THE RIEMANN HYPOTHESIS HAS BEEN SOLVED! OR HAS IT?
By Suvir Rathore
Recently, a renowned mathematician, Michael Atiyah, claimed to have solved the Riemann hypothesis, a 130-year old problem. A million dollars could be his as this problem is one of the seven millennium problems, of which only one has been solved (the Poincare Conjecture). If the proof is true, it could prove a lot of other theorems that are based around the single hypothesis, the most profound of which is the prime number distribution theorem, enabling mathematicians to find the distribution of all prime numbers. This has several consequences, for it could change banking altogether, since their security is related to the product of two very large primes. However. due to some of his recent proofs not being correct, there is much scepticism on this one.
So, what is the Riemann hypothesis? It was first proposed by Bernhard Riemann, who developed the Riemann-Zeta function. This is an infinite series shown as a summation function. Its range lies for values of s that are greater than 1, however using analytical continuation this can be extended for all values of s within the complex plane. Substituting s=-1 the series becomes the sum of all-natural numbers (1+2+3…) which gives a surprising answer of -1/12, however the process of analytical continuation means that this cannot be regarded as a suitable value, for the series diverge at these values rather than converging to an actual answer. For certain values of s, the series equals 0, this is apparent for every negative even number, and thus these zeroes are known as trivial. Riemann hypothesised that there were an infinite number of non-trivial zero’s in the strip of 0 < s < 1, with all the zeros being on the line y = ½, known as the critical line. This is known as his famous hypothesis, and so far, no one has been able to prove or disprove his hypothesis.
Computers have been running for a long time in search for a contradiction, yet none have been found with millions of values being tested. Although most mathematicians generally accept his hypothesis to be true, such assumptions cannot be made for the practicality of its uses, as it is used in many aspects of our lives. Leonhard Euler proved that this function can be written as an infinite product of primes – all the primes that exist, and thus this is very powerful for prime numbers are the building blocks of all number – like the elements of a periodic table, and thus it can prove many other unproven mathematical theorems. String theorists use the idea of the sum of the natural numbers with their answer being derived from this function, which in the future could shape our civilisation completely, especially our future of space travel.
In 1904, Hardy proved that there are an infinite number of zeroes on the critical line, as well as its equivalence to the Dirichlet eta function. Its congruence to the prime number theorem, and the fact that at least 40% of the roots lie on the critical line with symmetry means that it is of great importance that it is proven, which is why there is no reward for disproving it.
Atiyah won the field’s medal and the Abel prize, the two greatest awards available in mathematics, as well as having contributed massively to topology and theoretical physics, which was the inspiration for his prove, so there is a chance that it may turn out to be correct, although at the moment critics have claimed that his proof is not very coherent and makes wild claims. Many say that we need new mathematics to prove the hypothesis, as there is still so much that needs to be understood about the nature of primes and complex numbers in order to truly understand the problem.