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 have tried to solve this problem, but none have succeeded yet.

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

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.

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.

I will discuss this claim under the following headings:

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 the 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's 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.

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 is a theorem 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

• Discussion thread on Reddit Math
• The Hindu Article on this
• Official website for $1 Million Reward for the Proof of Riemann Hypothesis
• Reference book on Riemann Zeta function
  • Indian version
  • Global version
• Introductory textbook on The Riemann Hypothesis
• This Quora Q&A
• This Skeptics.SE Question

Frequently Asked Questions

Updates

1. The Author of the paper responded. (Jun 29, 2021 at 16:32 EDT) / Comments hidden.
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 was available on Math.SE. See the archived version. (Aug 29, 2022 at 00:12 EDT)
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, and 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)