Riemann hypothesis: Has one of the greatest mysteries of Math finally been solved?

In recent days, the world of mathematics has blurred the rumors that Sir Michael Atiyah, the famous Fields medalist and winner of the Abel Prize, would have solved the Riemann hypothesis.

If its proof turns out accurate, it would be one of the most important mathematical achievements for many years. In fact, this would be one of the most important results in mathematics, comparable to the proof of Fermat's last theorem of 1994 and the proof of Poincaré's conjecture of 2002.

Riemann's hypothesis is not just one of the big unresolved problems in mathematics, and hence a source of glory for the one who solves it, but one of the "million dollar problems" of Institute of Mathematics of the Earth (Clay Mathematics Institute). .

Riemann's hypothesis concerns the distribution of prime numbers, these integers can only be divided by themselves and one such as 3, 5, 7, 11, and so on. We know Greeks that there is an infinity of prime numbers. What we do not know is how they are distributed in integers.

The problem originates from the estimation of the function called "prime pi", an equation allowing to find the number of primes less than a given number. But its modern reformulation, by the German mathematician Bernhard Riemann in 1858, is related to the zeros localization of the zeta function of Riemann.

Visualization of the zeta function of Riemann. Jan Homann / Wikimedia, CC BY

The technical statement of the Riemann hypothesis is "the zeros of the Riemann zeta function that are in the critical band must be on the critical line". Even the understanding of this statement involves higher level mathematics courses in complex analysis.

Most mathematicians think that Riemann's hypothesis is very true. Up to now, the calculations have not revealed erroneous zeros that are not in the critical line. However, there is an infinity of zeros to check, and a computer calculation will not check much. Only an abstract proof will do the trick.

If, in fact, Riemann's hypothesis was not true, the current thought of mathematicians about the distribution of prime numbers would be very wrong, and we had to seriously rethink the prime numbers.

Riemann's hypothesis has been examined for over a century and a half by some of the greatest names in mathematics and is not the kind of problem that an inexperienced math student can play in his spare time. Verification attempts involve many very deep tools from complex analyzes and are generally very serious tools made by some of the best names in mathematics.

Atiyah gave a lecture in Germany on September 25, in which he presented the outline of his approach to verify Riemann's hypothesis. This schema is often the first announcement of the solution but do not believe that the problem has been solved – far from it. For mathematicians like me, the "proof is in the pudding", and many steps must be taken before the community pronounces the solution of Atiyah as correct. He will first have to circulate a manuscript detailing his solution. Then there is the tedious task of checking his evidence. It could take a long time, maybe months, even years.

Is Riemann's hypothesis Atiyah's attempt serious? Perhaps His reputation is remarkable and he is certainly able to get out of it. In addition, several other serious attempts to solve this problem have not been successful. At some point, Atiyah will have to circulate a manuscript that experts can check with a fine comb.

By William Ross, Professor of Mathematics at the University of Richmond. This article is republished from The Conversation under a Creative Commons license. Read the original article.