What is the Riemann Hypothesis REALLY about? 

One mistake in this video: It is not true that Riemann knew the first 3 roots and made the hypothesis. He had over 600 zeros of this type, because he used what is known to us all as the Riemann-Siegel formula rediscovered really long after Riemann's death (the zeros are really difficult to compute without a computer, and 80 years after death of Riemann the world of mathematics knew less than 100 first nontrivial zeros. It was quite a shock to discover that Riemann himself had calculated many more than the rest of the world)

“Ok guys, if this is the first time you heard of these ‘imaginary numbers’, let’s talk of a simple topic involving them: the Riemann Zeta Function” That was a hell of a leap!

My mind exploded when you showed how the Riemann Conversion of the subtraction of the pole in s = 1 and the non-trivial zeros of the Riemann Zeta Function approached the distribution function of primes 🤯

For those wondering, the zeta function has a reflection formula such that the zeros in the critical strip have reflection symmetry across the critical line. i.e. say if s = 0.49 + 100i is a zero, then so is s = 0.51 + 100i. And it's that zero with the real part greater than 1/2 that would mess up that x^(1/2) error bound.

Holy. Shit. This video is CRIMINALLY underwatched. Sharing it far and wide. I am a math phd (now in a different field) and, although I studied analysis, it is astounding that no one ever could explain to me, as well as you just did, how the Riemann Hypothesis actually matters to the study of prime numbers. Years of casual lectures and conversations. No one approached the explanation with your clarity. I have absolutely crazy respect for your ability to communicate this. Just. Wow.

I've been looking for something like this for a while. I always wanted a Riemann hypothesis video that went a bit more deeply into the math. The concrete examples were really helpful too; like doing the error calculation for pi(10^50) or showing the sum of the first 200 harmonics. Great stuff

Excellent video! I always found it kinda frustrating for math popularization that _the_ million-dollar question was not only so hard to explain (see the 3b1b video) but also even harder to understand why mathematicians care (I mean, you basically need an entire semester of analytic number theory to go through all the details of this connection between Riemann zeta and primes). Kudos to you for being able to boil it down with some incredible animations!

Explaining the basics of complex numbers and RH in one video. Man, you're a brave soul.

Without the knowledge from ring theory, people will never understand the true deepness of primeness as a general notion.

Thank you for making this. This is the only video I have seen that actually explains HOW the Zeta function actually contributes to primes in how it is constructed. All the other have just said "this allows you to know more about primes", but this was really clear and informative, thank you.

Thank you for another excellent video! ^_^ If I may attempt to add something of value, Robin's Theorem is a relatively easy to understand statement about an inequality whose truth for all positive integers greater than 7 factorial (5040) is equivalent to the truth of the Riemann Hypothesis. That is, sigma(n) < n * ln(ln(n)) * e^gamma The inequality holds that the sum of the divisors of an integer, n, is less than the product of n with the natural logarithm of the natural logarithm of n, as well as e raised to the Euler-Mascheroni constant "gamma", with the 27 exceptions of 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 36, 48, 60, 72, 84, 120, 180, 240, 360, 720, 840, 2520, or 5040. Robin's Theorem states that the Riemann Hypothesis is true if 5040 is the final exception, and is false if there are any more. So, for those really interested in that $1,000,000, here is another way to approach it.

This is the best video on Riemann hypothesis I've seen on YT. Congratulations on explaining it in-depth yet in simple terms.

I've seen many videos on this, and quite enjoyable they were too, but this is the first one that explains how all the bits fit together. Thank you!

Excellent. First video of yours I have seen. I am a retired electrical engineer that hung out with physicists my whole life. I find mathematics fascinating, luckily I have good math skills and I thought I knew a fair bit about the Riemann function and hypothesis. The stuff here is refreshing and new to me. Great job!!!

I'm very moronic when it comes to math but this was a joy to watch. I didn't fully grasp 5% of what you showed here but it made me want to understand more, seeing the prime frequencies emerge from the subtraction was absolutely beautiful, thank you!

I´ve been watching some math related videos lately and they´ve re-awakened my interest and curiosity about mathematics, physics and other related stuff, since they´re mostly entertaining and fun to watch, while being very informative a sparkling. I´m a civil engineer and wish at least some of my professors in college were like these youtubers. Thanks!

Totally watchable, well done in showing how to present clearly a complex subject to anyone, thank you.

Great video. The first thing I've seen that does a good job of explaining why even a single zero off the critical line would be disastrous for results that depend on RH.

Riemann must have been an incredibly brilliant and intelligent man, to have seen all this with only 3 zeroes and no computers to work with. What an amazing genius

Another nice connection between Riemann zeta-function and prime numbers is how the infinite sum of 1/n^s can be represented as an infinite product of 1/(1-1/p^s), where p goes over all prime numbers. Products of such kind are also known as Euler products.

Best and most accessible summary of the subject I have seen. Great graphics!

A really wonderful and valuable video. So many videos about the Zeta function skip how one interprets zeros to determine the number of primes below a given value -- this one does not make that mistake. Great work!

Amazing explanation, also great flow of ideas through the video. Also, nicely use of graphics ! Thanks for sharing this amazing knowledge with non-mathematicians! 🔥

Thank you, this helped me 'understand' the Riemann Hypothesis much better than anything else I've encountered.

I honestly have to thank you for this video as in no book or other video have I ever found such a clear explanation of what the whole endevour is about other than mentioning the fact that "if the RH is true, we'll know a lot about prime distribution"

This video is incredible. As an amateur math enthusiast (took nothing beyond ordinary differential equations), the mathematics behind the Riemann Hypothesis are well beyond me. This makes it much more approachable.

Fantastic video! I have watched multiple RU-vid videos on the Riemann hypothesis and this is the clearest and best one I've found at explaining precisely how the hypothesis relates to primes. Great job!

Best video on the topic (and I've seen many). Amazing work!

After watching at least 5 videos, I finally have a better understanding of the connection of the zeta func. to the prime numbers, thank you!

as a non-mathematician, I find it quite interesting and even mind-blowing. Thank you for your effort to present the material in an entertaining way.

4:25 is a bit misleading. This standard Mellin transform representation of the Riemann zeta function only converges for Re(s) > 1, like the standard series expression for zeta. Thus, saying that we are trying to find the roots of this representation is misleading since the zeros of zeta are all behind Re(s) = 1 in real part.

The most accessible explanation I've ever seen (from someone that has a bit of maths). Congratulations and thank you.

This is really one of the best explanation of RH, I need to watch it again and again, great job, many thanks and waiting for more.

This is the best Maths video I have seen on RU-vid. Well done.

Absolutely fantastic production quality! The sound, the animations, and… (what did I forget 🤔) Oh yeah! The *CONTENT* 🤣 Thank you for this gem! 🤗

This was very well presented, and honestly, I didn't think anybody would do it better than 3b1b, but you done did it...

If you instead add simple periodic waves for all the nontrivial zeroes (a Fourier transform), you get a result that has sharp spikes at all the POWERS of primes, that is, all p^n where p is prime and n is a positive integer (plus a continuous component). It is very odd.

Great work! This is the most amazing video about RH I have ever seen (i see about tens of video related to RH)! With reading the math bestseller Prime Obsession they are fully understand the meaning of the math problem and its beauty!

Very good video. Number theory is fascinating stuff. A very good read on this topic is Prime Obsession, a book about the technical aspects of the Zeta function, as well as Riemann's life.

Best video on this topic so far... thank you!

Incredible video. I now understand the importance of RH so much better. Thanks man

Best explanation I found on YT. Great job!

another amazing video, the animations with the harmonics were incredibly didactic - not to mention pretty! This should be shown in classrooms!

amazing video. the best video on the riemann hypothesis. I'm glad you didn't show it as a infinite series. I learned more from it that way. just a rlly good vid dude. idk I rlly enjoyed learning a bunch of new things

Fantastic explanation of concepts with a gentle guide to the symbols and now I am very curious to know more of the Reimann Converter...

Very good job. You had to talk fast, but you got a great deal of information out pretty clearly. The graphics were necessary and first rate. Never boring and you held your direction well by not running down every complication, but not ignoring them either.

Thank you! I've never seen this explained so well.

very well explained - first time i've understood any part of the Riemann Hypothesis!

Man oh man...brilliant presentation.....please keep up!!

Great video! I finally get how the zero's of the Zeta function relate to the prime numbers.

Thanks for this video. It really helped me understand more about this problem - although still a lot I don't yet fully get!

I have been looking for something like this for so long Thanks! 🙏

Fantastic video! Thanks for putting in the effort to make this :)

This is amazing! Hands down best video on the topic I've seen (and that means better than 3b1b which is saying something!)

Amazing video! Always amazed by the beauties of mathematics

Very good video. Nevertheless, there is one very important omission: the Euler product, which relates the primes to the zeta function. Saying primes and the zeta function are not linked from the start is misleading. In fact Riemann, in his original 8 page paper on the subject, begins with this amazing mathematical relationship. By extending the variable s to include complex numbers he arrives at his extraordary results. Historically it is after this work that it started to be called the Riemann zeta function. So from the beginning primes and zeta are linked. Without a doubt Riemann would have not gotten very far without this deep connection discovered by Euler. Everything springs from masterfully manipulating this mathematical identity. Another thing is that he gave little importance to the what later became known as a famous hypothesis, he does not say it is such a thing. Riemann simply mentions in passing that maybe all the complex zeros are on the line but quickly moves on, basically saying it is not the aim of his paper to find that out.

best video on youtube abou RH. Great visuals and explanation. Kudos to you sir

By far the best video on the RH on youtube! Thank you :)

7:54 I am a physicist and I don't think that the term "imaginary numbers" is misleading. Physicists use them a lot, but just because they make some computations easier. But they are never measured, unlike real numbers. Also even in quantum mechanics, where they appear in the wave function, one could use R² instead C to get rid of them as there is an isomorphism between R² and C. This isomorphism is used for example for all the diagrams of complex numbers used in this video (by identifying the real numbers with the x-axis and the imaginary numbers with the y-axis).

I've read Derbyshire's "Prime Obsession", which is about the RH multiple times, but you have insights here that I did not get from that excellent book. I'm really glad I checked this out!

Thanks for the video. It is both accessible and in-depth.

Thank you so much for making this !

You deserve millions of views!!❤

Legend... Thanks for putting this together

This was brilliant! Thank your for the great video

Omg. This the explanation that I have been looking for. Thank you.

this is a fantastic video! thanks so much for your effort. ❤

I have a very loosely formed idea that has been kicking around my head the last few days related to this. It came about while I was playing around with the idea of a "unit circle" contained within only the positive real numbers x-axis and y-axis with an diameter of infinity (centered at 1/2♾️,1/2♾️) and hence an infinite circumference. This was mostly just a fun little mental lark for me into investigating the intersection of unity, infinity, zero, the infinitesimal, and their identity relationships, which then begin to branch into the possible relationships to primes and calculated precisions of pi when viewing the path along that circumference as the real number line starting from the points where the x or y coordinates equaled either 0 or infinity and calculating the arc lengths of sections of the that infinite circumference circle bounded by some whole number along either axis or working the other way from whole number arc lengths to where they fall on the axii. Since all of this has been primarily a thought experiment I began to get a bit into the weeds as far as the limits of my intuitive imagination so I need to begin working it out on paper to get a full picture and solidify some concepts I seem to be encountering, but the thought of the nontrivial zeros of the zeta function popped into my head unbidden several different times as looking like what an infinite circumference circle bounded arc length looks like when viewed with such a "unit circle", and I am starting to thing that if I take it seriously and take the time to work it out on paper and bring some other concepts into play like the complex numbers, natural log, etc. that I may be able to come up with at least an amateurish proof of why all the non-trivial zeros lay along the real part of 1/2 and that they indeed actually must neccessarily do so. Anyone think this is worth pursuing further, or no?

what a great goddamn video. So well edited too!

Great Job!!!! very Informative with well-explained easy method.

Absolutely stunning video.

This is the best video on this topic !!!

A great explanation of the problem!

Best Zeta explainer yet. 👍

Excellent presentation.

i've watched prolly 10 videos explaining the Riemann Hypothesis and this is my favorite one. Very well done!!!

This is a fantastic video. I’m not a mathematician, but I’ve been curious about RH for awhile now from a layman’s perspective, mainly a result of reading a book called The Humans and then going down the RH rabbit hole. I’ve read several articles and watched several videos, and I think I had a reasonably good layman’s grasp of the Riemann prime counting function, but one thing I couldn’t get a handle on was just why a real part ½ was so important (as opposed to just any non-trivial zero). This video, with its graphics, does a really good job of simplifying things for people like me - it really does prove the statement that a picture is worth a thousand words. One thing - the video focuses on what happens if zeros are found with real part > ½, but doesn’t really address what happens if < ½. I think it would help to explain that, as John Chessant pointed out in his comment, and to use his wording, “zeros in the critical strip have reflection symmetry across the critical line. eg. say if s = 0.49 + 100i is a zero, then so is s = 0.51 + 100i. And it's that zero with the real part greater than ½ that would mess up that x^(½) error bound”. And to expand on that a bit more (hopefully someone will correct me if I’m wrong about this), at 12:04 in the video, waves with the same imaginary part but different real parts have the same frequency but different amplitudes. So not only would the wave with the real part larger than ½ mess with the bound, I think when you add the smaller wave too (as described at 12:37), that would mess it up even more. (Again, I hope I’ve got that right.)

This is the best. Kudos!!

What an amazing video! Thank you sooo much ❤❤🙏🙏

Congratulation! excellent explanation

best math video ever! amazing work! thanks!

Really nice video! I love it!

Amazing video. For those who speak spanish, Mates Mike has a very good video on RH as well, which I feel would complement this one very nicely

Awesome explanation!

2:00 pi of n makes sense because pi indicates periodicity (count) of wave cycles. Primes appearing in the harmonic series is another clue on why pi is useful here. Primes themselves are numbers that can't be divided other than themselves, so it would be natural that they appear at each higher harmonic frequency since they can't be imbedded in simpler ratios.

It’s actually not because they couldn’t choose another letter but, prime in Greek is spelled πρώτος. Great video and content!!

Another great video!

Really liked how acessible the video is

Excellent video. I heard and read a million times that the zeta function had "something to do" with primes but watching the sum with the "Rieman converters" approach the prime distribution function was really my aha moment that brought everything together. One detail: the definition of the Riemann converter contains a function μ(n) that is not defined, unless I am missing something even after re-watching multiple times. What is μ(n)?

Thank you, sir. Fantastic!

fantastic video, loved it

Tiny detail: imaginary numbers are not the same as complex numbers. Imaginary numbers are multiples of i. Complex numbers are combinations of a real and imaginary part. I.e. imaginary numbers are complex numbers with real part zero.

This is the first video to make me understand how everything is really connected.

I literally had my mind blown. Good video.

"you certainly have the right to be mad at me for just claiming I exist without any explanation" 🤣

8:25 "C ≅ R[X] / (X² + 1)" should be with a *forward* slash (quotient of the polynomial ring by the principal maximal ideal (X² + 1)).

Great video! Very interesting

The integral representation shown here is actually derived using the infinite series formula for the zeta function, so it is also only defined for Re(s) > 1.

These are some of the crispest animations I have seen in my life, bravo.

This deserves like a million views