Euler’s Pi Prime Product and Riemann’s Zeta Function

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What has pi to do with the prime numbers, how can you calculate pi from the licence plate numbers you encounter on your way to work, and what does all this have to do with Riemann's zeta function and the most important unsolved problem in math? Well, Euler knew most of the answers, long before Riemann was born.
I got this week's pi t-shirt from here: shirt.woot.com/offers/beautif...
As usual thank you very much to Marty and Danil for their feedback on an earlier version of this video and Michael (Franklin) for his help with recording this video..
Here are a few interesting references to check out if you can handle more maths: J.E. Nymann, On the probability that k positive integers are relatively prime, Journal of number theory 4, 469--473 (1972) www.sciencedirect.com/science/... (contains a link to a pdf file of the article).
Enjoy :)

[@Kabitu1]

"Euler pushed that infinite sum to the limit.." You deserve a beating for that joke.

[@koenth2359]

Even better, he said 'to the absolute limit' to make it more complex. Is that good enough for a total beat-up?

[@christopherellis2663]

From the Inner Mind, to the Outer Limits!

[@tejnadkarni131]

That was a Parker Square of a joke

[@davutsauze8319]

@@tejnadkarni131 Ah, I feel like you're a traitor, even though I do like Numberphile too

[@jameshoffman552]

This channel and 3Blue1Brown have fantastic visuals that are very helpful. Both also have excellent, clear presentation. No other mathematics channel I've found come close to your skill in communication, which obviously involves a great deal of work to create all the graphis.

[@xarmanhsh2981]

Best channels for insomnia so relaxing

[@bigmouthfisheyes]

The expressions 6n +/- 1 produce all prime numbers greater than three, and many more composite numbers. If we knew exactly where the composite numbers would appear in these sequences, we could infer the location of all of the prime numbers. Am I understanding this correctly? Of what use would this be to anyone?

[@naomioverton7893]

My mind is so blown that I might cry a little later at the indescribable beauty of what I just learned. Thank you for existing Mr. Mathologer.

[@aaronleperspicace1704]

"Take the differences between 1 and the square of reciprocal of every prime number. Multiply them out to each other and take the reciprocal of this product. Multiply this reciprocal by 6 and square root the result. The value thus obtained is the ratio of the distance all around a circle to the distance across it".

[@hybmnzz2658]

@@aaronleperspicace1704 something interesting about this is how mystical pi seems when it is classically defined in terms of circles. No wonder hardcore analysts think of pi as defined based on infinite series and do not appeal to geometry.

[@giladzxc17]

I love how this shows yet another intuitive explanation on 1is neither a prime nor a non-prime.

[@viharsarok]

By all means it's a non-prime.

[@PC_Simo]

@@viharsarok So, it has other factors, beside itself and 1 (which is itself)? I’d like to know more about these mysterious factors. 🤔

[@rudilapa6569]

Professor, these are perfectly paced. They don't go so fast that I lose the chain, but never so slow that it becomes unexciting. It's high level, so there's room for mulling over what's needed to ground the proof. There's always hints of ways to take it further. And the use of the animated background blackboard to do the steps while you explain the ideas keeps it flowing. If you offered this type of freewheeling 'class' at your university it would be packed. And PI is related to the primes?! This is at least as amazing as e^(i*π).

[@Mathologer]

Glad these videos work for you exactly as planned, and thank you very much for saying so :)

[@maurolocatelli3321]

14:00 with the primes < 1.000.000 you get 3.14159254713, 6 decimals correct! (Used Python...)

[@PrincessEev]

This might be my favorite video of yours yet. So much amazing information - especially how the prime product connects to the zeta function. That has always confused me - and I was probably too lazy to look it up - but that it's so simple is just ..beautiful in a weird way. XD

[@davutsauze8319]

"especially how the prime product connects to the zeta function. That has always confused me - and I was probably too lazy to look it up - but that it's so simple is just ..beautiful in a weird way." Ah, me too, I'm glad that I understand it now

[@reggyreptinall9598]

I solved the Riemann Zeta function! I posted the solution, check it out!

[@reggyreptinall9598]

I solved and proved the Riemann Zeta function. I just posted the solution check it out!

[@MrQwefty]

Mate, your pressentation is STELLAR! Seriously, the transitions, coloring, everything! Great job :D

[@abdullahalmosalami2373]

12:00 Woah!! That was awesome!!! I just finished taking my Random Processes course as an Electrical Engineer last semester, and I'm so glad I watched this video after having taken that, cuz that was brilliant!

[@pro4skill]

Please tell me how it was relevant! I'm doing math and physics research

[@KazimirQ7G]

I liked this format... with you floating around the equations. Much better than you presenting into a board, or scaring me with sudden sound effects. Keep it up!

[@calma5869]

I was deeply astonished by Euler's geniusity!! Thank you for introducing his research to me

[@darklink1113]

Your graphics make these lectures. I foresee a time when all math is taught like this. I can't follow half of it, but I can at least follow. Thanks for these

[@MrWkuling]

Thank you... beautiful video, your explanations demonstrate deep understanding and passion at the same time!

[@yash96819]

Great video, amazing explanation of how Euler arrived at that form involving just primes.

[@enzogiannotta]

You are my favourite math channel. I learn a lot from your videos!!!

[@christophersewell6611]

Awesome video! This channel is probably my favourite math subscription :)

[@richardschreier3866]

Thanks for another fun video! It is great that you still make the time to do this work even though the crush of the school term is once again upon you. Looking forward to more fun with zeta in the next instalment!

[@Mathologer]

If only I had a bit more time. It would be great to be able to make more of these videos :)

[@xdingo93x]

Fiiiinally, now I understand how all the prime numbers got in there! Thanks for this video :)

[@FaissalChahrour]

many thanks for this video. it's the first tim i find the Euler's formula so beautiful and could really understand them

[@joelhaggis5054]

I love the Euler product formula. Glad you did a video on it.

[@davidherrera8432]

Beautiful, I'm seeing stars and little numbers arround my head, this is the stuff that make me love math!

[@anitamathur2695]

David Herrera Loma?

[@rustyshackleford1964]

Probably my favorite video! Thank you mathologer!

[@SimonFrank369]

IMHO your math videos are some of the greatest to be found on KZbin!! Thx a lot for your work, Mathologer! And how come you speak German that perfectly!??

[@Meine_Rede]

Hello Mathologer, thanks you so much for your video, which is the most impressive one I have seen in regards to the relation between Euler's Pi Prime Product and the Riemann Function. Just one small remark: I think on 6:21 it should say 1/27^z + ... and not 1/28^z + ... as stated in the video, simply because 3*9 = 27.

[@koenth2359]

I think 32 years ago I was playing about with numbers and I discovered this relation between the prime product and the integer power sum (for real s>1) from a sort of sieve argument. I did not know this was a known result, nor had I ever heard of the Euler prime product or the Riemann Zeta function, but thought it was an interesting result. It was in the days before you could just google things. My father persuaded me to hand over my scriblings, so he could show it to a friended mathematician. Of course I got it back with the flaws in my 'proof' pointed out, without any hint that the result was known. My intuition was right, but I had no idea what a proof required.

[@valor36az]

Amazing explanation I have been trying to understand this forever

[@deathlord109]

I really appreciate your effort in doing these videos :)

[@1DR31N]

Nice and neat explanation. I would have wished to have a Math teacher like you when I was at school.

[@yairalkon4944]

An absolutely great video. Keep up the amazing job!

[@javierrivera7685]

Awesome video, very instructive and clear. I specially liked the probability connection of the zeta function

[@Mathologer]

Glad you like it :)

[@eliesimsch3491]

Hi Mr. Mathologer(for lack of a better name), while I was doing my math homework today, I remembered a problem my math teacher gave me and my friend in 5th grade after we'd finished the work a little early. The problem was as follows: the king's daughter is to be married, and to choose the prince, the king has decided to seat the knights around a very big round table. Then he will start at the first knight, and say "you live." To the second knight, he says "you die." The last surviving knight is the one who wins. The challenge was to make a formula so that you could find which seat you wanted to sit at, to live(given between 1 and infinite knights). I made a little progress, finding that (obviously) even seats are bad, and that the good seat starts at 1, then counts up odd numbers to 2 higher than before, then resets to zero, but I only had a little time and I'm not good at math. In retrospect, I don't think I've heard of any similar problems, which is why I still find it curious.

[@Mathologer]

+Elie Simsch This problem is called the Josephus problem :)

[@erikcools891]

brilliant video, as always. Keep up the good work.

[@Mar184]

8:02 wow, that pronunciation was perfect - well done!

[@pyrrho314]

your channel and videos are awesome, thanks!

[@-fitzy-3335]

Really great video man, keep it up :)

[@balern4]

I have watched many of your videos and I think this was the best one.

[@mrsebakuna]

beautiful story like always, love your presentation. strong logic and very valuable historical facts.

[@abdeljalilpr2033]

Salamo aalaikom..you are posing a very important maths issues ..i want to say that resect you and you have to continue ..respect from morocco

[@obsoquasi]

very nice presentation. This is how math should be taught! Herzlichen Dank!

[@rhc-weinkontore.k.7118]

Toll erklärt und sehr instruktiv. Ich danke dafür!

[@PhilipBlignaut]

Just love your videos!

[@martinullrich655]

Thanks a lot for this wonderful lesson!

[@Hythloday71]

AWESOME insights ! Expanded my knowledge / understanding INFINITELY ?

[@mattdarcy6891]

unreal video keep it up man i love this

[@tutordave]

Very informative. Thanks!

[@gerardomoscatelli8584]

Hello there, great video. I used a fast function I found using a boolean sieve to list primes in Python and listed primes up to n=1,000,000,000. There are 50,847,534 primes and the last one is 999,999,937. Using this list estimated pi with Euler product at 3.1415926536126695 accurate up to the 9th decimal. Can't find more than up to n=1,000,000,000 prime, memory error. It would be interesting to graph how the accuracy of pi evolves with the number of primes.

[@Mathologer]

Greetings from beautiful New Zealand. If you happen to be in Auckland today as part of the Maths craft festival I'll be giving a talk about the best ways to lace your shoes at the Auckland Museum at 5.15 p.m. Come and say hello :) www.canterbury.ac.nz/news/2017/maths-craft-festival-2017-hits-auckland.html As usual if you'd like to support the channel please consider contributing subtitles in your native language.

[@JTX8000]

Welcome to Nz, hope you have an enjoyable stay

[@JRush374]

Please do some videos on fractional calculus!

[@Mathologer]

+Josh I'll put this topic on my list of things to ponder :)

[@michaels4340]

Mathologer I'm not sure whether your proof (with conclusion at 5:16) is complete. Is that really a contradiction? Even if one function is bounded above by another, couldn't they still converge to the same value, as long as the difference between the sums converges to 0?

[@AnarchoAmericium]

You should consider doing a video taking infinite products to infinite fractions (which is again due to Euler), especially condiering you already have videos on infinite fractions you can reference.

[@maciejkozowski6063]

one of the best youtube chanel about math

[@billbill1235]

You should do a full video on the reimann heipothesis

[@jennyone8829]

Thank you for existing 🎈

[@zubmit700]

Best math channel!

[@nezihokur]

I watch your videos with a lot of wow!s. Great fun!

[@Mathologer]

Great, looks like "mission accomplished" as far as you are concerned :)

[@holyshit922]

13:16 To approximate pi i used Euler's formula for acceleration of series convergence to the arctan(1) After acceleration each iteration sets one bit of approximation (10 iterations gives three correct digits)

[@AngryArmadillo]

Fantastic video!

[@jweezy101491]

This was a really great video.

This is so satisfying!

[@satyanarayanmohanty3415]

You are amazing sir.

[@silasrodrigues1446]

For this, you've just got another subscriber!

[@nowhereman8564]

Riemann what a genius was involved in general relativity and prime numbers

[@AltonMoore]

This is a really good channel.

[@sarojsi890]

mind blowing excellent sir keep it up

[@xyz.ijk.]

This is even better the second time around.

[@hamidkh5488]

Thank you very much .

[@aaronr.9644]

Awesome video!

[@jacksonstarky8288]

The number of views of this video compared with, say, the number of views of your Harmonic Series/Gamma video suggests to me that this video is criminally underviewed... which might be why it seems like we're still waiting for that Riemann follow-up video you talked about at the end. I would love to see something like what you hint at, explaining the connection between the Riemann zeta function and the Euler gamma constant.

[@alexandersanchez9138]

10:20 The killer instinct. Mathologer isn't afraid to strike at the heart.

[@tvalladaress]

Wonderful work. Congratulations! By the way, what software do you use to make this animations? I'd like to use it to prepare my lectures. Sorry for my wrighting. English is not my mother language.

[@diegomontalvo9173]

Great video!

[@ronrubin5601]

Hi - your video lecture is truly amazing. I am thinking about adding a video component to some physics courses I teach at Boston College in the US. Could you advise as to what software you use to create these amazing lectures. Thanks a ton if you can!

[@travelion5359]

title seen - and already hyped

[@Achill101]

I've been writing questions in my comments so far, but I want to say also Thank you, Thank, Thank you, for your videos. It's a delight to watch them, especially this one. I hadn't heard it before, because I'm interested in geometry and calculus, not in primes, but I must admit this is a beauty. Now, to my question - sorry I can't let it be ;-) I would be grateful for an answer by everyone, of course, not only by Burkhard. @12:13 - it was said: "at least two of the ingredients of this proof need a little bit more justification and, well, can you tell which?" I cannot tell it now. Could anybody tell me, please, which two ingredients that could be?

[@Achill101]

I like this video a lot. At @9:46 the video seems a bit unclear of what is required to show for a second time that there are infinitely many primes, using equation: pi^2 / 6 = 1 / ( (1 - 1 / 2^2) * (1 - 1 / 2^3) * (1 - 1 / 5^2) * (1 - 1 / 7^2) * ... * (1 - 1 / prime(max)^2) ) with an assumed maximum prime number, prime(max). On the right side is a rational number, and it would follow that pi^2 is rational. But that pi^2 is irrational is a stronger statement than pi being irrational, like the square root of 2 is irrational, but its square, 2, is rational. Therefore, it's not enough to show that pi is irrational, which is difficult enough, but we would have to show that pi^2 is irrational for showing that there are infinitely many prime numbers. I personally find the proof using zeta(1) more beautiful.

[@forestpepper3621]

Comment on 10:45 in video. The concept of a "random integer" is tricky. When you say, "choose one of these items randomly", if there are N items, then usually each item is chosen with probability 1/N. However, the word "randomly" assumes that you have defined a probability function on the items. 1/N is the the "uniform" probability function, but a different probability function would make it more likely to select certain items than others. *PROBLEM*: The integers can not be assigned a uniform probability function, with all integers equally likely, because there are infinitely many. The sum of the probabilities assigned to all items must equal 1, and if all integers had the same probability, these probabilities would sum to infinity. So, in choosing an integer "randomly", you must specify which probability function has been defined on the integers. There is an unlimited variety of probability functions that may be defined on the integers.

[@saahilmehta]

I remember our math teacher gave this to us in a probability question(Find the probability that 2 randomly chosen natural numbers are coprimes). The answer is 6 by pi square

[@pierrecarrette4976]

Illuminating video. Have you ever shooted the video on the alternate inverse squares series? Any link? Thanks.

[@sylviaelse5086]

When z = 1, the manipulations required to get the prime product result require multiplying infinity by integers, and then subtracting one infinity from another, and finally dividing one by zero, just for good measure. The conclusion at around 9:27 that there are infinitely many prime numbers is certainly correct, because we know that to be true, but I'm not comfortable that reasoning there is valid.

[@manuelrodriguez4952]

One error, minute 6:24 the 28 should be a 27. Nice video, thanks for the info.

[@zixuan1630]

I really hoped that you could prove the Riemann Hypothesis.....

[@marekvlasak4648]

😂 No, its ur chance bro.

[@SquirrelASMR]

That was really cool

[@oneofmant]

amazing video!

[@jimjackson4256]

It is about time I subscribe .

[@zhibinren1904]

This is amazing

[@dylanmelvin5206]

Thanks

[@dushyanthabandarapalipana5492]

Thanks !

[@jimcarroll779]

Thank you

[@yiliang2813]

Amazing!!!

[@IllumTheMessage]

Beautiful

[@vikaskaushik8171]

Ultimate beauty of maths ......

[@josephyoung6749]

one of your easier to follow videos for me (just a simple layperson)...

[@pronounjow]

Looking forward to the next video! I think I'm finally starting to understand the Riemann Zeta Function!

[@dlevi67]

Wait until it becomes complex... (or rather, its argument becomes a complex number)

[@buddhirajabuddhiraja6319]

Love it.

[@weidox]

Great. Only that I'm unable to find the next video which was advertised at the end.

[@samuraimath1864]

9:37 Even Euler "saw" that one coming :D

[@matthaiosghanim562]

I've seen no videos regarding the odd powers of this function. Can you do a video on it?

[@Patapom3]

Love you!

[@FreemonSandlewould]

Did you ever make the Zeta function analytic continuation video???

[@bennoheilmann4049]

Hi mathologer, i love your videos! The pi^2/6 is beautiful. Do you know if there is a pretty value (free from infinite sums or products except maybe pi, e or the golden ratio) for zeta (3)?

[@kennethflorek8532]

Euler later did find a fairly simple derivation that gives, for even numbers only, the value of zeta as a rational number times that power of pi. In other words, zeta(102) is an explicit rational number times pi to the 102 power. It was a bit of mathematical good luck that left Euler the possibility to turn up the value for zeta(2). I can't imagine that Euler, and other mathematicians since, haven't done more than ponder whether something like it exists for odd numbers, like 3, but I have never seen any such natural thought expressed in print, pro or con. From that, I take it that whatever they have done shows no indication that it is a rational number times that odd power of pi, or any power of pi. It is possible, though, that professional mathematicians consider such speculations as the realm of crackpots, and do not care to be classed as such, and that explains their reticence. However, sometimes it is mentioned that, not so many years ago, it was proved that zeta(3) is irrational. The nice formula for even values of zeta can be turned into a formula for odd powers, but it would involve the odd Bernoulli numbers, which are zero (beyond 1), and powers of i, which would make zeta for odd numbers seem to be both imaginary and 0. How could you fix that? It is the kind problem you know Euler would love.

[@claudiodeluca2357]

Beautifull!!!

[@tim1878]

Good job.

[@user-cb6rp1ed8k]

6:22 it's supposed to be 1/27^z not 28

[@BehavingBeaver]

כמה משעמם לך אחי

[@dsfdsfsfdsfdsfd]

dmagen1994 אותה כמות שמשעמם לכולנו 🙂

[@TiloDroid]

lol, made the exact same mistake in a math exam. I also assumed 28 was dividable by 3

[@sunshinedaniela8572]

Yeah

[@justitroyal7032]

Like I want it to be 69 so I am replying

[@martin622292]

brilliant