Who Was Bernhard Riemann?
Bernhard Riemann, in full Georg Friedrich Bernhard Riemann, (born September 17, 1826, Breselenz, Hanover [Germany]—died July 20, 1866, Selasca, Italy), German mathematician whose profound and novel approaches to the study of geometry laid the mathematical foundation for Albert Einstein’s theory of relativity. He also made important contributions to the theory of functions, complex analysis, and number theory.
The Riemann Hypothesis
If one wants to compare Ramanujan's genius and his life to another equally noteworthy genius, it would be Riemann Georg Friedrich Bernhard Riemann (1826-1866). Riemann was a frail and shy child who had been brought up in utter poverty. His brilliance was recognized by his teacher Carl Gauss, the all time great, 'prince of mathematics'. Like Ramanjuan, Riemann died early, leaving behind his works that continue to add value to 21st century physics and mathematics. The immensity of Riemann's contribution to physics and mathematics can be appreciated through Einstein's General Relativity Theory, where Riemann's revolutionary ideas about geometry and higher dimensions led to the prediction of space-time having multiple dimensions. Riemann geometry (as opposed to Euclidean Geometry), introduced the idea of defining each point in space with a collection of numbers (tensors), to determine how much space was bent or curved. Riemann suggested that each point in space needed a collection of 10 dimensions to describe its properties. This path-breaking idea was later used by Einstein in his theories to predict the true structure of the warped space-time fabric. This is strikingly similar to Ramanujan's independent prediction about the 10 dimensions of space-time through his equations.
One of the many contributions of Riemann was the Riemann Hypothesis, in the field of prime number distribution. Even to this day, the Riemann Hypothesis remains one of the unsolved problems in mathematics, carrying a 1 million USD prize for the person who proves or disproves it. The Riemann Hypothesis is difficult to understand for simple souls. But let's try and make sense of its significance and impact on the modern world, if ever his hypothesis is proved. Earlier in this book, we have touched upon the importance of prime numbers and their crucial role in real life applications, especially in securing confidential data from malicious attacks. Just to recap, a prime is any natural number (1,2,3, etc.), that is divisible only by itself and 1. For example, 11 cannot be divided by any number other than 1 and 11, so 11 is a prime. The beauty of a prime number is that all natural numbers from 1 to infinity are built from primes. So primes are like the building blocks of natural numbers, just like bricks are for a house. Since primes cannot be divided further, they are also called atoms of the number system. As we learnt earlier, one of the deepest mysteries of prime numbers is that there is no formula available to generate a prime number. We also don't know how these primes are distributed across the number line. Further, we have no idea how many prime numbers lie below a given number. A great many mathematicians, including Euler, Legendre, Gauss, and others, have tried to come up with a formula that can predict how many primes occur below a given number. But all these were only approximates. Riemann went a step ahead and actually gave a solution to this problem in terms of a hypothesis that bears his name.
Relentless efforts are on to prove the Riemann Hypothesis. The stakes are very high and if proved, the Riemann Hypothesis will change the way we do business. It will strike disaster for the Internet, making it no longer safe. The encryption technique we spoke of in the Quantum Mechanics chapter, will no longer hold good. The safety of internet-based business transactions depends on the randomness of prime numbers. Given a very large complex number, it is very difficult to break it up into two prime factors. It takes decades for very powerful computers to break a code using prime number-based encryption techniques. The implications for businesses that use this coding technology, will be immense and bank accounts and personal details will be seriously compromised Several mathematicians around the world have dedicated most of their lives to finding the proof behind the Riemann Hypothesis The latest modern high speed computers are being programmed to check if Riemann's prediction about occurrences of primes is
true. The Riemann Hypothesis already works for trillions of values, that is to say, the hypothesis seem to work as per its predictions. But in mathematics, the proof cannot be accepted until it works for all the numbers up to infinity. In other words, what is needed is a general proof that works for all values from zero to infinity. No wonder David Hilbert, the great German mathematician regarded as one of the most influential and universal mathematicians of the 19th and early 20th centuries, said, "If I were to awake after having slept for a thousand years, my first question would be: Has the Riemann Hypothesis been proven?".
Coming back to the theme of this chapter about Nature's Language, it would be appropriate to say that nature seems to have embedded in itself the kind of abstraction that is being unravelled layer by layer, by the power of human imagination. Riemann's life was a sad analogy to Ramanujan's, with limited resources. He died at the age of 33. How much mankind would may have progressed if these two geniuses had lived a full life without being struck down by illnesses that are treated so easily today. So much treasure is hidden away in Ramanujan's notebook and Riemann's one volume of a few pages filled with equations and derivations, that posterity will one day look back in disbelief and admiration.
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