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@tajujithurr4276

2 ай бұрын

😅😅😅😅😊😅00

@tajujithurr4276

2 ай бұрын

Ooo😊😊😊poooo

@tajujithurr4276

2 ай бұрын

6:55 😅😅😅😊😊😊😊😊 7:01 7:02

@tajujithurr4276

2 ай бұрын

Po99oooo😅ooo😊99889ppp😊p😅oo99😊😊😅9😊😊9😊o😊9😊 12:03 oo😊o😊ooo😊oooo9ooo0😊ook o9😅op9ook 😊o99😊9p99popolice p😊😊9😅

The nature of humanity is just that every so often someone accidentally invents the Riemann Hypothesis again.

@jamieashworth_

Жыл бұрын

😂😂

@scriptorpaulina

Жыл бұрын

🦀

@guilhermecarneiro4711

Жыл бұрын

yep lol

@GuyNamedSean

Жыл бұрын

It's sort of like how π keeps showing up even when you don't see a circle anywhere near.

@TrackpadProductions

Жыл бұрын

@@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" 🤣

@thesenate5956

Жыл бұрын

Once again making people think its normal summation, but its not

@john_g_harris

Жыл бұрын

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.

@MrAlRats

Жыл бұрын

@@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.

@denisdaly1708

Жыл бұрын

Classic..

@lunatickoala

Жыл бұрын

@@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.

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.

@TimMaddux

Жыл бұрын

So you’re saying Matt is kind of a Parker Yitang Zhang

@hafizajiaziz8773

Жыл бұрын

@@TimMaddux exactly

@Abedchess

Жыл бұрын

🤣🤣🤣

@ophello

Жыл бұрын

He *makes *breakthroughs

@DavidSartor0

Жыл бұрын

@@ophello Haha, thanks.

In the future we'll refer to "Zhang Numbers" : arbitrary values that allowed us to make headway in various proofs.

Man, Numberphile has covered all of the simple math topics. These kinds of videos are HEAVY

@akshayvibhute97

Жыл бұрын

I finally feel a little bit better seeing someone else feel the same.

@ra99nano21

Жыл бұрын

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")

@TristanCleveland

Жыл бұрын

I recommend the 3Blue1Brown video on the riemann zeta hypothesis for background here. It is visually beautiful.

@ryanjohnson4565

Жыл бұрын

“This is HEAVY, doc” -Marty McFly

@RunaWorld

2 ай бұрын

Wow it’s Verlisify! The search for Siegel zeroes so hard they call it Verlisify. Verlisify isify whoo whoo

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

@tan_x_dx

Жыл бұрын

His age is a mathematical constant, rather than a variable.

@joeyhardin5903

Жыл бұрын

idk man, hes aged a bit since his smosh days

@crackedemerald4930

Жыл бұрын

He's asymptotically aging

@Silenthunter199

Жыл бұрын

He is probably a Youkai lol

@robind506

Жыл бұрын

A healthy even diet, with an odd snack here and there

Very astute product placement, Tony! I ordered your book when it was originally announced on Numberphile and thoroughly enjoyed it.

A link between the Twin Prime Conjecture and the Reimann Hypothesis? Numberphile really knows how to stop me working on my thesis!

@theludvigmaxis1

Жыл бұрын

Same here! My thesis is in fluid dynamics but this is way more interesting to me

@ffc1a28c7

Жыл бұрын

There are already connections. By the nature of the riemann zeroes generating the prime number theorem, you get twin prime conjecture somewhat easily.

@denisdaly1708

Жыл бұрын

What's your thesis on? Hope you are finding it interesting.

Imagine mathematicians were like song artists. Twitter post: "New RH proof dropping on December 21st, 7 PM EST. Don't miss it"

@theflaggeddragon9472

Жыл бұрын

This actually does happen on sites like Math Overflow

@u.v.s.5583

Жыл бұрын

Hey, dude, check this out! This stuff is fire! Read it while on shrooms, it will blow your mind!

I watched your Royal Institute lecture on very large numbers last night and it was only when I multiplied the SMALLEST thing I could think of (an angstrom) by a googol, did I learn that a googol angstroms is approximately 100 trevigintillion light-years long... and THAT made me see just how huge that (relatively tame) large number is. Thanks Professor!! 👍😎🎄🇦🇺

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.

@UnknownYTName

Жыл бұрын

What's the source on that first bit? How critical is the mistake?

@billcook4768

Жыл бұрын

Remember that Wiles’ proof of Fermat’s Last Theorem had a mistake. Give it time and we’ll see.

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 !

Yitang is an absolute genius and a legend

More videos like this, please! This was fantastic.

No one’s gonna talk about the fact his mouse is plugged into the wall socket?

@h00db01i

Жыл бұрын

nice one but of course it's plugged into the keyboard

Thank you, your videos are always well worth the time to watch!

Can we all appreciate how the style of video hasn't changed in forever.

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!

@SeanCMonahan

Жыл бұрын

It's a mistake. ϗ is the ligature for the Greek word "kai" which means "and." It is similar to the ampersand "&" in English.

@heavenlyactsatheavycost7629

Жыл бұрын

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.

@SeanCMonahan

Жыл бұрын

@@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."

@jaoswald

Жыл бұрын

Padilla also wrote a script xi when he should have written zeta.

@theflaggeddragon9472

Жыл бұрын

@@jaoswald Probably thinking of the completed zeta function.

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

@imeprezime1285

Жыл бұрын

What r u talking about?

@Xirrious

Жыл бұрын

Yes you can ! Do it if you love math

@gauravbharwan6377

Жыл бұрын

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

@xkjw7019

Жыл бұрын

@@gauravbharwan6377 You wanna be a mathematician too, bro?

@weserfeld4417

Жыл бұрын

R u kidding me? This is number theory. Ofc he's very talented. He was concidered the best back in the school

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.

@StriderGW2

Жыл бұрын

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

He's so proud of the video from 10 years ago, he still has the 2012 calendar.

I like this channel alot, its better than white noise and helps me sleep. No joke, super helpful.

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" ?

@scares009

Жыл бұрын

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

@AvntXardE

Жыл бұрын

If it is a Siegel zero for one Dirichlet character it doesn't mean automatically it is one for another.

@jagatiello6900

Жыл бұрын

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.

Wow. I mainly love how this can prove the twin prime conjecture to be true. Its very exciting actually

@gabrielrhodes9943

Жыл бұрын

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.

@JM-us3fr

Жыл бұрын

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.

@VoodoosMaster

Жыл бұрын

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?

@jagatiello6900

Жыл бұрын

@@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.

@VoodoosMaster

Жыл бұрын

@@jagatiello6900 Ohhh got it, thank you. Then it's not as exciting as I imagined lol

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.

He managed to get through a whole 3 mins before mentioning Euler XD

5:20 ive never seen a chi written like that before

@SeanCMonahan

Жыл бұрын

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."

This is so cool - thanks for the video!

12:07 "Seagul" Zero is in quantum state. Now we have "Seagul" Zero and Schrodingers Cat.

I ve been waiting a month for this!

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.

Finally something about zeta/l -functions

One thing is for sure. Yitang Zhang is a beast!

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.

I'd have to watch this video Graham's number of times to fully understand it

Prof Tony luvs his numbers

Love the office window

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.

Fascinating!

It came to me the thought that the Riemann-hypothesis could become the equivalent of the fifth Euclidean postulate but for number theory.

2024 in the answer makes me think this is an Olympiad question 2 years from now.

@gauravbharwan6377

Жыл бұрын

😂😂😂😂

Great stuff

This is something I've got to watch again. But not tonight.

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).

@timseguine2

Жыл бұрын

death knell*

@oldvlognewtricks

Жыл бұрын

“death nail” made me laugh… Mutant offspring of “death knell” and “nail in the coffin” 😂

@mauricemaths

Жыл бұрын

@@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...

Oh boy! Any advancement in number theory involving Riemann excites me.

@u.v.s.5583

Жыл бұрын

Did you know the following fact about Riemann and primes: Riemann's hands each had a prime number of fingers!

@theherk

Жыл бұрын

@@u.v.s.5583 Using the term "digits" would have been more correct and a double entendre. Missed opportunity.

I can only understand like 10% of the whole video. Still watch it

When D=1, ie the twin prime conjecture, c/Log(1) is undefined. Where would I check to find a Seigel zero?

Your letter zeta ζ looks like ξ

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.

8:26 isn't it the existence of infinite siegel zeros (one for each dirichlet character) that implies the twin prime conjecture which Roger Heath-Browns theorem says?

@japanada11

Жыл бұрын

Yes - and technically speaking, the concept of "a Siegel zero" is not well-defined (you can always choose a small enough constant c so that any given zero is more than c/logD away from 1). You need an infinite collection of zeros that converge to 1 very rapidly in order to call the whole set a *collection* of siegel zeros.

I didn't understand a word. But I appreciate the enthusiasm!

well explained

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.

So if i understood this correctly, the existence of Siegel-zeroes doesn't disprove the Riemann-hypothesis, but the generalized Riemann-hypothesis. So the Riemann-hypothesis could still be true.

@btf_flotsam478

Жыл бұрын

And, of course, his work supports the generalised Riemann Hypothesis anyway.

@gauravbharwan6377

Жыл бұрын

No wrong

@Peregringlk

Жыл бұрын

As far as I understood, the Riemann-hypothesis is a special case of the generalized one. If you disprove the generalized one, you disprove every one of its special cases, so the Riemann-hypothesis is then false.

Class love this channel icl

I didn't understand half of that but I'm happy for the progress on Riemann hypothesis :)

Knowing that, how many pots of paint does Paul need to paint his wall?

@Dominexis

Жыл бұрын

How many watermelons did Matt have?

What does zero divided by zero equal? “The jury is still out!” 😊

Nice badge Tony solidarity!

I like that this guy is embracing having the most controversial numberphile video

"There's a more general version of the Riemann hypothesis called the generalized Riemann hypothesis. It's the Riemann hypothesis but generalized."

@AvntXardE

Жыл бұрын

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.

@u.v.s.5583

Жыл бұрын

No, it is so called in honor of the famous mathematician Bernhard Generalized Riemann (1967-1975)

Love these. Maths as a detective story.

what a legend

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!🙏🙏

Siegel Zeros? More like "Super knowledge that mind blows!"

I was confused for a moment because I was conflating the Twin Prime Conjecture with the Riemann Conjecture.

I was awarded the title of "Numberphile" once by Google lol.

A little hard to follow at times but fascinating nonetheless.

Mochizuki may have proved non-existence of Siegel zeros.

If c is any number, isn't (c / log D) unbounded?

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!

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.

@btf_flotsam478

Жыл бұрын

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.

@AvntXardE

Жыл бұрын

@@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 am not sure I even want to ask how exactly you get to a proof that says "either there are no segel zeros or the twin prime conjecture must be true". Did not know some theorems where playing by Highlander rules.

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 🌞👍🏻

Maybe you could explain how the Siegel zero proves that the Riemann hypothesis is false. Also I would like to know more about these Dirichlet functions and why the focus is on the characters.

@AvntXardE

Жыл бұрын

A Siegel zero disproves a different statement, the generalized Riemann hypothesis (GRH), not the actual Riemann hypothesis (RH). The GHR is a statement for a class of functions (so called Dirichlet L-functions).

@KenHilton

Жыл бұрын

The generalized Riemann hypothesis states that all nontrivial zeros of the generalized Riemann zeta function have a real part of 1/2. The real part of Siegel zeros is within a certain distance of 1, instead of 1/2; if they exist, they would be nontrivial zeros whose real parts are not 1/2, which would contradict the GRH (and thus disprove it).

@Peregringlk

Жыл бұрын

Because the Riemann hypothesis says that all (non-trivial) zeros lies on the 1/2 vertical line of the graph. If you find a non-trivial zero outside the 1/2 vertical line, then the Riemann hypothesis is false by definition, and a siegel zero is precisely a non-trivial zero outside the 1/2 vertical line (specifically, one very close to 1 as the video explains).

There is an elementary statement about the RH related to the growth of the mertens function.

at 1:46, Tony writes what looks like a Xi instead of a Zeta. Am I wrong?

In the heath Brown conjecture, can both be true?

Okay, so what is bigger, the last two twin primes or tree(3)?

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.

land owl seagull zeros, great name! :P

Such a pretty chi symbol @5:20. I didn't even recognize it.

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.

LANDau-SIEgel zeroes? Surf-n-Turf Zeroes!

Anyone notice the subliminal blinds in the background representing the Riemann strip? 👀

To reiterate my questions about the -1/12… why is shifting a duplicated series underneath by one allowed or taken versus any other equal foul? Since when do we take an average of answers when a function gives more than one?!?!

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.

@docsy4529

Жыл бұрын

That is correct If the GRH is true, the RH is true But the converse is not the case.

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.

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.

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?

Casually dropping his Book in the background ;)

Riemann-Siegel zeros

Finding a zero off the line would be cool.

Does it mean that the twin prime conjuncture is false? I don't think the Riemann's hypothesis is false.

Exciting stuff if it proves true.

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.