The Riemann Hypothesis

The Riemann Hypothesis is one of the Millennium Prize Problems and has something to do with primes. What's that all about? Rather than another hand-wavy explanation, I've tried to put in some details here. Some grown-up maths follows.
More information: www.claymath.org/publications/...
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CORRECTION: The functional form of the zeta function is a reflection around the point 0.5, not the line x=0.5. But you can think of that as a reflection in the x-axis followed by a reflection in the line x=0.5.
For example, if we start with zeta(x + iy) and reflect it in the x-axis we get zeta(x - iy). Reflect that in the x=1/2 line we get (junk)*zeta(1 - x - iy). And reflect that in the x-axis again we get (junk)*zeta(1 - x + iy). When zeta(x + iy) = 0 so are all the others.
The zeros then look like this en.wikipedia.org/wiki/File:Zer... That's four zeros for the price of one.
The zeros are still symmetric around the x=1/2 line. The reflection of zeta(x + iy) is (junk)*zeta(1 - x + iy). In the video I mixed up zeta(1 - x + iy) with zeta(1 - x - iy).
But the point is the same, the error in the Prime Number Theorem is minimised if all the zeros lie on the line x=1/2.
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Some people have asked about my final claim that you don't get the prize for a counterexample. Here is the rule from the Clay Institute:
"if a counterexample is proposed, the SAB will consider this counterexample after publication and the same two-year waiting period as for a proposed solution will apply. If, in the opinion of the SAB, the counterexample effectively resolves the problem then the SAB may recommend the award of the Prize. If the counterexample shows that the original problem survives after reformulation or elimination of some special case, then the SAB may recommend that a small prize be awarded to the author. The money for this prize will not be taken from the Millennium Prize Problem fund, but from other CMI funds."
In other words, there are no guarantees either way.

[@andersjaevel]

The proof of the Riemann Hypothesis is clear and we leave as an exercise to the reader.

[@ItachiUchiha-ns1il]

andersjaevel damn that’s good one lmao

[@69erthx1138]

You need a couple of pints and few less text books mate.

[@user-xh9pu2wj6b]

Some good Landau right there.

[@yashhkotecha2647]

Trivial*

[@RodelIturalde]

@supadox Because often, the proof is rather easy to make if you did understand exactly the proofs and methods the textbook used to prove some other thing just before. And it is actually a good excercise to come up with ones own proofs. Though it is extremely hard i think.

[@72ackerman]

This is a great explanation of the Riemann hypothesis. I like how you presented it in historical context, broke it down into parts, and explained its relationship to other aspects of mathematics. Very critically, you presented the equations, you explained the intuitive portions to non-mathematicians, and you did it with the enthusiasm that is difficult to glean from dry textbooks. Love your KZbin posts and your appearances on Brady's videos. Keep up the great work.

[@singingbanana]

+72ackerman Thank you very much!

[@72ackerman]

+singingbanana Also, you finally taught me what that big Pi means before an equation, which I had not known before! (Math looks less like hieroglyphics now).

[@singingbanana]

+72ackerman Ha, I'm glad.

[@christosvoskresye]

+72ackerman The symbol is used for continued fractions. Well, not really, but I like the idea of using actual hieroglyphics for math.

[@christosvoskresye]

+christosvoskresye And, of course, it doesn't show.

[@hummanuhblubberrumma]

I believe that we just got -1/12'd again.

[@GermanSnipe14]

"It was like the mathematical equivalent of a mic drop. Peace out, that's it. I'm done" ahahaha I love you Dr Grime

[@dhidhi1000]

I find it interesting that I watched this video about 2 years ago when I was 17, and I remember I didn't understand somethings. Now I'm on college, and I decided to watch this again, and surprisingly enough, I got a lot more of what he said. It's nice to see our own development in practice.

[@nicolasd6391]

I am looking forward to understanding this:)

[@pakan357]

I feel exactly the same way. Although I failed both of my Math classes, I learned quite a lot.

[@tggt00]

+Dhiego Bersan Same here except I was 14 when I first watched this and now I'm 16 and I understand everything perfectly.

[@darmodak1773]

+tggt00 bahahahahahah

[@tggt00]

What's so funny?

[@Popexssj]

I love to eat my dinners and watch this dude. His videos are usually long enough for me to finish my food. Too bad I don't understand a single thing he's saying.

[@singingbanana]

I love that I'm somehow a strange part of your daily rituals. I have a similar habit.

[@CafeAlpha]

There's a Japanese club for families to learn a bunch of languages, and the funny thing is that the guy who started the club requires people who join, as their first assignment, read Heisenberg's book on quantum theory. He says that it won't make any sense to you, but that if you're going to learn new languages you're going to have to get used to reading things that don't make sense to you... Though, obviously, some of his members did come to understand Heisenberg's book because they published a book that goes through every single step of proving that Schrodinger's wave equations are equivalent to Heisenberg's version. It was written by students who had just learned it, on the theory that you're better at remembering all the details and explaining something that you've just learned. And it's very Japanese, full of cute little cartoons and asides like how to draw a cartoon the physicist De Broglie, who's face is shaped like a peanut.

[@Snijele]

Popexssj same here bro. :)

[@nge1301]

I dropped out of grade school and almost got this! It got hard after 0:08 though.

[@lightclock9761]

It is brilliant. Clear explanation of Reimann Hypothesis is given, and it is inspiring.

[@FelipeBudinich]

Aaaand, it might be easier to earn a million dollars selling hotdogs :D

[@SD-de4do]

it all depends on how you go about it ;)

[@deeptochatterjee532]

Felipe Budinich Or just borrow a small loan of a million dollars

[@cameronspalding9792]

Deepto Chatterjee And you’d have to pay it back within a long time interval

[@j.vonhogen9650]

Just rob a bank, or build one!

[@technowey]

Great lecture. Your deep knowledge is apparent, but you're also an excellent communicator. Thanks for the videos you've made.

[@ML-uu5ik]

You did an awesome job explaining this. I have been reading/watching videos on this for the last couple days trying to find a sufficient explanation of what in the world the connection is between the zeta function and prime numbers, and you explained it well enough that I am sufficiently satisfied. Thank you!

[@AliElamraniElhanchi]

You sir have a talent for teaching and for communicating information the best way possible. I thank you with all my heart for these explanations and these videos. Keep up the good work!

[@BradleyRobinson]

James, sincere thanks for following through on this request by interested viewers like me, I am familiar with the content but the nuances continue to grow over time. The internet access to information sources like these youtube channels is a stunning contrast to when I was first introduced to it reading a paper back copy of "Stalking the Riemann Hypothesis by Dan Rockmore" around 2005 "by candlelight". Watching this now three or four times over two days was again similarly juxtaposed personally.." very exciting" and a thoroughly helpful addition to the initial "incredible read" experience. The interval between then and now is remarkably influenced and is clearly given me access to real structure by better understanding the mathematics, especially in the margins of addition and multiplication. The theorem's wide appeal to many pursuits is in itself remarkable. In all the video here is "Exquisitely arranged material and lucidly presented with admirable understanding and enthusiasm in a skilled detailed explanation". So much of what excites me about mathematics is how much there is to learn and the many open questions still abound in this wild field... exploring the unknown... and its good to leave a trail!

[@shield543]

Dr James Grime, I salute you for finally making this video! :)

[@singingbanana]

The natural length of this video turned out to be 20 minutes. Normally they're only about 6 minutes. That proves something, but I'm not quite sure what yet.

[@paradoxica424]

singingbanana New proposition: Any attempt to explain a currently unsolved mathematical proposition/hypothesis will take about π times the expected length of time.

[@ClayCompton]

singingbanana It means this video should've been a three-parter.

[@LVBradley2013]

As a pure mathematics major, all I want is to sit down with you and learn even more. Wonderful video as always, Dr. Grime.

[@Boomshicleafaunda]

I'm really happy to see a new video on your channel. Numberphile doesn't always go into the same depth as you do on this channel.

[@ChongFrisbee]

You, sir, are a gentleman and the gratest math comunicator ever. Thank you.

[@JohnSmith-cy8hq]

"..that Riemann decided to rip off, I mean study." I coughed up my chocolate milk.

[@Epicaq]

Whenever I feel too clever, I just tune in on this channel and I feel stupid again. I love how passionate Jim feels towards math. Makes me passionate about it, even though I'm no strong mathematician. Keep it up :)

[@KannanNambiar]

strikingly beautiful, amazing clarity, please do more

[@santafucker1945]

i've discovered a truly remarkable proof which this comment box is too small to contain.

[@liamcarman6256]

Link or it didn't happen

[@mikhailmikhailov8781]

Ferma, pls, no more jokes

[@MrMlindsay]

wait isn't the comment box now infinite. I think that I have either killed a frog or your proof doesn't exist.

[@hemuln]

i think he dropped dead right after writing this. sadly, this theorem is now called the kingslayer's last theorem, and George RR martin is going to sue.

[@santafucker1945]

beary the bear LOL

[@RBuckminsterFuller]

I studied math for a while a couple of years ago and I almost understood all of this. I'm so proud of my brain.

[@taherajna]

same feeling XD

[@eamonnsiocain6454]

You take those topics which were always presented to me in the driest possible ways while I was working on my Maths degree and make them fascinating.

[@hillwin10]

This guy makes even the most daunting subjects seem approachable. It's so much easier to stay excited and learn like this. I wish all teacher were like this, maybe then more people would be interested in maths (perhaps then, people who are naturally drawn to maths and physics wouldn't feel so isolated *sighs*).

[@OhDannyBoy512]

I feel ridiculously out of my depth watching this video. I only feel as though I partially understood this because you led us through it in such a logical and sequential order! Thank you for the brain strain! :)

[@singingbanana]

This one isn't for everyone. I'll do something more accessible next time.

[@michaelbauers8800]

I know this post is old, but don't feel too badly IMO. I have read a book on this, and studied a lot of math in college, and it's not simple to follow. I think I would have been equally lost had I not read a book on this, and remembered the equations. I was confused a few times by the book ( Derbyshire,) and this video is actually simpler, but it's dense. I would benefit from watching it again, I think.

[@twizzler2682]

@@singingbanana The Riemann hypothesis is a distraction The True lie on the Sum of all Natural Numbers The answer to this problem of -1/12 is that Jesus come back to say to everyone "i have 12 Apostles"

[@philyking]

When you say log(x) are you referring to log_e(x)? It seems no one wants to keep consistent notation with it. Also, I'm happy you finally did a more intense video, but I find it sort of funny how people are acting like it was a super difficult video when you glossed over all the actual difficult details. Even the residue calculations can get horrendous.

[@singingbanana]

I sympathise, even the statement is more advance than most people are familiar with. It is the natural log.

[@luismontesdeocafis]

Is a great explanation of a non trivial topic, a wide background is needed but is understood with your guide! Great work! Thank you

[@jcdavid1974]

Beautiful complete explanation. The visuals are very well timed and helpful. Thank you.

[@ShonkyLegs]

I can't believe I just found this channel. I can't believe this is the channel name. I can't believe you have Little Professor as your banner. I'm very pleased.

[@rangedfighter]

great to see you back with a video :D

[@sradtke314159]

Excellent lecture - it brings light into a dark place. Quite enjoyable!

[@alexanderjones1129]

By far the most understandable and interesting explanation of the Riemann Hypothesis.

[@harryc2549]

I love this guy. Always very clear. Great video.

[@InvitingShores]

I cannot state having understood, but I'm greatly inspired to learn more to understand it finally. Thanks James and don't let the gap between consecutive videos be as big as those between big prime numbers! ;-)

[@PuglyWont]

Thanks for making this video. I had the Riemann Hypothesis explained by a professor once, (in a most excellent way too, outside with sidewalk chalk)... but I forgot much of the explanation. I'll have to watch this a few more times to really have it all sink in. I'd like to see more videos like this with the right level of detail without getting bogged down in the formulas too much. I especially like pointing out the important parts of involved formulas.

[@tkk3852]

Good job my brother. If you are still there,thank you. I never understood riemann until today. You made it simple

[@prasadtavva]

This has been solved by a Professor in India recently.

[@arsenelupin123]

Thank for the crash course, James.

[@rangedfighter]

I've read cash course at first

[@kevroy314]

If I came to the UK could I take you out to lunch or something? Through your fantastic presentation style (and I really mean you, not just numberphile) you've managed to get my mathematically agnostic SO presenting proofs to strangers at restaurants. Between this feat of magic and your continued ability to inspire my own insterest in maths beyond a lowly undergraduate minor, you've quickly become one of my favorite people on the Internet. If not lunch, please let me know some other way I might support you!

[@jainaadi]

Brilliant video!! Thank you so much !! I have been trying to understand this for months!

[@gingerfeest]

Don't tempt me to switch my major to mathematics.

[@singingbanana]

***** Perfect reply :)

[@nahidhkurdi6740]

I Wish I had the opportunity to do so when it was my time!

[@Jag0n03]

Picturing Riemann doing a mic drop and saying "Peace out, y'all" had me dying.

[@anonymous1361015]

Thanks for the video, I like the way you explain things with so much enthusiasm

[@SolPhoebusApollo]

I can definitely see why people ask you to explain that one. Sucks about no million for disproving the hypothesis. Great return to posting videos! Those formulas hurt my eyes and that's awesome.

[@hawk0485]

the person who proves it, if he/she has watched this video should go back and leave a comment like: "lol, proved it."

[@ZonkoKongo]

yea gonna do that, just give me some time

[@giuseppeforte8687]

Looks like someone did it

[@Ishuzu]

i have no idea what i just watched but i still enjoyed it.

[@jeffreyluciana8711]

The 5 travels down the grouping of 6 from the 5th position to the first position to the 25 and cancels out the right prime to 24. The 7 travels to the right through the first position back to the first position of 48 and cancels out 49. The squares of primes cancel out other primes. That is why the number of prime numbers less than a certain number follows the inverse of the log

[@MindLessWiz]

Wow thank you for taking the time to make this!! Finally I know what it's all about. :)

[@CompuBOOT]

Are you aware that there is a direct correlation between your enthusiasm and emotions attending this explanation and the actual rigor used in the ramping up of your description? It's almost melodious. What if numbers and math have human emotive and fantastical elements to them? What if math is an endless song or a dance that we can't sing or sway to because we lack the dimensional articulation? Thus it behooves us to merely express things mathematically using one-dimensional verbal or written symbolic gesturing. This idea is only partially reflected in the quiet passion of too few mathematicians or scientists. John Nash understood it.

[@singingbanana]

***** Sounds good to me.

[@doceigen]

+Peter McMillan Agreed. Especially where he spoke about proving that the zeta function has no zeros on the line of the real part of S = 1. Hurrah. Likewise from the WIKI, "In a lecture on prime numbers for a general audience, Fields medalist Terence Tao described one approach to proving the prime number theorem in poetic terms: listening to the "music" of the primes. We start with a "sound wave" that is "noisy" at the prime numbers and silent at other numbers; this is the von Mangoldt function. Then we analyze its notes or frequencies by subjecting it to a process akin to Fourier transform; this is the Mellin transform. The next and most difficult step is to prove that certain "notes" cannot occur in this music. This exclusion of certain notes leads to the statement of the prime number theorem. According to Tao, this proof yields much deeper insights into the distribution of the primes than the "elementary" proofs." Maths are the tools, but music is the spirt.

[@donutazarausg1623]

+doceigen chaos theory proves the relation between you and me, music and me, mathematics and me, using mathematics.

[@louischo2701]

Yay youre making videos again.

[@Wildpfad]

Thank you for this video, I really enjoyed it. Hopefully you'll do more videos like this on interesting problems.

[@philipmylan654]

Even though I didn't understand this enough to understand the Riemann Hypothesis, I still learned a lot from it. More of this so-called 'maths' stuff, please! :)

[@LynneSkysong]

I really think it's pretty crappy that you don't get the prize for disproving it. Wouldn't it be funny if someone disproved it and kept it to themself since there wasn't an incentive to share? Of course, that's impossible because anyone that dedicated to mathematics wouldn't be able to keep it a secret.

[@zzasdfwas]

You'd still be famous.

[@BobbieTheFish]

Oh yeah, I forgot to tell everyone, I disproved this years ago. It was so disappointing that I burned down my office with all my research.

[@rangedfighter]

I'm not 100% sure if he just made this part up or not, or where I could search to get a reference to that fact.

[@BiophysicalChemist]

What? That's incorrect. A proof OR disproof of the Riemann Hypothesis is a solution to the problem. The Millenium prize awards 1 million dollars to anyone who provides a solution to the Riemann Hypothesis, not just an affirmative proof.

[@leonardromano1491]

I think you would become the most loved and at the same time most hated person in maths.

[@ThatSpazChick]

I do not have the education necessary to watch this.

[@gameragedad8953]

Sad thing is, I have the education necessary to watch this, but I don't have the patience to listen to him. I feel like I'm listening to a lecture from a professor asking myself the same question I did then "Why the hell do I even care about this?". No offense to the presenter, he is a great guy and math is his thing and I'm glad he can get excited about it. He's also made some other cool videos that have kept me entertained, but this one is for the egghead nerds for sure.

[@ThatSpazChick]

GameRage Dad I'm the opposite. I have no idea what he's talking about but I love listening to it nonetheless. Most of the reason may be his adorable, awesome accent.

[@ThatSpazChick]

khaled khunaifer You're hurting my brain.

[@mikecheng104mike]

khaled khunaifer yeah... you missed the complex analysis part, we all wish we can LOL

[@ThatSpazChick]

***** Seventeen with the math ability of an eight-year-old.

[@Nai61a]

This is the first of your vids I have seen. I'm a new subscriber; you're obviously barking mad; I don't understand a word. I LOVE it!

[@singingbanana]

They're really not all like this. There was that one where I made a straw kazoo.

[@0730Ender]

This is certainly the most insightful video I've found on the properties of the Riemann Zeta function. I'd just wished you could go more slowly over some of the steps, but they can be filled in with some effort. Great job!

[@connorgibes709]

Every time I watch this, I understand it a little bit more.

[@hirayahana]

What if...the answer to Riemann Hypothesis is the friendship we found along the way?

[@paolaisapunkrocker3149]

This may have just saved my IB HL Mathematics IA! I am so glad I found this amazing video.

[@trejkaz]

When I saw this was about the Riemann Hypothesis, I wondered if the 1+2+3+4+5+6+... = -1/12 thing would find its way in here as well, and was not disappointed.

[@JamesSpeiser]

Looking at graphs of Riemann Zeta, would it be wrong to infer there are some fractal properties of primes?

[@amirabudubai2279]

Probably, but you should rephrase that to prime properties of fractals. Primes show up in almost every aspect of math; at this point it would be more impressive to prove something is not related to the primes.

[@RokeyGames]

I'm a physics student, I have a lot of math as well and I really like math, but I didn't get any of this :P I still really liked the video. James can really make such a thing really enjoyable!

[@0101771434]

I love maths, and Dr james! you made me love maths even more. With love from egypt

[@321joeyboy]

Thank you for putting so much effort and enthusiasme into your videos!!! It's because of you and numberfile that I love math's!! (Sorry for my bad English)

[@singingbanana]

Thank you!

[@romanr9883]

fine ill do it, hold my beer.

[@aeriumsoft]

i thought you drank soda

[@abstractapproach634]

Done yet?

[@markkeilys]

yo, this beer is getting kinda heavy... or at least the glass is.

[@user-mv4oh8yp1y]

30 years later... "Hey father, why are you always holding the beer?" "He's still solving it... Oh my dear"

[@genericteenager8341]

Surely OP will deliver

[@mgregggphone]

How did I miss you channel before? I really like seeing you on Numberphile. Subscribed.

[@singingbanana]

Hello!

[@99hourvideos]

Ok, it took me a few minutes of goggling to realize you're not "Numberphile", just a part of it! But let me tell you, you MAKE Numberphile what it is! Love your enthusiasm and videos!

[@friedrichhayek3683]

They paying one million dollar for this proof is like me paying 0,15$ for Lobster/Champagne dinner.

[@kot_robot8205]

there's a very interesting book "Prime Obsession" by John Derbyshire, it is all about primes, zeta function, the riemann hypothesis, and Riemann himself :D

[@lPlanetarizado]

yeah very interesting book, it also talks about the connection between HR and physics

[@seanoneill2098]

Thank you, been wondering how complex numbers were needed to do PNT proof, am hoping to learn more about this material

[@MadaxeMunkeee]

Wow, this was very easy to follow. Granted, I just got my maths degree and I'm running through a book on the Prime Number Theorem now, but still I think you did a great job. You also gave me more motivation to keep ploughing through all these proofs lol. Thanks!

[@singingbanana]

Thanks. This one does have a higher threshold, but you were the ideal viewer!

[@TomFoster1996]

Could you do a video on each of the clay millennium problems?

[@cosumel]

This is by far the easiest of the six to understand. The rest of them, particularly P vs NP, are so abstract, just the formulae used to write them require a PhD in migraine medication.

[@rosskrt]

@@cosumel i know i'm late as you can possibly be, but anyways. P vs NP is by far the easiest to explain. You can even explain it to a child. Now try that with the RH. It gets abstract and complicated when you start working on it, basically like all the other Millennium Problems.

[@aleggs6019]

in tenth grade atm, almost got the jist of it, hoping I'll be able to understand better as I learn.

[@RSLT]

Great Video. Very well explained the R.H. principles. Fantastic Job!

[@xXxBladeStormxXx]

He's BACK!!!!! We missed you James :)

[@TheSpinTensor]

I have discovered a truly remarkable proof of this hypothesis which this KZbin comment is too small to contain.

[@retepaskab]

I have discovered a counterexample but it was marked as spam.

[@Solitaan]

Even though I have almost no idea what he was talking about, I still feel a little bit smarter just watching it xD

[@paul1964uk]

Thank you James. This was a wonderful talk. It got me starting and stopping the tape excitedly and along the way you succeeded in filling some gaps in my understanding of this all-important problem. (Priceless!) Riemann was a part-time number theorist you say? But then he only stated the problem and couldn't hang around long (poor chap.)

[@javsmo]

This video summarizes in 20 minutes all odd-numbered chapters of "Prime Obsession". I liked!

[@raydredX]

19:17 That's... Lame, I mean I get why, you want to pay the guy for showing up, shake some hands and say some words and being a symbol for the problem. But still that's not how math works, solving the problem means either proving it, disproving it or proving it's "improvability".

[@aleksapetrovic7088]

If you disprove it they'll be bummed out

[@martijnvanweele6204]

Why am I even watching this? I understand about as much about math as a banana! ... *continues watching* ....

[@jupytr1]

Awesome!!!! An absolutely brilliant job explaining the Riemann Hypothesis. I didn't realize though that disproving the RH would fail to win the prize :)

[@sabassegovia3915]

I transferred from a community college and began my first quarter in an actual university in California (UC Davis). I transferred to study math but had no interest in it at all, it was just a way to move on. I wish I began watching your videos earlier, here and on numberphile. They've made some hard to grasp topics much more enjoyable and understandable, such as the video where you deal with the different types of infinity. I look forward to newer videos from you and everyone on numberphile.

[@ragnkja]

The harmonic series sounds nice, at least the first few terms. The seventh is a bit flat, though.

[@BiophysicalChemist]

Do the Hodge Conjecture next! :)

[@joeeeee8738]

I love your videos. Especially the numerphile ones. You should make one video for the 8 problems of the Clay Institute (and if you have time, about 23 Hilbert's Problems). This comment is from Argentina!

[@orestiskopsacheilis1797]

James Grime What a great and approachable explanation to what is perhaps the holy grail of research-no matter the field. I am about to start my PhD in Economics and cool vids on pure math, such as this, give me a bitter-sweet taste as I keep asking myself what if I had chosen a different academic path… Keep up the good work! By the way, do you think you can do a section with these Clay institute problems? Not just the famous ones, like P vs NP or Poincare conjecture (the one solved by Perelman) but even the rest -less well known ones! Thanks!

[@IoEstasCedonta]

9:20 - This isn't strictly true, since he didn't explain the restriction that the function must be analytic.

[@jupytr1]

Oh, sorry:( I hastily hit a thumbs down on you're comment w/o thinking. You are of course completely right.

[@trevormugalu3797]

10 years later, I'm still waiting for the proof that cat equals dog and dog equals cat

[@zimbabweian]

I am currently doing a doctoral degree in Physics where I use the Riemann zeta function all the time. I had no idea of its origin or how interesting it really was, just kinda used it as a tool up until now. Great video, really enhanced my interest on the subject!

[@singingbanana]

That's awesome - thanks!

[@Teld]

Awesome video. I haven't seen you deal with PDE's much, but I would love you to do a Navier-Stokes video like this one, thanks.

[@tabularasa0606]

Sadly been too long since I did maths like this. I'll leave the proof to you :)

[@lolzomgz1337]

11:23 -1/12?! NOT AGAIN!!!!!

[@IlBoiaChannelGamer]

lolzomgz1337 the curse of numberphile

[@joebrinson5040]

Clear, concise explanation. Thanks.

[@SiMyt848]

Thank you! I spend a lot of time speaking with my classmates (we are studing maths) about the series 1+2+3+... after the numberphile's video. I refuse their 2 "proof" (because I lerned in analysis 1 that 1-1+1-1+... doesn't converge, so it isn't equals to 1/2 and in the second video they set -1 in a formula (coming from a geometric series) where |x|

[@unvergebeneid]

Am I correct to assume that those symmetrical non-trivial zeros that you plotted do not in fact correspond to any actually known zeros because otherwise the whole Riemann hypothesis would have been disproven already? Isn't it a bit confusing then to just put them there?

[@matthewcooke4011]

Yes, you are correct. I guess he was just illustrating the fact that IF they were to exist as shown, they would have to be symmetrical about the ½ line. I agree that it may have been a little confusing to some people and he could have made it clear that he was showing hypothetical zeros. But a good video generally.

[@TheWeco]

At that point in the story, we had only achieved that information. It was only later with the Zeta function that we got the idea of them beeing on a single line. Furthermore, that proves a point. Never trust a mathematician, and never ever trust a picture! :)

[@freedomwarrior7734]

I've found an amazing proof of Riemanns hypothesis. But I'm too lazy to write it down.

[@littleflower787]

AnonymousIntellectual Isn't that what Fermat said when he supposedly proved his theorem. Please, tell us. Don't make mathematicians wait for hundreds of years for your wisdom. :)

[@ulmovalar3076]

AnonymousIntellectual u little fermat

[@michaelrobertson9870]

AnonymousIntellectual Best KZbin comment yet.

[@AkshayAradhya]

AnonymousIntellectual I have almost solved it but I have to go feed my cat.

[@theadammz4475]

AnonymousIntellectual You sure, Fermat? You don't want that million dollar?

[@davidsimor4271]

I'm learning about infinite sums at university nowadays im learning chemical engeneering at Budapest. But i just got in love with this kind of maths and i was amazed by riemanns state about the Conditionally convergent sums ! i wish i could count better with complex numbers to do tricks like he did with the dzeta function :D i mean thats amazing that you can get the sums of divergent series i mean if the dzeta function is extended by this way and it has such a huge impact on the real numberline and the prime numbers then i'd say that the sums that can be got out of divergent series can be really useful in other sciences :D!

[@TheMauror22]

The best video about the Riemann hypothesis on the internet!!!

[@EpicUnderscoreJdog]

Is it a british-ism that you call 1/3 "a cubed" instead of a "a third"? The other ones seem the same as North American usage. He does this twice around 5.40.

[@rezwhap]

I'm British and it's not something I recognise. I'd even say he misspoke, but it could be a regional thing.