A mathematician just solved the 160-year-old problem worth $ 1 million

The famous mathematician Michael Atiyah claims that he developed a proof of the Riemann hypothesis, a 160-year-old math problem worth $ 1 million.

This is not the first time this mathematician claims to have solved a major problem in mathematics. However, he never publicly revealed the evidence for his results.

Michael Atiyah, a mathematician who won several senior awards in mathematics, gave a lecture at the Heidelberg Laureate Forum in Germany on Monday to explain his evidence of the Riemann hypothesis . This theory was first coined by Bernhard Riemann in 1859. Michael Atiyah said that the numbers returned a value of zero when used as an argument to a given function - but he declined the supply. Provide evidence for your results.

Mathematician Michael Atiyah.

As Atiyah pointed out in his talk, Riemann's hypothesis was "digitally verified for millions of people and millions of computers that you can think of, but there is no concrete evidence." However, this hypothesis has great practical value for mathematicians because it explains the strange distribution of primes in other mathematical calculations.

If Atiyah's proof is correct, it will be a big shock for the math community because during the past 160 years, the evidence for the Riemann hypothesis has become one of the most puzzling problems in mathematics. Since 2000, the Clay Mathematics Institute has offered a $ 1 million prize to mathematicians who can publish their results on this issue in a prestigious journal and wait two years for mathematicians. Other may object. Although Atiyah proved his proof on Monday, it was not accepted for publication.

Atiyah said in a lecture in Heidelberg: 'The Riemann hypothesis has been proven, unless you are the type who does not believe in the evidence of contradiction. In that case, I have to go back and think again. But people often accept evidence of contradiction, so I think I deserve to receive the prize. '

Many mathematicians expressed doubt about the reasonableness of the thesis and argued that Atiyah made such statements to prevent his thesis from collapsing when supervised or unpublished.

If Atiyah's evidence is not accepted, this is not the first time a mathematician has claimed to have unlocked Riemann's hypothesis and failed. In 2015, a Nigerian professor named Opeyemi Enoch also claimed to have provided evidence for the Riemann hypothesis but the whole evidence turned out to be a fake. Unlike Enoch, Atiyah has won both the Fields Medal and the Abel Prize, which can be likened to Nobels for mathematicians.

According to Markus Pössel, a German astronomer at Atiyah's lecture, it is too early to make a judgment as to whether Atiyah's evidence is correct.

Pössel said: "Those who are experts in this field do not have enough information to evaluate that claim. Specifically, Atiyah used a rare function he called 'Todd's function' after a I don't know if that function exists in Atiyah's statement, but it is certainly reasonable to be cautious in this. '

What is the Riemann hypothesis?

In 1859, mathematician Bernhard Riemann proposed a hypothesis about when a particular function returns a zero. Hypothesis has some practical applications in mathematics, such as an explanation for the strange distribution of primes only divisible by itself and one.

Riemann's hypothesis is about the values ​​used in the zeta function, creating a series of convergent or divergent numbers depending on the value of s - called the argument of the function - in the following string:

Zeta Riemann function.

Riemann's insight is that the zeta function can also be extended to complex numbers, which is a combination of virtual and real numbers. (Quick explanation: A complex number is a number with the form a + bi, where a and b are real numbers, i is a virtual unit, with i equal to the square root of -1. Example 3 + 5i is a number complex.)

According to Edward Frenkel's explanation in a video on the Numberphile, if you put a real number into the zeta function, such as '2 ' you will get the string '1+ 1/4 + 1/9 + 1/16 + .' . The more numbers added to this sequence the closer the string to a given total is called the limit. If the chain approaches the limit, it is considered a convergent sequence.

On the other hand, if a number like '-1' is used as an argument for the zeta function, it returns a string '1 + 2 + 3 + 4 + 5 + .'. This type of sequence is not limited because the sum of the numbers continues to be larger and is known as a divergent series.

Riemann argues that if a complex number is used as an argument to the zeta function, this leads to a convergent sequence. When certain numbers, such as real numbers, are used as input to a zeta function whose argument is a complex number, it returns a value of zero.

Some input examples are quite easy to explore. For example -2, -4 and -6 will return zero. But what Riemann hypothesized is that if half is used as a real number for the complex argument of the zeta function, then any virtual number that it joins will also return zero. Therefore 1/2 + 1i, 1/2 + 2i, 1/2 + 3i, etc. all will return.

'Which value makes zeta function equal to 0?' Frenkel said in the video Numberphile: "That's a million dollar question."

The evidence that Atiyah claims to answer this question is based on something he calls 'Todd's function' , named after mathematician, Atiyah's former teacher JA Todd. As Pössel pointed out, the novelty of this function is the source of many mathematicians' skepticism about Atiyah's evidence.

If Atiyah hopes to receive a $ 1 million prize for solving the millennium problem, according to the Clay Institute of Mathematics is one of the seven hardest problems, the 'Todd' function will be subject to close scrutiny. of other mathematicians in the next two years.

So far, only one of the seven millennium issues has been resolved, despite dozens of solutions to various issues that have been proposed. This speaks to the difficulty of problems at hand and the importance of peer assessment in mathematics. Even if Atiyah's evidence eventually missed the mark, his solution would certainly be at the forefront of some of the world's best mathematicians in the next few years. pointed out, the novelty of this function is the source of many mathematicians' skepticism about Atiyah's evidence.

If Atiyah hopes to receive a $ 1 million prize to solve this millennium problem, the name given to the seven most difficult math problems according to the Clay Mathematics Institute, the function Todd will have to bear close supervision of other mathematicians. two years.

So far, only one of the seven millennium problems has been solved, although dozens of solutions to various problems have been proposed. This speaks to the difficulty of these issues and the importance of peer assessment in mathematics. Even if Atiyah's evidence eventually lost points, his solution will surely be at the forefront of the world's best mathematicians in the next few years.