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「リーマン予想」がついに証明された？89歳の数学者アティヤさんがたったの5ページで証明
クッパ姫騒ぎの最中に、理系界に走った衝撃をまとめました。
･160年間未解決で、証明すると1億円貰える問題である、リーマン予想の証明が発表された
･証明は89歳の数学者アティヤさん
･論文がたったの5ページで完結
･他の研究の｢おまけ｣として証明された
リーマン予想とは
数学において、リーマン予想（リーマンよそう、英: Riemann hypothesis, 独: Riemannsche Vermutung）は、リーマンゼータ関数の零点が、負の偶数と、実部が 1/2 の複素数に限られるという予想である。ドイツの数学者 Bernhard Riemann (1859) により提唱されたため、その名前が付いている。名前は密接に関連した類似物に対しても使われる。例えば有限体上の曲線のリーマン予想。リーマン予想は、英語表記 Riemann hypothesis の直訳であるリーマン仮説と表記したり、RH と略すこともある。
ただし彼は以前、別件の証明で大きなミスを犯した経験があり、
今回もそのような穴が発見される可能性は十分にあります。
しかし、彼が偉大な数学者であることに変わりはありません。
Incidentally, here is the proof. You're welcome. #HLF18 https://t.co/cXe2mPyFfB
On volumes of published work, Euler definitely beats Riemann, 73:1. #HLF18 https://t.co/ajuBHy81m3
"Solve the Riemann hypothesis, and you become famous; if you're already famous, you become infamous." #HLF18 https://t.co/cii8Mvz4LR
Today's #HLF18 film tip is "The man who knew infinity," which Atiyah says is enjoyable even if one is not partial to prime numbers. https://t.co/MkKaIkaVQt
Those of us not still thinking about why Hardy was holding an umbrella in sunshine are now invited to think about the different collar styles of Hadamard and Selberg, and also about immortality. #HLF18 https://t.co/pZmYFGCjHT
The most succinct beautiful formula in mathematics. The mathematical equivalent of "To be or not to be", short and deep. #HLF18 https://t.co/Q7l0SNdG16
Afraid that at this pace, though enjoyable, we might indeed be heading for "I have found a beautiful proof, but it is too long to fit into this 45 minute talk". #HLF18
The plural of genius might be genii, but it sounds confusingly like Djinn. Also, there is no conservation law for geniusses. #HLF18
Also, the button for the laser pointer was unfortunately close to the button for jumping back to the first slide. We're back on track now, though. #HLF18
When you do something, you want to do things beautifully. Sometimes, this involves choosing a kyrillic letter such as "zh". #HLF18
Now we are getting to the meat of the matter, the Todd function. A "very very clever function" that maps Euler's equation to its quaternionic generalization, and is defined by an infinite iteration of exponentials. #HLF18 https://t.co/4xm9GTOrXA
Infinite iterations are dangerous territory, but von Neumann tells Atiyah it's OK, and Atiyah is taking von Neumann's word for it. So that's apparently alright then. "Magnificent animal", the Todd function. #HLF18
Atiyah didn't set out to prove the Riemann Hypothesis. He was trying to derive the fine structure constant (yes, that thing from physics). The RH was just a bonus. #HLF18 https://t.co/vaPpOgot19
After Atiyah had found the fine structure constant value, he put his feet up, opened the champagne, and realized he had accidentally proved the Riemann Hypothesis. #HLF18 https://t.co/dIqBps5VV5
The audience is getting some homework. #HLF18 https://t.co/SXz98IMXwK
First question from audience: will Atiyah claim the Millenium problem prize? Atiyah says: yes, he deserves the prize. #HLF https://t.co/vNsGibie0H
Uh-oh. Question from audience about when this will be published. Atiyah claims ageism, says his papers will be rejected before he is too old. This is problematic. #HLF18
Also, Atiyah claims that people are unfairly dismissive of new ideas - in this case, the weird infinite series of series thingy that defines the Todd function. #HLF18
So, let's not get carried away. Next, we need a write-up of the proof. Say, something like this: t.co/il50fmY9MD (thanks to Thilo Kuessner for the pointer). #HLF18
Then, a peer-reviewed article. Peer review in pure mathematics tends to be rigorous - after all, you cannot ask a peer reviewer in physics to repeat the experiment described, but you *can* ask a mathematics reviewer to check each step of a proof. #HLF18
And needless to say, in this case, a reviewer had better be very, very thorough. That pesky Todd function really has it coming, for sure. Von Neumann or not. #HLF18
Also, the Clay Institute has very appropriately set the bar even higher. Not only does your proof need to be published in a peer reviewed top journal. They then wait 2 years. #HLF18
みんなの反応
日本語の解説
ついにリーマン予想が証明された！？
t.co/1BkM4f6OL4
１） 証明の論文はたった５ページであること。
２) 証明は微細構造定数を導出するという物理学上の研究から、副次的に得られたこと。
リーマン予想は経過でおまけで証明できて、微細構造定数を導出したということだけど、こういうのは２年くらいのうちに反論が出てこないかどうかという公開査読みたいなもんなんで、まだわからん。本当だったら数学が成り立つ世界では常に物理常数は同じということになる
なんかアティヤ先生が論文中で「リーマン予想の証明にはZFは不十分で選択公理を必要とする」とか書いてるみたいなことが目に入ったんですけど、そこだけは確実な誤りですよね？ リーマン予想はΠ_1文なので、Shoenfield絶対性から、もしZFCでRHが証明可能なら必ず選択公理なしでもRHは証明できるので。
数学界最大の難問と名高い『リーマン予想』が証明された(かも)と話題ですが、ポイントをまとめておきます
①物理の「微細構造定数」を導出する過程でリーマン予想が解けちゃった
②その証明がたったの5ページで書けちゃった
4時間ほど前（09/24 09:45 CEST)、 Heidelberg Laureate Forum という会議において、「150年間未解決だったリーマン予想をついに証明した」という爆弾発表が行なわれたらしい。本物だろうか。 t.co/BhSgkow4Cf
反論も
TL上で話題のリーマン予想について、Todd 関数は
T(γ)/γ = T(π)/π
かつ
T(π)=1/α=137.035999...
を満たすとされているのに、実際に数値計算してみると
T(π)=1/α= 0.160259...
となってしまうので、Atiyah卿の主張はおかしい、という声が上がっている。
t.co/f2nP7cxtm3
Here is Atiyah's lecture: on the Riemann Hypothesis
t.co/i2ujyEzfsF
Here, apparently, is his paper:
t.co/BFXu2TYhli
It refers extensively to this much longer paper, where he attempts to compute the fine structure constant:
t.co/bB0YdyibVA
@johncarlosbaez Erm, combining Atiyah's Eqs. 1.1 and 7.1 in that PDF gives a number that is very, very different from the fine structure constant... or its reciprocal... https://t.co/I1tUHWAI8Z
Tomorrow Atiyah will talk about his claimed proof of the Riemann Hypothesis.
It's all about "the music of the primes". Here the function that counts primes < n is being approximated by waves whose frequencies come from zeroes of the Riemann zeta function.
(continued) https://t.co/lzNDyXo3Im
The Riemann zeta function is given by this simple formula when the complex number s has Re(s) > 1. Then the sum converges! But we can "analytically continue" the Riemann zeta function to define it for other values of s, and that's where the fun starts.
(continued) https://t.co/E6fUwdfNJF
The Riemann zeta function is zero for some numbers with 0 < Re(s) < 1. These are called the "nontrivial zeros" of his zeta function. Riemann computed a few and hypothesized they all have Re(s) = 1/2.
t.co/6aA5tSGq4m
Riemann found a formula for the number of primes < n as a sum over the nontrivial zeros of the zeta function. My first tweet shows the sum over the first k nontrivial zeros.
So, if the Riemann Hypothesis is true, we'll get a better understanding of primes!
(continued) https://t.co/CbTvsCV1VE
I bet that Atiyah's claimed proof, if and when he writes it up, will not convince experts.
In 2017 he claimed to have a 12-page proof of the Feit-Thompson theorem, which usually takes 255 pages:
t.co/5N2WDiJ00U
He showed it to experts, and... silence.
(continued)
In 2016 Atiyah put a paper on the arXiv claiming to have solved a famous problem in differential geometry. The argument was full of big holes:
t.co/STpqByhBMG
So, I'm not holding my breath this time.
But of course I'd be happy to be wrong.
(the end)
おまけ
【数学のトリビア】数学の世界には解くと賞金1億円がもらえる問題がある
出典元：YouTube