easwaran. I have emailed Clay Mathematical Institute to know about their stand on this. I’ll update once any reply follows. Update received. See update point 9. (Jun 29, 2021 at 20:12 EDT)
  3. A closed discussion on this is available on Math.SE: https://math.stackexchange.com/questions/4185354/is-kumar-eswarans-proposed-proof-of-the-riemann-hypothesis-correct
  4. The Wikipedia page for Riemann Hypothesis is now locked to new edits. (Jun 29, 2021 at 12:36 EDT)
  5. I talked to some of my peers in the same field, they have pointed out that the claim has been ignored for a while due to theoretical issues. When the paper was first published in 2018, sufficient attention was provided but various flaws were detected in the proof. (Jun 30, 2021 at 08:30 EDT)
  6. Others have pointed out that the paper was submitted to Researchgate. Such projects should be published to an international paper for the proper reviews to happen as standard international math magazines approve such claims only after a discussion. (Jun 30, 2021 at 08:32 EDT)
  7. Reddit links fixed. (Jun 30, 2021 at 08:32 EDT)
  8. Added The_Stutterer’s response to the claim above. (Jun 30, 2021 at 09:51 EDT)
  9. Received a word from ClayMath, the problem is still unsolved and open to discussion. K Easwaran is invited to send his research to the institute and others in the community. (Jun 30, 2021 at 19:00 EDT)
  10. The official webpage of the Millennium Problem page for Riemann Hypothesis states that the problem is Unsolved. (Jun 30, 2021 at 23:55 EDT)
  11. Softened the language a little bit. (Jun 30, 2021 at 23:55 EDT)
  12. Added additional resources, including Quora and Skeptics.SE (July 2, 2021 at 00:02 EDT)
  13. Added FAQ for generally interested readers. (July 9, 2021 at 08:20 EDT)

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20 comments add your comment

  1. Dear Mr. Gaurav Tiwari, I saw this page by accident. I wish to inform you that, I have refuted Garry Herrington’s objections long ago ( Nov 2018). Please see the project log page in my Research gate. The link is:
    https://www.researchgate.net/project/The-Dirichlet-Series-for-the-Liouville-Function-and-the-Riemann-Hypothesis/update/5d238c27cfe4a7968db31f1e
    If you read my note you will perceive that I do know that the Liouville lambda function is a deterministic arithmetic function,
    but I say that if you take large consecutive values of the Liouville function, lambda(n), the sequence behaves like a random walk (or coin tosses taking H=1 and T=-1). So if you add N consecutive values the absolute value of the sum L(N) behaves like:
    C. N^(1/2) thus proving RH. To actually prove my assertions I had to actually prove several theorems which is the heart of the paper. These theorems relate to proofs of (i) equal probability of 1 and -1 of the lambdas (ii) independence (ii) un-predictability and (iv) non-periodicity of the lambda sequence. Then and only then, one can say that the sequence of lambdas behave like coin tosses (or a random walk). I have done all that in the paper, you must read it! Once this is done (ie. all the required theorems proved) the application of Littlewood’s Theorem and the theorem on the iterated logarithm proves RH. I must say you were extremely hasty to draw your conclusions that the paper is a Lie! I will make no more comment but leave it to the public at large to judge if I deserve such condemnation. Regards Kumar Eswaran

    • Where is your proof that the sequence behaves “like” a simple random walk? What does that even mean? The only thing you have asserted so far is that your proposed sequence is a realization of an SRW, and, truth be told, any sequence of integers differing by 1 that starts at 0 is a realization of an SRW, and they’re all equiprobable in the sense that they all have probability 0.

      The asymptotic properties of the SRW are given by the central limit theorem, and state a result about convergence in distribution, not convergence for any particular realization of the SRW. They do not prove anything about any particular sequence. It seems like you do not understand probability theory.

      Your criteria for independence (unpredictability, acyclicity) are not nearly sufficient to show independence. Brownian motion at t+1 is unpredictable from simply knowing the values up to t, but you still gain a lot of information about B(t+1) from B(t) (E[B(t+1) | B(t)] = B(t)), and indeed, they are not independent.

      Your theorem showing the equal proportions of -1 and 1 is laughably incorrect. Showing that a bijection exists between the two says nothing. And your method of showing such a bijection is needlessly complicated. There are an infinite number of natural numbers with an odd number of prime factors (by construction, 3^1, 3^3, 3^5, so on). The same can be said of those with an even number of prime factors. Therefore, by Cantor-Bernstein-Schroder, there exists a bijection between them.

      That says nothing about probabilities. One can say the same of perfect squares and even numbers.

      You deserve condemnation for making press releases and lecture videos to amateurs instead of debating the truth with mathematicians, convincing them through proof, and letting the work speak on its own merits.

  2. Continued from previous comment:
    My papers have been thoroughly reviewed by many and the Expert Committee has put all the comments on the Institution Webpage along with my replies in the form of an E-Book, without any redactions. The E-book (200 pages) has been uploaded so that everybody can see and make their own judgement. I am giving a link to it: https://sreenidhi.edu.in/pdffls/A_Report_on_Riemann_Hypothesis_updated(25-06-21).pdf
    With Best Wishes
    Kumar Eswaran

  3. Dear Mr. Gaurav Tiwari, I saw this page by accident. I wish to inform you that, I have refuted Garry Herrington’s objections long ago ( Nov 2018). Please see the project log page in my Research gate. The link is:
    https://www.researchgate.net/project/The-Dirichlet-Series-for-the-Liouville-Function-and-the-Riemann-Hypothesis/update/5d238c27cfe4a7968db31f1e
    If you read my note you will perceive that I do know that the Liouville lambda function is a deterministic arithmetic function,
    but I say that if you take large consecutive values of the Liouville function, lambda(n), the sequence behaves like a random walk (or coin tosses taking H=1 and T=-1). So if you add N consecutive values the absolute value of the sum L(N) behaves like:
    C. N^(1/2) thus proving RH. To actually prove my assertions I had to actually prove several theorems which is the heart of the paper. These theorems relate to proofs of (i) equal probability of 1 and -1 of the lambdas (ii) independence (ii) un-predictability and (iv) non-periodicity of the lambda sequence. Then and only then, one can say that the sequence of lambdas behave like coin tosses (or a random walk). I have done all that in the paper, you must read it! Once this is done (ie. all the required theorems proved) the application of Littlewood’s Theorem and the theorem on the iterated logarithm proves RH. I must say you were extremely hasty to draw your conclusions that the paper is a Lie! I will make no more comment but leave it to the public at large to judge if I deserve such condemnation. Regards Kumar Eswaran
    P.S.: The unedited comments of the reviewers and Experts are available in the form of an E-Book please download:
    https://sreenidhi.edu.in/pdffls/A_Report_on_Riemann_Hypothesis_updated(25-06-21).

    • With all due respect. You aren’t a Mathematician as you don’t have any degree in mathematics. You are a physicist turned computer scientist who has not done any research in pure math except this unpublished paper. If your proof was correct people like Manjul Bhargava, Peter Sarnak etc would approve. The committee you set up didn’t have a single mathematician let alone a number theorist.
      If a proof at this level was correct it would have been already published in Annals of Mathematics by IAS, Princeton. You are a joke to the mathematical community of India the same country which gave birth to people like K. Chandrashekharan, Narasimhan, Seshadri, Harish-Chandra and many many more.

    • Firstly, it does not make sense to call it an ‘expert committee’ when exactly one of them is a professional mathematician. The opinions of all other laymen in that ‘export review’ is moot. They have absolutely no relevant expertise.

      Secondly, the report grossly misrepresents the actual reviews of the few relevant mathematicians involved: Prof. Ken Roberts and Prof. SR Valluri, Prof. WladislawNarkiewicz, and Prof. German Sierra. All mathematicians have outright said that your proof is wrong, and yet, the summary of the report claims that they support you, which is a blatant lie! Review 1 by Roberts and Valluri says they are unsure of your proof, Review 2 by Narkiewicz says that you did not solve it correctly, and offers consolation by saying that there are many who did not manage to solve it, and Review 3 by Sierra flatly proves your work wrong.

      Thirdly, no, no one needs to go through your ENTIRE proof to be proven wrong. One logical fallacy in one part of the proof is enough for it to be dismantled. That has been done repeatedly since 2018.

      Finally, CMI has confirmed that the RH is still open and the Indian media (unsurprisingly) jumped the bandwagon to declare you a genius.

  4. Riemann hypothesis is NP complete problem. I will prove this by my theory. Sir did you know P vs NP problem 1 million dollars questions

  5. Thank you so much for trying to correct such spurious disinformation. Keep up the good work! :)

  6. If Mr. Kumar Eswaran has a proof of Riemann Hypothesis, he must submit his RH proof to reputed Math journals like Annals of Mathematics. It will be subject to peer review by experts in the field, and if it passes all the litmus tests, it will be published. Then, he will be awarded Clay Institute’s 1M $ prize for his commendable achievement. Until then, we wait.

    We wish Dr. K. Eswaran a grand success,
    N. Ganesan
    Senior Scientist, Loads and Dynamics
    Houston, TX, USA

    http://nganesan.blogspot.com

  7. I have not read the proposed proof in question, but perhaps I should expose a few things that happen in academia, behind closed doors:

    A proposed proof of a big problem like the RH by an “unknown” author will *almost* surely never be tolerated by the so-called big journals, irregrdless of whether it is valid or not. In fact, most editors of these journals will not even read such a paper from an unknown author. I abandoned my own dreams of solving the P vs NP problem 15 years agox after going through similar hurdles. I then left academia totlly, and joined industry, a decision which I have never regreted.

    • I disagree with your statements here. When Yitang Zhang published a ground breaking paper, it was published by Annals of Mathematics right away and shot to fame right away. Until then, he was not a well known figure in mathematics community. If Dr. Easweran has got the correct proof of this RH, he can send it to Annals of Mathematics and/or notify renowned mathematicians about his work. Perelman did the same when he proved Poincare Conjecture. what stops Dr. Easweran to do this. His proof was rejected in the year 2018 as flawed. Why not people like Manjul Bhargava , Terence Tao, Peter Sarnak etc are not notified about this proof. Michael Atiyah claimed similarly and he presented it in an international conference. His work was rejected as flawed. I don’t understand why this Dr. Easwaran is not making his work known to experts in the field. There was a similar publicity stunt done in Nigeria. Now India. What a joke.???

  8. Indian media was too quick on this (but not quick enough, it took 5 years for them to notice his work):

    • Indian Media did not know how to spell even the name “RIEMANN” correctly,..

  9. Dear Sir / Madam,

    The reason I am posting here is that out of my personal exploration and curiosity I did come across and found my own intuitive solution to the Riemann Hypothesis in the year 2017, posted it on facebook @ https://www.facebook.com/photo/?fbid=10219716192124274 and later published the proof in my book Any Body Can Derive Everything From GOD (Geometry Of Dimensions) https://abcdefg.co/publications . I request you to kindly review my proof as well and see if it is valid. I am really looking forward to hearing from you. Please email me @ sridattadev@gmail.com or connect with me @ https://www.facebook.com/sridattadev.kancharla .

    Singularity
    Of
    Universal
    Light,
    Sridattadev Kancharla.

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