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The Riemann Hypothesis – The Millennium Prize Problem

One of the most significant unsolved mathematics problems has found its solution after 161 years later. Yes, the problem has been declared as the Millennium Prize problem worth $1 million two decades ago and is rewarded from the Clay Mathematics Institute of Cambridge, Oxford, England. Indian Mathematician Claims To Solve 161 Year Old Problem Carrying 1 Million Reward, “the Riemann Hypothesis”.

Dr. Kumar Eswaran, a mathematical physicist at Sreenidhi Institute of Science and Technology (SNIST) has claimed to have found the proof for the 161-year-old unsolved mathematical problem, the Riemann Hypothesis (RH).

Riemann Hypothesis is Considered the top-most mathematical problem of the top 10 unsolved mathematical problems by American mathematician Stephen Smale.

What is Riemann Hypothesis?

In layman terms, the Riemann Hypothesis helps in counting the prime numbers. The Riemann Hypothesis arose from the work of famous mathematician Carl Friedrich Gauss in the 19th century.

The Riemann hypothesis is deeply rooted in number theory and the Swiss mathematician Leonhard Euler, laid its foundation in the 18th century. In 1859, Bernhard Riemann expanded Euler’s work to develop a mathematical function. Euler’s Riemann hypothesis leads to a deeper connection between positive integers, prime numbers, and imaginary numbers. These connections are responsible in multiple areas of mathematics, including analytic theory and cryptography.

Prime numbers are of great importance in diverse ways today. Mathematicians are uncertain about the pattern in which these prime numbers occur. They study the different ways where numbers can be connected according to their interest. Carl Gauss, a German mathematician, kicked off Interest in this problem. He wanted to know if there was a simple way to find the number of prime numbers between any two numbers. For example, how many prime numbers are there between 100 and 1,000?

Riemann Zeta Function

prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers, e.g., 2, 3, 5, 7, etc. They play an important role, both in pure mathematics and its applications. The distribution of such prime numbers among all natural numbers does not follow any regular pattern. However, the German mathematician G.F.B. Riemann (1826 – 1866) observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function

 ζ(s) = 1 + 1/2s + 1/3s + 1/4s + …

called the Riemann Zeta function. The Riemann hypothesis asserts that all interesting solutions of the equation

ζ(s) = 0

lie on a certain vertical straight line.

Euler: a mathematician without equal and an overall nice guy

Leonhard Euler’s Last Contribution to this Topic

Euler had formulated a mathematical series called Z (s), such that:

Z (s) = (1/1s) + (1/2s) + (1/3s) + (1/4s) + …

He found that Z (2) – i.e., substituting 2 for s in the Z function – equaled π2/6, and Z (4) equaled π4/90. At the same time, for many other values of s, the series Z (s) would not converge at a finite value: the value of each term would keep building to larger and larger numbers, unto infinity. This was particularly true for all values of s less than or equal to 1.

Euler was also able to find a prime number connection. Though the denominators together constituted the series of positive integers, with a small tweak, Z (s) could be expressed using prime numbers alone as well:

Z (s) = [1/(1 – 1/2s)] * [1/(1 – 1/3s)] * [1/(1 – 1/5s)] * [1/(1 – 1/7s)] * …

“It means that if we want to think about questions related to prime numbers, we can use this Z (s) to translate them to questions about regular old positive integers, which are much easier to deal with.” Thomas Wright, a number theorist at Wofford University, South Carolina  


Kumar Eswaran’s Research

According to Eswaran, it is easy to count the number of prime numbers from say 1-20 but the same becomes a problem when calculating the prime numbers for big numbers like till one million or 10 billion and more. 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 cautious to put the paper through a detailed peer review.


The hypothesis was important to prove as it would enable mathematicians to exactly count the prime numbers – Eswaran.

With over thousands of downloads, the review commenced in February 2020 went on for a year, according to the institution. An expert committee looked into the proof Eswaran developed. The committee comprised eight mathematicians and theoretical physicists.

Conclusion of the Committee

More than 1,200 mathematicians took part in an open review in which – the referees would willingly have their names and institutional affiliations revealed openly. Also, nothing should remain hidden from the other experts to see. Soon after, seven international scholars responded to the call.

After scrutinizing the comments of the seven competent reviewers and the responses of the author, the committee came to the conclusion that Eswaran was determined proof of the RH is correct.

Professor M Seetharaman, former professor of the department of theoretical physics at the University of Madras, one of the members of the committee who reviewed the RH proof by Eswaran. He said, “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.”

“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, Sreenidhi Institute of Science and Technology, Hyderabad.

However, for the past 161 years, proof of the hypothesis has eluded the efforts of the most famous mathematicians across the world. The Riemann Hypothesis (RH) holds great importance to well-known mathematicians. The proof of RH opened doors to numerous theorems that were dependent on the truth of this hypothesis, the SNIST said in its statement.

The great mathematician Carl Friedrich Gauss (1777-1855) had written a formula that predicts the number of prime numbers below any given number. Euler discovered the zeta function using only real numbers. Georg Friedrich Bernhard Riemann (1826-1866) improved the formula by using entirely original methods involving the calculus of functions of a complex variable, according to the statement. And now, Dr. Kumar Eswaran hence proved the 161-year-old toughest problem.



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