Riemann Hypothesis is considered a holy grail of mathematics since its first publication. First devised in 1859 by Georg Friedrich Bernhard Riemann, this theory is yet to be solved until or unless the recent claims of solving it are true. Riemann Hypothesis was a fantastic piece of mathematical theory which was published in a famous paper Ueber die Anzahl der Primzahlen under Feiner gegebenen Grosse (“On prime numbers less than a given magnitude”).
This hypothesis revolves around an essential category of mathematical numbers, the prime numbers. Great mathematicians like Euclid, Euler, Gauss, Legendre, Hadamard, and de le Vallee Poussin, all have made huge contributions in this field. Prime numbers have no regular intervals. You can not predict the next prime number unless you study other figures along the way.
It was postulated that instead of looking forward, looking backward might lead us to the answer. That is exactly what Riemann attempted to achieve. This result was Riemann making one of the biggest steps in our understanding of prime number theory since antiquity. No one was able to match that insight for more than 160 years. The Clay Mathematics Institute explains, “[Riemann] observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function:-
?(s) = 1 + 1/2s + 1/3s + 1/4s + …
[This is] called the Riemann Zeta function. The Riemann hypothesis asserts that all interesting solutions of the equation:-
?(s) = 0
lie on a certain vertical straight line.”
In simplified terms, this relates to the distribution of prime numbers but doesn’t explain it. A more in-depth explanation of this goes out of the scope for now, but Jogen Veisdal, a Ph.D. fellow at the Norwegian University of Science and Technology, has given a very informative overview. His work formed the main focus of the prime number theory and was the main reason for the proof of prime number theorem in 1896. Even though it is complicated but what it tries to solve is very simple. Instead of trying to find where prime numbers were, Riemann attempted to find their nature. This is a sort of holy grail of mathematics. Marcus du Sautoy of Oxford University said, “Most mathematicians would trade their soul with Mephistopheles for a proof.”
Mathematicians are unable to determine the exact values, but they want to know how good their approximations are. This is the problem Reimann was trying to explain in his 1859 paper. If this hypothesis is correct, it will guarantee a higher bound on the difference between existing approximations and the real value. It will tell if prime numbers are as problematic as they look like to the students today. The hypothesis tackles hundreds of other concepts as well while its core is connected with the distribution of prime numbers. This may seem like a lot of trouble for nothing, but when you realize that huge organizations like NSA hires number theorists to research this field, you will conclude that there surely is something vital about it. Prime number theorem was purely theoretical, but it has also started to find real-world applications in our digital world. However, these rely on some properties of prime numbers to allow multiple signals to work on the same frequency band.
Prime number factorization is a commonly used practice in encryption techniques like public key encryption systems. They use large semi-primes to secure the encryption. To break it, you will need to find the prime factorization of the sizeable semi-prime number. This means that two or more prime numbers when multiplied together result into the original number. When small prime numbers are used, it is simple to crack the technique, but it gets hard when the numbers get large. The prime numbers have a non-linear distribution and the process followed to use the method is a trial and error process. Some problems remain beyond our abilities to solve. In the field of mathematics, these are called the Millenium Prize Problems.
They consisted of seven problems which were identified by the Clay Mathematics Institute at the turn of the new millennium. Following are the famous millennium problems:
Yang-Mills and Mass Gap
P Versus NP Problem
Birch and Swinnerton-Dyer Conjecture
The prize to solve these problems is $1 million cash each. However, the real award is the ever-lasting fame and respect from your mathematician peers. Till now, only one of the original seven has been solved, which is Poincare Conjecture. A Russian mathematician Grigori Perelman answered this in 2003. He made a career out of solving math problems and made significant contributions to Riemannian geometry and geometric topology. When it was made official that he met the criteria for the Clay Millenium Prize in 2010, he rejected the prize money saying that his contributions were not more than Richard S. Hamilton. The Riemann hypothesis might become the next one to get solved if the recent news turns out to be correct. It looks like a 90-year-old retired mathematician might have a solution which has been hidden from his peers for 160 years. His claims will be verified at the Clay Mathematical Institute first, but this implies that Riemann hypothesis has also been solved.