Visualizing the Riemann zeta function and analytic continuation

  • Am Vor 4 years

    3Blue1Brown3Blue1Brown

    Unraveling the enigmatic function behind the Riemann hypothesis
    Help fund future projects: www.patreon.com/3blue1brown
    An equally valuable form of support is to simply share some of the videos.
    Special thanks to these supporters: 3b1b.co/zeta-thanks
    Home page: www.3blue1brown.com/
    Posters/shirts for this visualization at 3b1b.co/store

    Music by Vince Rubinetti: soundcloud.com/vincerubinetti...

    Check out some of Vince's other work here: www.vincentrubinetti.com/

    For those who want to learn more about complex exponentiation, here are a few resources:
    - My video on the topic: de-film.com/v-video-mvmuCPvRoWQ.html
    - Mathologer's: de-film.com/v-video--dhHrg-KbJ0.html
    - Better Explained: goo.gl/z28x2R

    For those who want to learn more about the relationship between 1+2+3+4+... and -1/12, I'm quite fond of this blog post by Terry Tao: goo.gl/XRzyTJ

    Also, in a different video "What does it feel like to invent math", I give a completely different example of how adding up growing positive numbers can meaningfully give a negative number, so long as you loosen your understanding of what distance should mean for numbers: de-film.com/v-video-XFDM1ip5HdU.html

    Interestingly, that vertical line where the convergent portion of the function appears to abruptly stop corresponds to numbers whose real part is Euler's constant, ~0.577. For those who know what this is, it's kind of fun to puzzle about why this is the case.

    ------------------

    3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with DE-film, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).

    If you are new to this channel and want to see more, a good place to start is this playlist: goo.gl/WmnCQZ

    Various social media stuffs:
    Website: www.3blue1brown.com
    Twitter: twitter.com/3Blue1Brown
    Patreon: patreon.com/3blue1brown
    Facebook: facebook.com/3blue1brown
    Reddit: www.reddit.com/r/3Blue1Brown

Chiranjivee subedi
Chiranjivee subedi

Analytic= Angle preserving

Vor 5 Stunden
Yash Pal Goyal
Yash Pal Goyal

there's a glitch in the animation. the frame shown at 543.97s, which means just move 2 frames behind 9:04 (i.e. 544s), is a corrupt frame

Vor 20 Stunden
Christina Awakened
Christina Awakened

🤯🤯🤯

Vor Tag
Connor Kearley
Connor Kearley

🥵🇸🇧🇳🇿

Vor 2 Tage
Abraham Lim Ken Zhi
Abraham Lim Ken Zhi

Could you give a full introduction to Riemann Zeta function? Let more people know about it so that the chance of it being proved increased

Vor 2 Tage
samarth wadhawan
samarth wadhawan

He lost me exactly qt 00:03

Vor 5 Tage
Corn Holeium
Corn Holeium

Great channel thanks for your work 🙌🙌 the episode about Euler's equation was truly remarkable.

Vor 6 Tage
ilillili
ilillili

그래서 리만 제타함수가 뭔데 이 씹덕아

Vor 14 Tage
Rene Ticona
Rene Ticona

Lo siento pero parece que en bolivia 🇧🇴 , Beimar López encontró la fórmula simple a este problema del milenio. Saludos desde Bolivia 🇧🇴.👍

Vor 16 Tage
esilence
esilence

18:09 i am still waiting for that video you mentioned about why and how it carries info re the primes!!!

Vor 18 Tage
Javi JA
Javi JA

I would like to know your opinion about the formula developed by Beimar Wilfredo Lopez Subia for the Riemann conjecture.

Vor 19 Tage
Valerii Radchenkov
Valerii Radchenkov

That graph looks a lot like a smith chart... Strange, what could an infinite sum of complex reciprocal exponents have to do with an easy geometric way to calculate circuit parameters?

Vor 21 Tag
Nic Ivanov
Nic Ivanov

And why, after the right half-plane, to extend the function in the same two-dimensional dimension if everything points to a cliff like a funnel, where negative numbers are the inner side this funnel?

Vor 22 Tage
kidd lee
kidd lee

12:20 can't help crying. How beautiful the math is!

Vor 24 Tage
Jason Preneveau
Jason Preneveau

All i want to do is watch a movie and I'm here watching math videos. 3b1b math videos specifically

Vor Monat
Aniruddh XIE 2K21
Aniruddh XIE 2K21

Hey I think this sum 1+2+3+..... Is called the Ramanujan Sum if I am correct

Vor Monat
ᄀᄉᄂ
ᄀᄉᄂ

머냐 저거 정의 내리고 물리에 적용하면 공전궤도읽고 은하간 간섭까지 계산 가능한거냐?

Vor Monat
Torpedo Time
Torpedo Time

느금마

Vor Monat
Adam Muse
Adam Muse

Reimann was brilliant

Vor Monat
ole86
ole86

Riemann Euler Gauss = holy trinity

Vor Monat
OOF OOF
OOF OOF

Uh……. What

Vor Monat
monasa
monasa

RiGHt??

Vor Monat
Indonesia Bisa
Indonesia Bisa

Thank you for doing great work in promoting advanced mathematics. The world needs more people like you, Sir.

Vor Monat
BafBaf
BafBaf

Cool

Vor Monat
ydhakal
ydhakal

No question that the explanation is beyond just awesome. But, every time the zeta function for s = -1 equals -1/12, I lose interest. What’s missing here ? Please suggest if I should study something to understand this.

Vor Monat
Gregory Petersen
Gregory Petersen

Looks an awful lot like RF antenna design

Vor Monat
Sameer Sehgal
Sameer Sehgal

13:40 is L'Hopitals Rule

Vor Monat
Tycho Photiou
Tycho Photiou

Grant I have great admiration for you. I absolutely love your animations. You are a brilliant teacher. But please please please leave out that annoying background music - its totally unnecessary and really annoying and distraction. I would prefer the occasional sound of your workmen in the background than that annoying music.

Vor Monat
Hermes A V Urquijo
Hermes A V Urquijo

I usually not comment YouTube videos but that was an amazing one, congratulations 👏.

Vor Monat
king frozen
king frozen

Teaching mathematics is easiest thing in entire world , just hand a person a ruler and compass and eventually after years he will come up with complex analysis

Vor Monat
Parizat Bhatia
Parizat Bhatia

you lost me at 19:00

Vor Monat
Parizat Bhatia
Parizat Bhatia

i was having mini adrenaline rushes throughout the video! I loved every second of it.

Vor Monat
Parizat Bhatia
Parizat Bhatia

Question: I like to think of -infinity as an infinity for negative numbers and because we see infinity coming out from a lot of different approaches. It makes sense right? What do you think? please let me know

Vor Monat
퀄대신물량뻘댓증식중
퀄대신물량뻘댓증식중

신기한 숫자 12 인공위성 관련해서 수식이 안 풀리면 이 숫자를 대입하면 풀리는데 이게 왜 그런지 의문이라고...

Vor Monat
Kasim Hanif
Kasim Hanif

10:05 Damn, 3b1b just said damn

Vor Monat
Quanta_relet um
Quanta_relet um

Well how did they know the output of the function is minus one above twelve when s = -1? 'Cause the function for real part less than one is not defined. Who gave this result? Was it Ramanujan?

Vor Monat
Reddblue
Reddblue

Ah yes, simple kiddie stuff really.

Vor Monat
유키؟
유키؟

그래서 이게뭔지

Vor Monat
Varol Cavdar
Varol Cavdar

Thanks!

Vor Monat
mCSenkei
mCSenkei

Hi. I am here because i am looking for video about bitcoin, i dont remember topic of this video but it was telling that bitcoin moves like light. And there was a loot beautiful charts with a loot of lines explaining why it is move like this. can anyoe help please? thank you

Vor Monat
Chandu Chevala
Chandu Chevala

Finally it is solved by an Indian mathematician🥳

Vor Monat
machinegun3455 Gaming
machinegun3455 Gaming

The Riemann Zeta function on 1/2 + i*t is actually a sum of "3d-sine (e^iwt)" waves with the overtones having frequencies of log(b)*w, where w is the base frequency and log(b) is the natural log of b, where b ∈ ℤ and b ≥ 1.

Vor Monat
아옹이
아옹이

4×7=27?????

Vor Monat
Mauro Manucci
Mauro Manucci

sure the left-hand analytical function has got an equation for it (being that the whole thing we are seeing is just 2 functions one next to the other). Has anyone got a quick link to that?

Vor Monat
templarseries
templarseries

Brilliant video, I just wish you'd have covered the analytic extension, the trivial zeroes and - 1/12 are bothering me.

Vor Monat
Stuart Hollingsead
Stuart Hollingsead

10:57 This is like a fish looking up at the underside of the surface and seeing its reflection. The fish surmises that the opposite side of the surface is just like the water it lives in. It jumps out of the water and realizes it is quite different. But that original concept is still influencing thoughts. Theologians do this all the time when surmising about God. And the new age religions take the idea of the mirror quite seriously. And for that matter, so does the occult. Heaven looks nothing like this earth. It is a place of eternal flame. Hell is not a place, but a state of being. death is hell. You know nothing in death. Know the Truth of how terrible eternal death is. It is the only thing God fears. Now with this hand hold on the first step to wisdom, learn the Truth. Fear what God fears. And realize people that try to tell you that you live after death... they are liars and the Truth is not in them.

Vor Monat
sasoribi 1341
sasoribi 1341

nice!

Vor Monat
Dr. Devil
Dr. Devil

Can't the combination of different rules and laws of various theorem are unfit? I am always amazed of deep realms of mathematics but sometime i think that what if we are using the rules of limited dimension with the rules of next. Am i the only one who is amazed as well as confused of mathematics?🤔

Vor 2 Monate
Mr Sport Endurance
Mr Sport Endurance

Too easy ....should teach in primary school

Vor 2 Monate
Aphra Xiaojun
Aphra Xiaojun

amsr

Vor 2 Monate
Aneesh Srinivas
Aneesh Srinivas

Why is there only 1 extension

Vor 2 Monate
Dakota Thomas
Dakota Thomas

"I'm assuming you know about complex and how to calculate with them, and im assuming a little calculus as well.." Etc etc Me, the two time high school dropout: yeah bro we got this for sure

Vor 2 Monate
Shashank Gupta
Shashank Gupta

I couldn't understand, how did π²/6 come?

Vor 2 Monate
Master Oogway
Master Oogway

My mistake... I thought I was good at math and clicked this. Now I regret😶

Vor 2 Monate
Josue Martinez
Josue Martinez

Could there be more information if the circular shapes graph is then rotated along x or y to form a solid in 3d, could the points inside the holes formed hint at something?

Vor 2 Monate
baek choi
baek choi

Vor 2 Monate
James Fortune
James Fortune

I didn't hear the words 'conformal mapping'. 🙂 When positive values are added to 1, the sum will never reach 0. $1,000,000 please. 🙂

Vor 2 Monate
kpkdhar 36
kpkdhar 36

6:53 onwards, what does the animation looks like, growth of a vine or a leaf slowing growing or a small plant growing, in slow motion they show the same curves as this animation does. I think God created entire universe on prime numbers and what we don't see with eyes on imaginary complex numbers that we can't represent in 3 dimension.

Vor 2 Monate
Ph B
Ph B

Traduisez votre application en francais et vendez-la à l'Education Nationale française (ou au CNED) qui sont incapables d'en produire autant......malgré le confinement du COVID.....

Vor 2 Monate
wilson lu
wilson lu

The feeble feigned direction markedly shelter because distance only boast under a optimal knickers. satisfying, ugly desire

Vor 2 Monate
Bonnie Massaro
Bonnie Massaro

your video shows the truth and the hypothesis is invalid, no longer science, everything is a swirl, why put an imaginary line to it, it is happenstance, and why put numbers to it, the truth is the output, just keep adding dimensions and the physics will be shown :)

Vor 2 Monate
Kavorka Games
Kavorka Games

Everything you do is the best thing ever.

Vor 2 Monate
मनुष्य!(human)
मनुष्य!(human)

It is solved by an indian

Vor 2 Monate
EnvY
EnvY

Bistcord^quion. 78-0

Vor 2 Monate
Martin Piekarski
Martin Piekarski

Alright, if I understood correctly, the zeta function of -1 can amount to -1/12 and it's because the imaginary axis is applied. Without it, it collapses into regular sum of the natural numbers. Damn it, it's difficult to operate on complex numbers because the imaginary number is just so abstract and unsubstantial. Abstract concepts are a pain when used in such a rigorous field as mathematics.

Vor 2 Monate
Rizuken Nekuzir
Rizuken Nekuzir

It looks like the sphere projected onto the 2d grid from your quaternions video

Vor 2 Monate
Himanshu Rane
Himanshu Rane

Resently this was proved by an Indian ❤️

Vor 2 Monate
Devanshu
Devanshu

@ARUNABH MUKHERJEE checkout again plz

Vor 2 Monate
ARUNABH MUKHERJEE
ARUNABH MUKHERJEE

@Devanshu What nope? Check out Mr. Kumar Eswaran

Vor 2 Monate
Devanshu
Devanshu

Nope

Vor 2 Monate
P Bh
P Bh

Proved!!

Vor 2 Monate
Aytidaymuos@31
Aytidaymuos@31

Indian mathematician K Eshwaran solved the hypothesis

Vor 2 Monate
Anand Sharma
Anand Sharma

It has been proved.

Vor 2 Monate
Shabd Sahni
Shabd Sahni

Why did the video end abruptly? Weren't you gonna show the derivative of the zeta func? :( cue the cc....

Vor 2 Monate
Pietro Gagliardi
Pietro Gagliardi

It still does, in the final few seconds. There used to be a sponsorship spot in between, but it was removed. Sadly the animation of that derivative was cut with it.

Vor Monat
Mohammed Abdul Qayyum Khan
Mohammed Abdul Qayyum Khan

To all the people blindly commenting that it has been solved - the answer is No. The proposed proof has been published but it has not been accepted by any journals and experts in this field apparently due to some major flaws. The mystery remains unsolved.

Vor 2 Monate
Devanshu
Devanshu

It has a major mathematical flaw

Vor 2 Monate
Divyadeep Singh
Divyadeep Singh

solved today

Vor 2 Monate
Anirudh R. - 091
Anirudh R. - 091

Coming back to this after the Reimann Hypothesis was proved by Dr Kumar Eswaran

Vor 2 Monate
Utkarsh Srivastav
Utkarsh Srivastav

Anyone here after Dr Kumar Eswaran solved the reiman hypothesis......proud moment for all Indians 🇮🇳🇮🇳🇮🇳

Vor 2 Monate
Viraj Goyanka
Viraj Goyanka

Who is here after when an Indian Physicist solved the problem..

Vor 2 Monate
Nikhil Sai
Nikhil Sai

There have been reports saying that the Riemann Hypotheses has been proved, could you make and explanation video for the same?

Vor 2 Monate
Harrith V Rajesh
Harrith V Rajesh

An Indian mathematician just solved it!!!

Vor 2 Monate
Mevada Vraj
Mevada Vraj

I heard that it is solved ?

Vor 2 Monate
James Armstrong
James Armstrong

Math is the art of taking easy questions and moving out of the foundations that made it easy.

Vor 2 Monate
BT19ECE064_Manoj M
BT19ECE064_Manoj M

Dr.Kumar Eswaran of Hyderbad claims to have solved the Riemann hypothesis , I am leaving it here for future discussions on subject @3Blue1Brown

Vor 2 Monate
IExist
IExist

I still feel cheated. Like stop writing 1+2+3+4+5+6...= -1/12 unless you actually figure out the connection

Vor 2 Monate
advay kumar
advay kumar

Who's here when it has been solved?

Vor 2 Monate
Mevada Vraj
Mevada Vraj

Me

Vor 2 Monate
pratyaksh tyagi
pratyaksh tyagi

✋🏻

Vor 2 Monate
Gerardo Tobar
Gerardo Tobar

Grant, you still owe us a video on the relation to primes, don't you?! I'm sorry to dare asking but your work is so amazing that even if I already understand it, I need to visualize it in the way yo visualize things!

Vor 2 Monate
null
null

lol numbers are meant to represent divisions of a whole of parts.

Vor 2 Monate
Cypress Hodgson
Cypress Hodgson

How do you compute the critical strip in a concrete way. I’m working on doing this in Desmos, (which is a mathematicians hell because Desmos doesn’t have the slightest clue how to deal with complex numbers) and I wanted to be able to calculate it. The extension shown only works for numbers with real component less than 0.

Vor 2 Monate
인생검색
인생검색

가끔 생각나면 보러옴

Vor 2 Monate
Ryan Park
Ryan Park

wow...

Vor 2 Monate
편안해지
편안해지

와우

Vor 2 Monate
Pteromandias
Pteromandias

I only heard of it because of the book Cryptonomicon.

Vor 3 Monate
The Evil Mathematician
The Evil Mathematician

The Riemann-Zeta Function, which states ζ(s)=1/1^s + 2/2^s + n/n^s must be complex. For even negative numbers, s= -2k, meaning k is real. Then, 1/1^-2k+ 2/2^-2k+..., which implies k is a complex number; k=a+bi. But this is a clear contradiction! Therefore, the Riemann Hypothesis cannot exist!

Vor 3 Monate
The Evil Mathematician
The Evil Mathematician

Do I get one million dollars for disproving the Riemann Hypothesis?

Vor 3 Monate
David Wagner
David Wagner

Yay for complex analysis! Thanks 3blue1brown!

Vor 3 Monate
정주원
정주원

The delicate deposit nutritionally arrest because accordion classically chop mid a overconfident feast. straight, flawless sparrow

Vor 3 Monate
aash syed
aash syed

NOW I SEE THAT SOLVING THE REIMANN HYPOTHESIS IS NOT HARD.

Vor 3 Monate
aash syed
aash syed

@Mr. Virtual nooo

Vor 2 Monate
Geoff Strickler
Geoff Strickler

This is the most easily understandable explanation of numerous things, including analytic continuation, complex analysis, and the continued zeta function that I’ve ever seen. And I say that as someone 50+ yrs old who has had a tremendous facility with math since I was at least 4 and generally taught myself most mathematics before I learned it in school, including many aspects of trig and calculus. And by “taught myself”, I actually mean figured it out on my own, not that I picked it up by studying a book. Books largely just connected my self-learning with accepted terminology, but did offer expanded views and results of what I figured out on my own. It wasn’t until calculus that I started encountering a significant amount about math that I hadn’t figured out on my own.

Vor 3 Monate
Roman Medvedev
Roman Medvedev

Fascinating, thank you so much!

Vor 3 Monate
Thomas JR.
Thomas JR.

BTW, SAYING THAT 1+2+3+...=-1/12, ONLY MAKES SENSE UNDER THE CONTEXT OF THE ZETA FUNCTION ANALYTIC CONT. THERE MAY BE OTHER FUNCTIONS WHOSE ANALYTIC CONT ASSIGNS A DIFFERENT NUMBER TO THIS SUM.

Vor 3 Monate
Thomas JR.
Thomas JR.

19:03 *DON'T BE NAIVE THOUGH, THEY WILL ONLY GIVE YOU THE PRIZE IF YOU'RE ONE OF THE NOTABLE ONES. IF YOU'RE A MODERN DAY RAMANUJAN, HAHA, NO, THEY WILL NOT GIVE YOU ANYTHING. THEY ARE WAITING FOR A RENOWNED MATHEMATICIAN'S SOLUTION, NOT AN ORDINARY PERSON'S SOLUTION, EVEN IF IT'S RIGHT.*

Vor 3 Monate
Thomas JR.
Thomas JR.

*WELL, THERE'S A LITTLE INACCURACY HERE. SINCE HE'S TALKING ABOUT COMPLEX FUNCTIONS, SAYING THAT LOG(x) IS ANALYTIC IS WRONG. IT IS NOT, IT'S NOT ANALYTIC AT 0 AND DEFINITELY NOT ANALYTIC WHENEVER |x-1|>1! THERE IS NO POWER SERIES A such that A == LOG(x) FOR ALL x. AN ANALYTIC FUNCTION IS EQUAL TO ITS TAYLOR SERIES EXPANSION EVERYWHERE.*

Vor 3 Monate
Thomas JR.
Thomas JR.

I totally had forgot why it's the real part of s that dictates if zeta(s) is finite or infinite. Thanks for reminding me.

Vor 3 Monate
Sidney Silva
Sidney Silva

La hipótesis de Riemann, Estimado profesor noble amigo del canal, 3 Blue1 Brown Interesante su explicación, sencilla y fácil de aprender, con todo mi respeto a todos los profesores, alumnos y amigos de este sencillo canal, vengo al público a manifestar que la Hipótesis de Rieman ha perdido fuerza, de vez en cuando en el pasado en la actualidad. era, con un AP (Progresión Aritmética) simple, y con dos fórmulas estandarizadas: afirmo con total veracidad, probando Científica y Matemáticamente; ¿Qué impacto causaría en el Universo de las Matemáticas al afirmar que algunos números que se mencionan a continuación no son primos? y los primos gemelos no existen? dos; 19; 41; 59; 61; 79; 101; 139; 179; 181; 199; 239; 241; 281; 359; 401; 419; 421; 439; 461; 479; 499; 521; 541; 599; 601; 619; 641; 659; 661; 701; 719; 739; 761; 821; 839; 859; 881; 919; 941; 1019; 1021; 1039; 1061; 1181; 1201; 1259; 1279; 1301; 1319; 1321; 1361; 1381; 1399; 1439; 1459; 1481; 1499; 1559; 1579; 1601; 1619; 1621; 1699; 1721; 1741; 1759; 1801; 1861; 1879; 1901; 1979; ahora la Hipótesis de Riemann se desmorona, el autor Sr. Sidney Silva, por lo tanto soy el poseedor del mayor número primo jamás encontrado, llegando a la casa de Nonillions ...... el autor de este descubrimiento único y majestuoso ... Sr. Sidney Silva.

Vor 3 Monate

Nächster

Fractals are typically not self-similar

21:36

Fractals are typically not self-similar

3Blue1Brown

Aufrufe 2 700 000

Pi hiding in prime regularities

30:42

Pi hiding in prime regularities

3Blue1Brown

Aufrufe 1 600 000

7,000 Year Old Toothbrush vs Future Toothbrush !!

4:52

The Riemann Hypothesis -- Science étonnante #62

16:59

The Riemann Hypothesis -- Science étonnante #62

ScienceEtonnante

Aufrufe 1 000 000

But how does bitcoin actually work?

26:21

But how does bitcoin actually work?

3Blue1Brown

Aufrufe 10 000 000

Riemann Hypothesis - Numberphile

17:04

Riemann Hypothesis - Numberphile

Numberphile

Aufrufe 4 700 000

Some light quantum mechanics (with minutephysics)

22:22

How to send a self-correcting message

20:05

How to send a self-correcting message

3Blue1Brown

Aufrufe 1 000 000

Body matching descripton of Gabby Petito found in Grand Teton

5:43

Ariana Grande Talks Demi Lovato Collab And Duet With Kelly

5:38

Wählt Hazel! Nicht Laschet!

1:03

Wählt Hazel! Nicht Laschet!

Hazel & Thomas

Aufrufe 50 129