Dr Eswaran, 74, works with the Sreenidhi Institute of Science and Technology in Hyderabad.
Hyderabad: It was a chance discovery, says Dr Kumar Eswaran, a mathematical physicist in the city, who claims to have found the solution to the Riemann Hypothesis, a millennium problem in mathematics. It was waiting for proof for 161 years, Dr Eswaran told this newspaper.
The Riemann Hypothesis, touted to be number one out of the top 10 unsolved mathematical problems, was designated as a millennium problem in 2000, with a reward of $1 million from Clay Mathematics Institute, Oxford, England, for anyone who could solve it.
"It was, in fact, back in 2016, that I first gave proof for the formula improved by the great mathematician Georg Friedrich Bernhard Riemann in the 1800s. I had put it on the web for open review and downloads after working on it for about six weeks. During 2018-19, I gave several lectures on the proof," Dr Eswaran, 74, who is with the Sreenidhi Institute of Science and Technology, Hyderabad, said.
The Riemann Hypothesis, in the simplest possible explanation, relates to how prime numbers are distributed. This has been a problem that has challenged mathematicians for generations. According to mathematicians who have grappled with the hypothesis, proof of the Riemann Hypothesis would have far-reaching consequences for number theory and for the use of primes in cryptography, and would automatically lead to the proof of numerous theorems which are dependent on the truth of this hypothesis.
In Dr Eswaran's methodology, the ‘factorisation sequence of numbers’ was shown to be like a 'random walk', he said. "This method used was actually not just number theories. I had to use techniques from random variables and probability as well," Dr Eswaran added.
An expert committee of scientists including Dr T. Ramasami, former secretary, Union ministry of science and technology, Prof. P. Narasimha Reddy, executive director, Sreenidhi Institute of Science and Technology, Prof. M. Seetharaman, former professor and Chair, department of theoretical physics, University of Madras, Prof. V. Srinivasan, former professor and dean, School of Physics, UoH, Prof. K. Srinivasa Rao, former senior professor, Institute of Mathematical Sciences, Chennai, Prof. M. D. Srinivas, senior Fellow, Centre for Policy Studies, Prof. Vinayak Eswaran, department of mechanical and aerospace engineering, IIT Hyderabad, and Dr Adindla Suma, associate professor, department of computer science, Sreenidhi Institute of Science and Technology conducted an open review of the proof for a year from February 2020 to conclude that Dr Eswaran had indeed proved the Riemann Hypothesis.
The committee invited more than 1,200 mathematicians to participate in an open review of the presented proof. The review was open in the sense that the referees had to be willing to have their names and institutional affiliations openly revealed, so that nothing is done anonymously, and nothing can be said that would not be openly available for all other experts to see.
Dr Eswaran said in his experience this had been the most fair and robust way to get the proposed proof assessed. The expert committee then examined the comments of the reviewers and the responses of the author. In their final meeting held on May 16, the committee concluded that Dr Eswaran’s proof of the Riemann Hypothesis is correct.
The committee also ruled that the proof, the referees’ comments, and the author’s responses in their entirety, without any redactions, should be published in the form of an e-book and made available to the public at large and then followed, a few months later, by a hard copy.