A real hard to crack

By Debkumar Mitra

A 159-year-old mathematical problem, whose solution has escaped the best minds, is attacked again. One of the world's best-known mathematicians, 90-year-old Michael Atiyah, said on Monday he found a simple proof of this diabolical riddle: Riemann's hypothesis.

In 1859, the German mathematician Georg Friedrich Bernhard Riemann put forward the hypothesis concerning the distribution of prime numbers (numbers which can only be divided by themselves and by 1. For example, 2, 3, 5, 7 , 11 …) among the positive integers (natural digits or "counting", for example, 1, 2, 3 …).

Mathematicians consider counting numbers as the building blocks of addition. Prime numbers are the building blocks of multiplication. Most theorems of number theory examine additive or multiplicative properties. But Riemann's hypothesis is an exception. Although we do not quite understand how, many mathematicians consider that the hypothesis is a fundamental link between addition and multiplication. But the elusive proof did not prevent mathematicians from using the hypothesis in thousands of documents. The latter has great implications for digital security.

Mathematicians usually do their best research before the age of 30. This weekend, Atiyah announced that he would share his simple demonstration on September 24 at the Heidelberg Laureate Forum, a conference for mathematicians and computer scientists in Heidelberg, Germany. skepticism. His recent attempts to solve some long-standing problems were considered insufficient. Yet nobody was willing to get rid of the winner of the Fields Medal, considered the "Nobel Prize" of the mathematician and the prestigious Abel Award.

The proof of Atiyah indeed seems elegant. He takes the help of a mathematical artefact known as the "Todd Function" to obtain a proof of Riemann's hypothesis using a method known as "contradiction" – where one starts with a hypothesis and then prove that it is wrong. The evidence must stand up to an independent review. And mathematicians are famous for finding holes in well-argued theorems. Bertrand Russell dealt a fatal blow to Friedrich Ludwig Gottlob Frege's work on the fundamentals of mathematics.

Now that the evidence is open, it is up to the experts to find inconsistencies. Until the first objections are raised, Atiyah's proof of Riemann's hypothesis is valid. It does not mean that any inconsistencies will be fatal to the evidence. A few decades ago, when Andrew Wiles claimed the proof of another difficult mathematical problem – Last Theorem (FLT) of Fermat – there were serious objections to the evidence. But Wiles could fix the damage and conquer the FLT. So, Atiyah will have a chance to review his arguments, in case an objection is raised.

Atiyah left a touch in the evidence. He expects that "a more general proof of Riemann's hypothesis is an undecided problem in Gödel's sense". He invokes here the famous "IncompletenessTheorem" of the Austrian mathematician Kurt Gödel, whose results can be summarized as follows: a) any coherent axiomatic mathematical system will contain theorems that can not be proved; (b) If all the theorems of an axiomatic system can be proved, then the system is inconsistent and therefore has theorems that can be proved both true and false.

Atiyah suggests that Riemann's hypothesis can not be proven.

Riemann's hypothesis is the "holy grail" of mathematics. This is one of the millennium problems of the Clay Mathematical Institute, with a $ 1 million prize to anyone who can prove it. Atiyah's affirmation can provoke a flood in a serene and solitary world of mathematicians. Over the past 110 years, generations of mathematicians have tried, with varying degrees of success, to come closer to a proof. But Riemann's hypothesis has hitherto remained open.

Now, Atiyah told the world not to try to prove it, because it can not be proven. The question that will probably arise is: what makes you conclude this?

From the evidence, it seems that he has developed a new mathematical framework to address a host of other issues. This framework, which Atiyah calls "arithmetic physics," has a "secondary product" in the form of the "proof" of Riemann's hypothesis. In a sense, Atiyah did not try to prove the hypothesis. It is the product of a "new theory" that he is trying to build.

Mathematics are full of stories of failure. But even these futile attempts have given rise to new techniques that have energized future research. Atiyah asks the mathematical community to rethink number theory. He says that Riemann's hypothesis is not provable. But it is too early to say if he is right. But mathematics is certainly richer with Atiyah's latest ideas.