The Search for Siegel Zeros - Numberphile
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- Опубликовано: 26 сен 2024
- Featuring Professor Tony Padilla. See brilliant.org/... for Brilliant and get 20% off their premium service (episode sponsor)
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Yitang Zhang strikes again... Discrete mean estimates and the Landau-Siegel zero: arxiv.org/abs/...
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6:55 😅😅😅😊😊😊😊😊 7:01 7:02
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The nature of humanity is just that every so often someone accidentally invents the Riemann Hypothesis again.
😂😂
🦀
yep lol
It's sort of like how π keeps showing up even when you don't see a circle anywhere near.
@@namelastname4077 You can spend all your time contemplating the miseries of life and inevitablility of death if you want - personally I prefer to spend mine getting excited about fun cool things
I love Tony's tongue-in-cheek statement "without any controversy at all, it is equal to -1/12" 🤣
Once again making people think its normal summation, but its not
Let's be clear about this. 1+2+3+... does not equal -1/12. The series is the result of a function definition that doesn't work at -1. However, it's true that there is another more complicated function definition that gives the same values where the first definition works, and also works at -1. It's that other function that has the value -1/12 at -1.
A theoretical physicist tries to calculate something and gets the result 1+2+3+... . They guess that maybe they used the wrong maths, and maybe the right maths would give that other function so the answer is -1/12. If experiments then agree with this prediction the physicist becomes famous; if not they shrug and try a different way to calculate it.
Edited : I typed +1 when I meant -1. Hey ho.
@@john_g_harris What 1+2+3+... equals, depends on your particular choice of how to assign values to infinite series. It's not possible to assign any finite value to it if you choose to adopt the standard definition but there are other definitions. The Ramanujan summation of 1+2+3+... does equal -1/12. Which particular definition is relevant to solving any particular problem can vary depending on the context in which the summation arises.
Classic..
@@john_g_harris Regularization of the Riemann zeta function at s = -3 is used in calculating the Casimir effect and more generally in quantum mechanics there's a fair amount of renormalization where techniques are used to get a finite sum from a divergent series to get actual results.
The argument that the sum of 1+2+3+ ... does not equal -1/12 because it uses a different method of getting the result comes up a lot. While it's important to recognize that yes, it doesn't mean "equals" in the same way as other "equals", this exact sort of thing has happened before. By the rules of basic arithmetic, the sum of a rational number and another rational number is a rational number. But take all the nonnegative integers and sum the reciprocal of their factorials and you get the transcendental number e. However, getting to this result, or for that matter getting the result of any convergent infinite series requires a different technique than basic arithmetic.
This is not a controversial result today because people are used to the concept of limits and zero, but in the time of Pythagoras or Archimedes, it would have been jus as controversial as summing the positive integers to -1/12. There's an apocryphal story that a member of the Cult of Pythagoras came up with a proof that the square root of 2 is irrational and that the Pythagoreans were so incensed with the result because it broke the rules that they believed in that they took him out to sea in a boat and returned without him. Archimedes came very close to inventing calculus but couldn't make the final conceptual leap because the Ancient Greeks did not believe zero. The idea of using limits to get a result and getting an irrational number from an infinite sum of rational numbers would have been quite controversial.
I've never been more confused by land-owls and seagulls, but I'm glad he's excited about them.
This is exciting to hear. It's evident Professor Padilla is passionate about these breakthroughs. Keep up the good work, Brady. Pete, your animations have been a game changer for this channel.
Man, Numberphile has covered all of the simple math topics. These kinds of videos are HEAVY
I finally feel a little bit better seeing someone else feel the same.
That's not true, it always have been a mixture of both hard and easy topics. Take the last 6 videos, for example, I would argue 3 are very "simple"/"easy" ("Making a klein bottle", "a hairy problem" and "cow-culus")
I recommend the 3Blue1Brown video on the riemann zeta hypothesis for background here. It is visually beautiful.
“This is HEAVY, doc” -Marty McFly
Wow it’s Verlisify! The search for Siegel zeroes so hard they call it Verlisify. Verlisify isify whoo whoo
Very astute product placement, Tony! I ordered your book when it was originally announced on Numberphile and thoroughly enjoyed it.
In the future we'll refer to "Zhang Numbers" : arbitrary values that allowed us to make headway in various proofs.
okay but real talk this dude's been with numberphile since the beginning and HASN'T AGED A DAY
Vampire? Fountain of Youth? Made a dark pact with the heathen maths Gods? Take your bets
His age is a mathematical constant, rather than a variable.
idk man, hes aged a bit since his smosh days
He's asymptotically aging
He is probably a Youkai lol
A healthy even diet, with an odd snack here and there
Yitang Zhang is like a more successful version of Matt Parker. He makes breakthroughs in important cases, but not to the point that was conjectured.
So you’re saying Matt is kind of a Parker Yitang Zhang
@@TimMaddux exactly
🤣🤣🤣
He *makes *breakthroughs
@@ophello Haha, thanks.
Imagine mathematicians were like song artists.
Twitter post: "New RH proof dropping on December 21st, 7 PM EST. Don't miss it"
This actually does happen on sites like Math Overflow
Hey, dude, check this out! This stuff is fire! Read it while on shrooms, it will blow your mind!
I am myself mathematician but doing topics far from these mathematics, and I feel really impressed by the incredible pedagogical skill of this mathematician ! Thank you Tony !
A link between the Twin Prime Conjecture and the Reimann Hypothesis? Numberphile really knows how to stop me working on my thesis!
Same here! My thesis is in fluid dynamics but this is way more interesting to me
There are already connections. By the nature of the riemann zeroes generating the prime number theorem, you get twin prime conjecture somewhat easily.
What's your thesis on? Hope you are finding it interesting.
Yitang is an absolute genius and a legend
Zhang such an inspiration, he clearly devoted his life to humble steady hard work. I wonder if anyone who loves math and works hard can eventually contribute to the world even if they aren’t naturally talented
What r u talking about?
Yes you can !
Do it if you love math
If love it it's possible, if you still have doubt then watch David goggins Then if you still don't go after it you will regret it
@@gauravbharwan6377 You wanna be a mathematician too, bro?
R u kidding me? This is number theory. Ofc he's very talented. He was concidered the best back in the school
Can we all appreciate how the style of video hasn't changed in forever.
From what I've heard it seems that unfortunately, the paper contains a mistake. It might be that Zhang or someone else will fix it, but it could be that it just can't be fixed.
Also, at 8:22 Tony says that if you can find a Siegel zero then the twin prime conjecture will be proven. It's not quite as simple as finding a single Siegel zero. The definition of Siegel zeros has this constant c in it, and for Heath-Brown's theorem you need to prove that for all possible values of c>0, there exists a Siegel zero.
What's the source on that first bit? How critical is the mistake?
Remember that Wiles’ proof of Fermat’s Last Theorem had a mistake. Give it time and we’ll see.
This needs a health warning!
There are so many rabbit holes that are signposted in this video, all of which look as if they would be fun to follow up.
A second health warning for being reminded that theories about primes link up to the sum of an infinite series of complex powers of numbers.
Dangerous stuff - keep it coming.
It truly is fascinating how long number theory reaches into other fields of mathematics in order to even begin to grasp the nature of primes
I don't know what the fancy character is used to depict a lower-case greek chi in the animation, but it definitely ain't a lower-case chi...
EDIT: it seems to be the greek equivalent of "&", dubbed "kai" (same pronunciation as Pr. Padilla's chi). Still wrong character, but leading to an interesting discovery in ancient abbreviations!
It's a mistake. ϗ is the ligature for the Greek word "kai" which means "and." It is similar to the ampersand "&" in English.
probably a mistake by whoever typeset the animation. the hand written letter is chi and as far as i can see that's the standard notation as well. interesting to see this letter tho; it's new to me.
@@heavenlyactsatheavycost7629 ϗ is a ligature for the Greek word "και," which means "and"! It's similar to how the ampersand (&) is a ligature "et," the Latin word for "and."
Padilla also wrote a script xi when he should have written zeta.
@@jaoswald Probably thinking of the completed zeta function.
I like this channel alot, its better than white noise and helps me sleep. No joke, super helpful.
No one’s gonna talk about the fact his mouse is plugged into the wall socket?
nice one but of course it's plugged into the keyboard
He's so proud of the video from 10 years ago, he still has the 2012 calendar.
Interesting stuff! One interesting corollary of the last point about Riemann Zeta tying into physics is that if a physics experiment behaves in an unexpected way in, it could be due to a failure of understanding the mathematics and not a failure of the theory itself.
Or in other words, if there's a weird experimental result that relies on certain interpretation of underlying mathematics, that could develop the mathematical theory as well.
Wow. I mainly love how this can prove the twin prime conjecture to be true. Its very exciting actually
Except Heath-Brown's theorem will almost certainly will never prove the twin prime conjecture, because the Riemann Hypothesis is widely believed to be true.
Siegel primes would essentially guarantee very large fluctuations in the sequence of prime numbers, so much so that primes would inevitably need to be close together every so often. However, fluctuations of the primes appear to be FAR smaller than even the Riemann Hypothesis guarantees, so this method will almost certainly not prove the Twin Prime conjecture.
But if I understand correctly, Heath-Brown's theorem states that if there are no Siegel Primes then the Twin Prime Conjecture is false. And they said it's widely believed that these zeroes don't exist. So doesn't that mean that it's also believed the Twin Prime Conjecture is false?
@@VoodoosMaster I think the inexistence of Siegel Zeros doesn't prevent the Twin Prime conjecture to be true. 08:05 The statement says that one of them has to be true, meaning that at least one of them is true if the other is false (but maybe both are true, hahaha).
However, both being false is not possible according to the theorem.
@@jagatiello6900 Ohhh got it, thank you. Then it's not as exciting as I imagined lol
If a "Siegel Zero" is found or proven to exist, is it "only" the "Generalized Riemann hypothesis" that fails or also the normal "Riemann hypothesis" ?
I think it would only disprove the generalised one, since we would know there's some generalised zeta function that has a non-trivial zero off the line, but it doesn't show that there's a non-trivial zero off the line on the original zeta function
If it is a Siegel zero for one Dirichlet character it doesn't mean automatically it is one for another.
In addition, the Riemann zeta function doesn't have real zeros inside the critical strip, so all of its non-trivial zeros are complex (i.e. not purely real). See e.g. Titchmarsh book on the RZF, p.30. Chapter 2, Section 12.
Although the RZF can't have Siegel zeros, this doesn't imply a thing about the original RH either, for there still could be off the line complex zeros somewhere inside the critical strip.
Brilliant, I love videos like this about ongoing math developments
12:07
"Seagul" Zero is in quantum state.
Now we have "Seagul" Zero and Schrodingers Cat.
5:20 ive never seen a chi written like that before
ϗ is the ligature for the Greek word "καί" which means "and." It is similar to the ampersand "&" in English, which is a ligature for "et," the Latin word for "and."
He managed to get through a whole 3 mins before mentioning Euler XD
Thank you, your videos are always well worth the time to watch!
I love the way he says, at 6:00, "we don't want to go into all the details here..." when in fact he completely lost me about 4 minutes ago. And the video still has 10 more minutes to run.
One thing is for sure. Yitang Zhang is a beast!
I'd have to watch this video Graham's number of times to fully understand it
Finally something about zeta/l -functions
"There's a more general version of the Riemann hypothesis called the generalized Riemann hypothesis. It's the Riemann hypothesis but generalized."
It's fairly simple. Instead of the Riemann zeta function Sum (1/n^s) we look at the functions Sum (f(n)/n^s) for some additional function f(n) called Dirichlet character. If one chooses f(n):=1 then we get the Riemann zeta function.
No, it is so called in honor of the famous mathematician Bernhard Generalized Riemann (1967-1975)
It came to me the thought that the Riemann-hypothesis could become the equivalent of the fifth Euclidean postulate but for number theory.
I don’t know why, but I thought it was funny when he said the mathematician proved that there was an infinite amount of primes that differ by 70 million.
Watching this in 2024
Zhang did essentially the same thing as before. With the twin-prime conjecture he proved that there are infinitely many pairs of primes that differ by a number greater than 2 (so not exactly 2), and here he proved that there is a region where there are no Siegel Zeros, but that is smaller than needed for the full proof. I think this is the death knell for the existence of Siegel Zeros (if the proof holds of course).
death knell*
“death nail” made me laugh… Mutant offspring of “death knell” and “nail in the coffin” 😂
@@oldvlognewtricks Thanks for that! That will teach me to be more careful when using expressions! Well, English is my second language... I've corrected the error because it distracted from the point I try to make...
at 1:46, Tony writes what looks like a Xi instead of a Zeta. Am I wrong?
Isn't the non existence fo Siegel zeros "the same" thing as the Riemann hypothesis? It just feels like that the critical line (going from 0 to infinity) now "just" is projected onto that line part going from c/log(D) to 1.
This is something I've got to watch again. But not tonight.
Prof Tony luvs his numbers
I can only understand like 10% of the whole video. Still watch it
Love the office window
Thank you very much for this video. Most of the articles I read about this were written very poorly and were hard to actually figure out what was going on.
I ve been waiting a month for this!
I like that this guy is embracing having the most controversial numberphile video
What does zero divided by zero equal?
“The jury is still out!” 😊
The formulation at 8:12 is a bit unfortunate, as it can be read as if only one of the two statements is true, which is not what the Heath-Brown theorem states. It states that at least one of those statements is true.
Mochizuki may have proved non-existence of Siegel zeros.
Knowing that, how many pots of paint does Paul need to paint his wall?
How many watermelons did Matt have?
I am fairly convinced that it should be like this about Siegel zeros:
a) For each real Dirichlet character the corresponding L-function has at most 1 Siegel zero.
b) Heath-Brown proved if there are infinite Siegel zeros (meaning for each real Dirichlet character one), then the twin prime conjecture is true.
So the existence of one Siegel zero does not prove the twin prime conjecture.
It's impossible to have one Siegel zero- you just lower the constant until it isn't a Siegel zero. You need the infinite family to eliminate all possible constants.
@@btf_flotsam478 I guess this boils down on the definition of a Siegel zero. How do you define it? Is the Siegel zero defined in terms of multiple L-functions (meaning it's a zero for all L(s,\chi_q) for any \chi_q) or is it defined for one single Dirichlet L-function? I thought any exceptional zero that we can find for one specific Dirichlet L-function was called a Siegel zero and then we look at the collection of these zeros (for all Dirichlet characters) to formulate Heath-Browns theorem.
Or do you call these just zeros and then define the Siegel zero to be one zero for all L(s,\chi_q)?
I didn't understand a word. But I appreciate the enthusiasm!
If c is any number, isn't (c / log D) unbounded?
what a legend
1:45 A little nitpicking but that's a xi Tony writes there, not a zeta xD And at 5:22 the graphic uses a kappa instead of a chi, which Tony says and writes.
When D=1, ie the twin prime conjecture, c/Log(1) is undefined. Where would I check to find a Seigel zero?
If Generalized Riemann Hypothesis is proven to be right, then "ordinary" Riemann Hypothesis would automaticaly be proven right too, if I'm correct.
Sure. But GRH could be proven false while RH could still be true, right ? So, for example, there could be a Siegel zero AND Riemann Hypothesis could nonetheless be true (that would be an interesting possibility, IMHO).
Else there would be no point in distinguishing between the two conjectures, for what I understand.
That is correct
If the GRH is true, the RH is true
But the converse is not the case.
Fascinating!
OMG, after ten years, is the good professor now embracing analytic continuation (which he referred to previously as "spooky")? He may make a mathematician yet!
LANDau-SIEgel zeroes? Surf-n-Turf Zeroes!
I didn't understand half of that but I'm happy for the progress on Riemann hypothesis :)
Anyone notice the subliminal blinds in the background representing the Riemann strip? 👀
land owl seagull zeros, great name! :P
Nice badge Tony solidarity!
I'd love to see a video about how much our current understanding of primes would be completely broken if the Riemann hypothesis were to be disproved.
A "D" of 2 can generalize the sum to the Dirichlet Eta.
Siegel Zeros? More like "Super knowledge that mind blows!"
Such a pretty chi symbol @5:20. I didn't even recognize it.
I was confused for a moment because I was conflating the Twin Prime Conjecture with the Riemann Conjecture.
Chabelo UK.
Steven Siegel is an amazing world class action zero.
Any YT links for the relationship between the Riemann zeros and "energy levels of heavy nuclei" that Tony talked about? My searches are not getting anywhere. TIA!🙏🙏
This is so cool - thanks for the video!
8:20 : "if you find a Siegel zero, the twin prime conjecture has to be true". 11:45 : "i like to be one (Siegel zero), because then it disproves the Riemann hypothesis". These two statements contradict each other.
It should correctly be:
- "if you find infinite Siegel zeros (one for each Dirichlet character chi mod q), then the twin prim conjceture is true."
- "if you find a Siegel zero for the principal Dirichlet character, then you have found a zero for the Riemann zeta function above the 1/2 line (since there is an identity) and thus disproved the RH"
AWESOME EXPLANATION !!! when this result first came out all the "science news" articles about it had no decent explanations, they were all for the mathematically illiterate and basically useless. MANY THANKS for doing this one !!!
Also, I'm familiar with the fact that there are a great many pseudo-theorems of the sort "If the RH (or GRH) is true then XYZ", but was not aware there are any "if the RH (or GRH) is false then XYZ", describing some of the theorems in these two possible alternate worlds would be another great topic for you to cover.
How come those simply hypothesis get some rather large and unexpect upper bounds?
Or is that just a proof that all numbers are equal and we are bias towards smaller number?
Yitang Zhang is the modern day Hercules of mathematics.
Keep all the works of Hercules but replace him with an elder dude with negligible physical abilities. That's what Prof. Zhang is. And still he did it, and not just once. Amazing dude.
I was awarded the title of "Numberphile" once by Google lol.
If a Gull flies over the ocean, he is a Seagull....but if he flies over the bay, he is a Bagel....
And if he flies into a net, then it is simply a goal.
8:14 that’s not an exclusive or relationship for those wondering, both statements could be true, but at least one must be true.
well explained
I thought Seigel Zero's was the nickname of the bullet holes Bugsy Seigel would shoot into his victims
Nope, didn't get any of this. I'm thick as a whale omelette. Somehow I still got excited about this breakthrough. It''s weird.
Great stuff
Looking for zeroes. Nothing to it 🤣
Class love this channel icl
2024 in the answer makes me think this is an Olympiad question 2 years from now.
😂😂😂😂
1:47 I believe that's the letter xi, not zeta.
Zeta is the name of the function, not the character
@@dancurtis8476 the function got its name from the character...
Why cant this be checked by a computed? Is the function wildly oscillatory there or something?
Casually dropping his Book in the background ;)
I may have missed something, but why is the existence of Siegel Zeros so hard to be proved or disproved? As I understood, those roots are real ones (no imaginary part), what makes them (in theory) somewhat easy to be located numerically (should they exist). Is the zeta function wildly oscillatory in the neighbourhood of 1 (approaching by the left)? Maybe I'm too crude on this topic.
What's with the shape of χ on the transcription around 5:17? I thought that it was a kappa before I saw the handwritten version.
I did n-t quite understand that if it'd disprove the „Generalized Riemann Hypothesis“, it'd disprove the „Riemann Hypothesis“ as well - as I did not understand if the RH is „totally“ included in the GRH or if it is just one case of the GRH and those „Siegel-zeros“ could be found to be in other cases but not in the „special“ case of the RH.
Could some-one help? I'd appreciate it 🌞👍🏻
There is an elementary statement about the RH related to the growth of the mertens function.
not to be confused with Seagal zeros of course, which refers to the quality of the movies made by/starring Steven Seagal the last 30+ years
There's a conjecture that all Seagal numbers are Seagal zeros. So far all known Seagal numbers are zero but it's possible that a non-zero Seagal number exists that we just haven't found yet.
Give credit where credit is due (to his body doubles and ADR stand-ins)
The Heath-Brown theorem claims that "ATLEAST one of the two is true".
So if we do manage to find a Siegel zero, that would imply Twin prime conjecture is true.
But, if we prove that Siegel zero does not exist (which it most probably does not), won't prove or disprove Twin prime conjecture.
The twin prime conjecture might be difficult to prove, but at the same time you would pretty much expect it to be true. If is was not, there would be an n, so that for no prime p>n p+2 would also be a prime. As primes are pretty random, why shouldn't there be another pair of twin primes between n and infinity? The more exciting proves are the ones where your expectation is the opposite of what is proven.
The more exciting proofs are ones that cascade throughout maths and prove a bunch of other stuff - such as the twin prime conjecture.
Intuition isn’t always correct, u never know
Hold up… maybe I misunderstood. Does this mean Riemann and Twin Prime can’t both be true?😢
(Edit: spelling)