Heart of the Machine reporting, Heart of the Machine editorial Department.

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On September 20, a Twitter screenshot set the maths world on fire: Sir Michael Atiyah, the Fields and Abel Prize winner, was about to prove the jewel in mathematics’s crown: the Riemann conjecture. Sir Attiyah presented the proof of a 150-year-old mathematical conjecture to the world for 45 minutes today at the Heidelberg Laureate Forum 2018 in Germany, but the first 30 minutes were history, The proof is a one-page POWERPOINT… In fact, shortly before the conference began, a pre-print version of Sir Attiyah’s paper was reported to be available online, but Machine Heart was only able to identify a Link to Google Docs, originally from a discussion on Reddit.

Sir Michael Attiyah speaks about his proof at the Heidelberg Winners’ Forum in Germany today. Attiya is shown announcing that “I will present a simple proof, in a completely new way,” based on the related work of von Neumann (1936), Hitzerbruch (1954), and Dirac (1928).

Riemann hypothesis

In 1900, at the second International Congress of Mathematicians held in Paris, France, The German mathematician Hilbert made a speech called “Mathematical Problems”, listing 23 mathematical problems that he thought were the most important. One hundred years later, in 2000, the Clay Mathematics Institute held a mathematical conference in Paris, where mathematicians discussed and listed the seven most important mathematical problems, with a $1 million prize for each problem. Yes, this is the famous “millennium question”.

Of the 23 Hilbert puzzles and the seven Millennium problems, only one appeared at the same time, and that was the Riemann conjecture.

Lu Changhai, a well-known popular science writer in China, commented on the Riemann Conjecture as follows:

Riemann’s century-and-a-half record is far less impressive than fermat’s, which was solved after more than three and a half centuries, or Golbach’s, which has survived for more than two and a half centuries, but it is far more mathematically important than either of the more popular conjecture.

So, what is the Riemann conjecture?

The Riemann conjecture was developed by the mathematician Friedrich Riemann in 1859 in a short paper (only eight pages) submitted to the Berlin Academy of Sciences on the distribution of prime numbers. The distribution of prime numbers plays an important role in number theory, just as the concept of atoms plays a role in modern physics. Riemann found that the distribution of prime numbers was hidden in the zero-point distribution of a function. The function is the Riemann zeta-function:

The Riemann zeta-function

Riemann extended this function analytically to the whole complex plane, and pointed out that the non-trivial zeros of the Riemann zeta-function (i.e., the values of s not -2, -4, -6, etc., which are trivial zeros) are all 1/2 in real numbers. That is, these nontrivial zeros are all distributed on the line Re(z)=1/2 of the complex plane (the dotted line in the figure below).

In 1901, Helge von Koch pointed out that Riemann conjecture and strong condition prime number theoremEquivalent, where π(x) is the number of primes not greater than x. It has been proved that the theorem is true for all the original 1,500,000,000 prime numbers. However, it has not been proved whether all the solutions are valid for the theorem.

The Riemann conjecture is closely related to the distribution of prime numbers in number theory. Number theory is an important branch of mathematics. In number theory, the distribution of prime numbers is an important research topic. In other words, the confirmation of the Riemann conjecture also has a very important impact on number theory. In addition, it has been found that the Riemann conjecture is related to some physical phenomena, which has increased the influence of the Riemann conjecture in the field of physics.

The establishment of Riemann conjecture and its generalized form is the premise of many existing mathematical propositions. If the Riemann conjecture and its generalized form are proved, these mathematical propositions will become mathematical theorems. Conversely, if the Riemann conjecture is falsified, more than 1,000 mathematical propositions will become the “funerary objects” of the Riemann conjecture. To falsify the Riemann conjecture, one only needs to find a nontrivial zero that is not on the line Re(z)=1/2. Of course, no such zero has been found.

If you were to return to earth 500 years later, the first thing you’d have to ask is whether the Riemann conjecture has been proved or disproved. — Hilbert


Sir Michael Attiyah’s certificate

Michael Atiyah (April 22, 1929.22 -) is a famous British mathematician and one of the greatest mathematicians of our time. His main area of research was geometry, and in the 1960s he worked with Isado Singer to prove the Attia-Singer index theorem for the two fields of communication mathematical analysis and topology. Sir Attiyah was awarded the Fields Medal in 1966 and the Abel Prize in 2004, jointly with Singh.

Sir Attiyah, who challenged the Riemann conjecture this year, is 89.

Due to his age and status, some questioned Sir Attiyah’s claims after the Twitter screenshots surfaced. “Sir Attiyah has not been in the best of times, his previous propositions have not been proven, and he is 89 years old, an age few mathematicians would have achieved. American mathematical physicist John Carlos Baez (@JohnCarlosBaez) tweeted: Attiyah’s recent discoveries, such as his claim to show that there is no complex structure of a six-dimensional sphere, don’t hold water, and everyone who knows him well is embarrassed to discuss the reasons publicly.”

Of course, he has his supporters who think, “I think if anyone can get this done, it’s Attiya.” (Twitter from Matt Hunt, British applied mathematician)

Faced with skepticism about his ambitions in his later years, He once said: “I have won all the prizes I need. What can I lose? That is why I am taking risks that a young scholar would not dare.”

Whether skeptical or supportive, what Sir Attiyah called “a simple demonstration, a new approach” was finally unveiled today at the Heidelberg Winners’ Forum:

Due to the overload of the number of viewers, the live broadcast crashed, so the official live broadcast was changed to mobile phone. After the message reminder from the heart of the machine, the official live broadcast screen was finally focused on the PPT, which was touched.

First, Michael Atiyah introduces the history of prime number research and the relationship between prime numbers and the Riemann conjecture.

He also joked, “Solve the Riemann conjecture that you will become famous, but if you are already famous, there is a risk of notoriety.”

Atiyah spent a lot of time introducing Euler’s formula, not because of the beauty of connecting imaginary numbers and other elements, but also because linking the key ideas of Von Neumann and Hirzebruch could lead to a more general Euler expression, which was important to look at and prove the Riemann conjecture in a new way. Atiyah said: ‘Euler’s formula is the mathematical equivalent of Shakespeare’s’ to be or not to be ‘.

Why is the Riemann conjecture so interesting but so hard to prove? According to Michael Atiyah, there are three main aspects. First, prime numbers show local irregularity, but gradually show some rules. Secondly, it is very difficult to know the number of primes within N; And finally, many of these difficulties and puzzles can be explained by the Riemann conjecture, so even though it hasn’t been proven yet, there’s actually a lot of reasoning based on it.

There was a lot of speculation that Michael Atiyah would use quantum mechanics to prove the Riemann conjecture, but Atiyah stated in his talk that the proof of the Riemann conjecture was the Todd function:

Atiyah describes the relationship between TODD’s function and the Riemann conjecture, which was previously impossible to prove but which is now possible with new tools. The most important property of TODD’s function is that it develops an interpretation of the fine structure constant α.

And then, the highlight, 30 minutes of Riemann conjecture history, finally the moment of proof. All the proof, Sir Attiyah says, is on the next slide.

So what can we do after we prove the Riemann conjecture? Michael Atiyah says RH can be generalized to a variety of conditions, and is being demonstrated step by step. At the same time, we need to achieve numerical results for prime numbers, whose proof is very important for young researchers in mathematics, computer science, logic and physics, but the expectation of infinite expansion of RH is uncertain.

Sir Attiyah concludes with a summary of the tasks expected in the future: use the most powerful tools available; Test all well-known conjectures (proven and unproven); Determine which computations are valid (on the required time scale); Make sure we have time to finish.

That’s all of Sir Attiyah’s HLF18 talk, and the full video will be available later. Twitter was abuzz after the talk, with mixed views on whether the one-page proof could unravel the Riemann conjecture.

In fact, before the conference began, the Internet had circulated a claim that Sir Attiyah had released a pre-print paper. Machine Heart could only trace the paper to a Reddit discussion a few hours earlier, but one Reddit discussion user said that the pre-print appeared to come from an email list, But the original sender was not Atiyah, but a person who said he had received an email from Atiyah.

After the live broadcast, there was still no confirmation of the exact source of the paper.

Machine Heart has attached its pre-printed paper below for readers to study for themselves. This preprint paper is very short, the heart of the Machine only briefly introduces the introduction part, more content and proof need to see the original document.

In the ICM Abel lecture in Rio de Janeiro 2018 [1], I explained how to solve long-term mathematical problems that arise from physics. The key to these problems lies in understanding the fine structure constant alpha.


Details of the whole lecture are given in [2], and have been submitted to agenda A of the Royal Society. The solution technique developed in [2] is a novel fusion of the key ideas of Von Neumann and Hirzebruch. This exponential infinite iteration is a very complex and powerful technique with inherent simplicity.


The power and generality of these methods suggests that they should also solve other problems, or at least shed new light on those that are hard to solve. In the Abel lecture I gave on expanding ICM agenda, I speculated that the technology in [2] could lead to a new subject in Arithmetic Physics.


The Riemann conjecture (RH) asserts that ζ(s) has no zeros in the critical zone 0 < Re(s) < 1 and from the critical boundary Re(s) = 1/2. It is one of the most famous unsolved problems in mathematics and a formidable challenge envisaged in [1]. I believe this new tool will live up to this challenge and this paper will provide proof.


The proof relies on a new function T(s), the Todd function, named by Hirzebruch after my teacher j.A. Todd. Its definition and properties are given in reference [2]. But in Chapter 2, I’ll review and explain. In Chapter 3, I will prove the Riemann conjecture using T(s) functions. In Chapter 4, the Deus ex Machina section, I will attempt to explain the proof of the Riemann conjecture. Finally, in Chapter 5, AS expected, I will put this paper into Arithmetic Physics.

Here is the original document:



The plate paper a total of 5 pages, readers can click on the download: pan.baidu.com/s/1mT4oy1VN… .

Finally, if there are readers who have studied this proof, let us know in the comments (hahaha). Two days ago, 2018 Fields Medal winner Peter Scholze and mathematician Jakob Stix announced that Professor Mochizuki’s paper on the ABC conjecture did not prove the ABC conjecture…