Understanding the Riemann Hypothesis: Why it's a high-stakes puzzle

Someone wrote a sci-fi story suggesting that anybody who proved the Hypothesis would be promptly murdered by a consortium of bankers

One of the most intriguing “open” prob­lems in mathematics is called the Rie­mann Hypothesis. This was a conjecture made by the great mathematician Georg Friedrich Bernhard Riemann (1826-1866) in a paper he wrote in 1859.

In 1900, another mathematician, David Hilbert, listed 23 important problems to be solved in the 20th century. One of these, the Riemann Hypothesis remains unsolved. In 2000, the Clay Mathematics Institute offered $1 million for anybody who proved the hypothesis.

In 2016, Kumar Eswaran, a physicist at the Sreenidhi Institute of Science and Tec­h­nology, Hyderabad, claimed a proof. And now in June 2021, news reports appeared saying his paper of the proof had been validated by several mathematicians. But the Clay Institute says the proof is not correct and the problem remains open.

So, what is the Riemann Hypothesis?

Riemann was looking at a str­a­n­ge infinite series called the Zeta Function. His teacher, the lege­n­dary Carl Friedrich Gauss had a method of est­imating how many prime num­bers there wo­u­ld be between zero and any given number, no matter how large. While inves­tigating Gau­ss’ methodology, Riemann extended his analysis to com­plex numbers, and looked at the Zeta Function. In a 10-page paper, he listed many new dis­coveries about numbers, one of which he admitted he couldn’t prove. That’s the Hypothesis.

What are prime numbers?

Prime numbers are numbers that cannot be divided except by themselves (and by 1). For example, 2, 3, 5, 7, 11, 13, etc, are pri­mes. There are an infinite number of pri­mes. Gauss found a way to estimate the number of primes, and calculated every prime till 3,000,000. But nobody knows how to predict where exactly the next pri­me number will turn up. The largest kno­w­n prime number is 24 million digits long!

What are real numbers, imaginary numbers and complex numbers?

Real numbers include integers (1, 2, 3), the fractions (which can be written as ratios like 1/3), the irrational numbers (like square root of 2, which cannot be written as a ratio), the transcendental numbers (Pi for example). Imaginary numbers inc­l­u­de the square root of negative numbers. These are usually ex­pressed as multiples of “I”, which is de­f­i­ned as the square root of nega­tive one. Complex numbers are a combination of real and ima­g­inary numbers. Imaginary and complex num­bers are hugely important in very real engineering problems, involving electronics and rocketry.

So, what did Riemann do?

Riemann plugged complex numbers into Zeta functions. He found that, if the real part of a complex number was ½, or close to 1/2, the series summed to zero. Why? He didn’t know. Would this hold true for all complex numbers? He guessed so. That’s the Hypothesis.

How come it hasn’t been proved?

Nobody has figured out why this is true though the critical strip where zeros turn up has been narrowed down to 1/2. The Riemann Hypothesis has been tested with huge numbers, and always turned out cor­rect. But that doesn’t mean it’s always true. Literally, hundreds of theorems have been written, assuming Riemann was correct.

Why is it important?

It’s foundational to number theory. Rem­ember, Riemann was trying to make sense of the pattern of distribution of prime numbers. Now in practical terms, almost all modern cryptography is based on large prime numbers. Multiplying two known, very large prime numbers is a mechanical process, which computers can do quickly. So you can establish a code easily. Dividing a very large number to figure out if it is prime, or it has only two prime factors, is extremely tedious and long-drawn. This makes the code hard to break.

Anybody who proves the Riemann Hy­p­othesis will, almost certainly, have figu­red out a faster way, or several faster ways, to find prime numbers. This would make most cryptographic systems vulnerable. There’s a great deal more at stake than the $1-million prize. In fact, someone wrote a science fiction story suggesting that anybody who proved the Hypothesis would be promptly murdered by a consortium of bankers, and national security agencies!

Riemann’s Zeta Functions closely resemble the wave functions of quantum physics. Under­sta­n­ding how Zetas work could lead to a grea­ter understanding of quantum physics. (Incidentally Riemann Geometry is a cri­tical underpinning of Einstein’s Theory of Relativity, so, this mathematical genius has already made a huge contribution to physics.)