Renowned mathematician Michael Atiyah claims that he has developed a proof for the Riemann conjecture, a 160-year-old math problem worth million.
This is not the first time the mathematician has claimed to have solved a major problem in mathematics. However, he never made public the evidence for his results.
Michael Atiyah, a mathematician who has won several senior prizes in the field of mathematics, gave a presentation at the Heidelberg Laureate Forum in Germany on Monday to explain his proofs of the Riemann conjecture . This conjecture was first posited by Bernhard Riemann in 1859. Michael Atiyah suggested that numbers return a value of zero when used as an argument to a given function – but he refused to provide provide evidence for their results.
Mathematician Michael Atiyah.
As Atiyah pointed out in his talk, Riemann’s hypothesis “has been numerically verified for millions of people and millions of computers that you can think of, but without concrete proof”. However, this hypothesis has great practical value to mathematicians because it explains the strange distribution of primes in other mathematical calculations.
If Atiyah’s proof is correct, it will be a huge shock to the mathematical community because over the past 160 years, the proof for the Riemann conjecture has become one of the most puzzling problems in mathematics. Since 2000, the Clay Mathematical Institute has offered a million prize to the mathematician who can publish his results on this problem in a reputable journal and wait two years for the mathematicians to Others may object. Although Atiyah demonstrated his proof on Monday, it has yet to be accepted for publication.
Atiyah said in a lecture in Heidelberg: “The Riemann hypothesis has been proven, unless you are the type of person who does not believe in 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 the award.”
Many mathematicians have expressed doubts about the validity of the argument, arguing that Atiyah made such statements to prevent his thesis from collapsing under scrutiny or unpublished.
If Atiyah’s proof is not accepted, it wouldn’t be the first time a mathematician has claimed to have cracked Riemann’s conjecture 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 forgery. 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 who was present at Atiyah’s lecture, it is too early to make a judgment on whether Atiyah’s evidence is correct.
“Those who are experts in the field still don’t have enough information to judge that claim to be true,” says Pössel. In particular, Atiyah used a rare function he calls the ‘Todd function’ after a while. among my teachers. I don’t know if such a function exists in Atiyah’s claim. But it is certainly reasonable to be cautious in this.”
In 1859, the mathematician Bernhard Riemann made a hypothesis about the moment when a particular function returns a value of zero. The conjecture has a number of practical applications in mathematics, for example as an explanation for the odd distribution of primes that are only divisible by itself and one.
Riemann’s conjecture is about the values used in the zeta function, which produces a series of numbers that converge or diverge depending on the value of s — called the function’s argument — in the following series:
Riemann zeta function.
Riemann’s insight was that the zeta function could also be extended to complex numbers, which are combinations of imaginary and real numbers. (Quick explanation: A complex number is a number of the form a + bi, where a and b are real numbers, i is the imaginary unit, where i is the square root of -1. For example 3 + 5i is a number complicated.)
As explained by Edward Frenkel in a video on Numberphile, if you feed a real number into the zeta function, such as “2 ” you get the string “1+ 1/4 + 1/9 + 1/16 + …” . The more numbers that are added to this sequence the closer the sequence is to a certain total called the limit. If the series approaches the limit, then it is said to be a convergent series.
On the other hand, if a number like “-1” is used as an argument to the zeta function, it returns a string “1 + 2 + 3 + 4 + 5 + …”. This type of series has no limit as the sum of the numbers keeps getting bigger 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 series. When specified 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 of the input examples are pretty easy to explore. For example -2, -4 and -6 will return zero. But what Riemann hypothesized is that if 1/2 is used as a real number for the complex argument of the zeta function, any imaginary number it concatenates will return zero. Hence 1/2 + 1i, 1/2 + 2i, 1/2 + 3i, etc. will all return zero.
“What value makes the zeta function zero?” “It’s the million dollar question,” Frenkel says in the Numberphile video.
The proof that Atiyah claims to answer this question is based on something he calls the “Todd function” , named after Atiyah’s former teacher and mathematician, JA Todd. As Pössel has pointed out, the novelty of this function is the source of many mathematicians’ skepticism about Atiyah’s proof.
If Atiyah hopes to receive a million prize for solving the millennium problem, which the Clay Mathematics Institute says is one of the seven hardest problems, then the “Tod’s function” will come under scrutiny. closely by other mathematicians over the next two years.
To date, only one in seven millennium problems has been solved, although dozens of solutions to different problems have been proposed. This speaks to the difficulty of the problems at hand and the importance of peer review in mathematics. Even if Atiyah’s proof eventually misses the mark, his solution is sure to be at the forefront of some of the world’s best mathematicians over the next few years. As pointed out, the novelty of this function is the source of many mathematicians’ skepticism about Atiyah’s proof.
If Atiyah hopes to receive a million prize to solve this millennium problem, the name given to the seven most difficult math problems according to the Clay Mathematics Institute, the Todd function will suffer. the close supervision of other mathematicians. two years.
To date, only one in seven millennium problems has been solved, although dozens of solutions to different problems have been proposed. This speaks to the difficulty of these problems and the importance of peer review in mathematics. Even if Atiyah’s proof eventually loses ground, his solution will surely be at the forefront of the world’s best mathematicians in the next few years.