In mathematics, either you are correct or incorrect. It has no room for ambivalent claims like half-correct or partially mistaken. For this reason, mathematics enjoys the reputation of being notoriously elusive even to the most prolific minds. Between 1913 and 1915, for example, Albert Einstein had to work ceaselessly, sometimes making mistakes that took him months to detect and correct, while struggling to hammer out the geometrical formulation of his masterpiece: the general theory of relativity. Even more formidable challenge held sway over Bernard Riemann, one of the greatest among the giants of mathematics, when he found himself asking: how many prime numbers are there less than a given integer? With this seemingly innocent question, the brilliant Riemann had hurled himself into the heart of one of the most difficult problems of all time, currently enshrined under his name: “The Riemann Hypothesis”.
Riemann and his hypothesis
“The great book of nature,” said Galileo, “Can be read only by those who know the language in which it is written. This language is mathematics.” There are few geniuses who understood, mastered and tamed this language better than Bernard Riemann. In his brief career of less than 15 years, he revolutionized number theory, transformed geometry and topology, and laid down the foundation for the n-dimensional mathematical space. This idea of n-dimensionality eventually helped launch a revolution in quantum field theory and statistical mechanics about a century later. Needless to say, Riemann, with his unprecedented innovations, was centuries ahead of his own time.
Born on September 17, 1826, Riemann was a humble man but a ferocious genius. He was known for his shy and introverted, sometimes annoyingly so, personality. Of course, this had something to do with his impoverished background. After all, he was born and raised in a poor family living in rural Prussia of the time. In any case, he had the tenacity of an untamed lion when it came to commanding the bizarre symbols and ruthless rigor of mathematics. The very fact that within his withdrawn personality there was a mind that inhabited the power to transform the whole of complex analysis is miraculous. And it all began with the Basel problem.
The Basel problem was first posed by Jacob Bernoulli more than a century before Riemann was born. The problem was to find the sum of the infinite series called the Basel series. Members from the Bernoulli family had already proven that this series, unlike the harmonic one, was convergent but the task of chasing that exact limit proved to be elusive. During the 18th century, this problem became so popular that anyone who solved it would find himself rise to the status of the greatest mathematician in Europe. As it turned out, it took the greatest mathematician of all time to bring this beast into the cage. The man was Leonard Euler and the year was 1735.
The Byawastha-Awastha Hypothesis
One hundred and twenty-five years after Euler’s breakthrough, Riemann contemplated the same Basel series through a completely new perspective. It is important to note, however, that there are some important differences between these two minds. No doubt, both were extraordinary mathematicians but while Euler was a prolific thinker Riemann was a revolutionary one. While Euler extended the horizons of different branches of mathematics, Riemann created new ones where none had existed before. While Euler solved one of the most difficult problems of his time, Riemann created the hardest one of all time. This is the reason why when Riemann looked at the Basel series, he discerned a pattern which even Euler had not. What Riemann did to this series led to the birth of analytic number theory and the pattern he noticed became one of the most elusive hypotheses in the history of mathematics: The Riemann Hypothesis.
An obsession
The Riemann Hypothesis is directly connected to the Prime Number Theorem (PNT) which was developed by Carl Friedrich Gauss, another titan of mathematics. Six years after publishing his paper ‘On the number of prime numbers less than a given quantity’, Riemann passed away, leaving the enigma he unleashed untamed.
Prime numbers are the most mysterious of all integers. They have eluded and tormented some of the best mathematicians of history. After Euclid showed that there are infinitely many prime numbers, the most obvious question that came to the fore was – ‘Is it possible to find a formula that would generate the total number of primes less than a given integer?’ Gauss tried, but only partially succeeded. Partial success, however, is no success in mathematics. Then came Riemann and did, in his short life, what nobody before had done to prime numbers.
The Holy Grail ahead appears simple to non-mathematicians. After all, it’s a question even a 12-year-old can ask. It is this gentleness of the problem that has attracted so much attention since it saw its first dawn. One of the most dynamic geniuses, who got obsessed by this problem, was David Hilbert. He proposed that the Riemann Hypothesis is the most important scientific challenge which needs to be solved to make a leap towards a deeper understanding of the prime numbers. Thanks to Hilbert, this hypothesis is also called Hilbert’s Eighth Problem. In the ensuing years of the twentieth century, Riemann’s unfinished legacy became a story of obsession which gripped most of the mathematicians of that age. One of the most successful was GH Hardy of England. He attacked the hypothesis and proved that there are infinitely many numbers which satisfy the requirement. Of course, he made extraordinary progress on the problem but that does not prove the hypothesis. Mathematics, as one knows, needs a complete answer. In fact, he had not proven that all numbers satisfy the Riemann Hypothesis. After all, infinity does not mean all.
As of 2024, the Riemann Hypothesis still remains as elusive as it was more than a century ago. Clay Mathematics Institute receives countless papers claiming to have proved the hypothesis but, so far, none has succeeded to pass the test of mathematical rigor. It still remains one of the most difficult problems of all time.
Conclusion
Mathematics is the heart of all sciences. In many ways, mathematics symbolizes the economic prosperity of any nation. During the 19th century, Germany was the most powerful country in the world because it was home to some of the greatest mathematicians of all time. At present China, India, America and Russia are producing great mathematicians and no surprise they are the most powerful countries of the 21st century. Therefore, Nepal too should focus on strengthening its mathematics so that we too can make ourselves leaders of the world in scientific innovations.
Riemann’s enigmatic hypothesis is yet to be tamed and one can hope that someone from Nepal might be able to look at it in an entirely new way and crack it.
