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

Uploaded September 2017 | Updated September 2026, 2 weeks ago

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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/beautiful-pi

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) sciencedirect.com/science/article/pii/0022314X72900388 (contains a link to a pdf file of the article).

Enjoy :)

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

$Pi is IRRATIONAL: animation of a gorgeous proof NEW (Christmas 2019). Two ways to support Mathologer Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details) This video is my best shot at animating and explaining my favourite proof that pi is irrational. It is due to the Swiss mathematician Johann Lambert who published it over 250 years ago. The original write-up by Lambert is 58 pages long and definitely not for the faint of heart (http://www.kuttaka.org/~JHL/L1768b.pdf). On the other hand, among all the proofs of the irrationality of pi, Lamberts proof is probably the most natural one, the one thats easiest to motivate and explain, and one thats ideally suited for the sort of animations that I do. Anyway its been an absolute killer to put this video together and overall this is probably the most ambitious topic Ive tackled so far. I really hope that a lot of you will get something out of it. If you do please let me know :) Also, as usual, please consider contributing subtitles in your native language (English and Russian are under control, but everything else goes). One of the best short versions of Lamberts proof is contained in the book Autour du nombre pi by Jean-Pierre Lafon and Pierre Eymard. In particular, in it the authors calculate an explicit formula for the n-th partial fraction of Lamberts tan x formula; here is a scan with some highlighting by me: http://www.qedcat.com/misc/chopped.png Have a close look and youll see that as n goes to infinity all the highlighted terms approach 1. Whats left are the Maclaurin series for sin x on top and that for cos x at the bottom and this then goes a long way towards showing that those partial fractions really tend to tan x. There is a good summary of other proofs for the irrationality of pi on this wiki page: https://en.wikipedia.org/wiki/Proof_that_π_is_irrational Todays main t-shirt I got from from Zazzle: https://www.zazzle.com.au/25_dec_31_oct_t_shirt-235809979886007646 (there are lots of places that sell HO cubed t-shirts) lf you liked this video maybe also consider checking out some of my other videos on irrational and transcendental numbers and on continued fractions and other infinite expressions. The video on continued fractions that I refer to in this video is my video on the most irrational number: https://youtu.be/CaasbfdJdJg Special thanks to my friend Marty Ross for lots of feedback on the slideshow and some good-humoured heckling while we were recording the video. Thank you also to Danil Dimitriev for his ongoing Russian support of this channel. Merry Christmas!$

500 years of NOT teaching THE CUBIC FORMULA. What is it they think you cant handle?

2000 years unsolved: Why is doubling cubes and squaring circles impossible?

$The Pigeon Hole Principle: 7 gorgeous proofs Lets say there are more pigeons than pigeon holes. Then, if all the pigeons are in the holes, at least one of the holes must house at least two of the pigeons. Completely obvious. However, this unassuming pigeon hole principle strikes all over mathematics and yields some really surprising, deep and beautiful results. In this video I present my favourite seven applications of the pigeon hole principle. Starting with a classic, the puzzle of hairy twins, we then have a problem with pigeons on a sphere, a pigeon powered explanation of recurring decimals, some party maths, a very twisty property of the Rubik’s cube, a puzzler from the 1972 International Mathematical Olympiad, and, finally, what some people consider to be the best mathematical card trick of all time. 00:00 Intro 01:49 Chapter 1: Hairy twins 06:46 Chapter 2: Five pigeons on a sphere 08:16 Chapter 3: Repeating decimals 13:14 Chapter 4: Partying pigeons 17:00 Chapter 5: Repeating Rubik 22:20 Chapter 6: Pigeons at the Olympiad 26:18 Chapter 7: The best mathematical card trick ever 31:24 Supporters Here are some links for you to explore. A scanned copy of Récréation mathématique: Composée de plusieurs problèmes plaisants et ... by Jean Leurechon on Google books. For the hair puzzle check out page 130) https://tinyurl.com/3b6amaxk The Pigeonhole Principle, Two Centuries Before Dirichlet by Albrecht Heeffer and Benoit Rittaud A very nice article about the origins of the pigeon principle and the hairy twins problem. Also features an English translation of the relevant page in Récréation mathématique https://tinyurl.com/hpkcuepx The 4/5 pigeons in a hemisphere puzzle was problem A2 of the 63rd Putnam competition in 2002 https://prase.cz/kalva/putnam/psoln/psol022.html Why are repeated decimals fractions? Watch this video on why 9.999... =10 for a big hint https://youtu.be/SDtFBSjNmm0 Or just skip straight to the answer https://en.wikipedia.org/wiki/Decimal_representation#Conversion_to_fraction If you dont own a Rubiks cube you can use this simulator to test what happens when you repeat some algorithms (move the faces using the keyboard) https://ruwix.com/online-puzzle-simulators/ The website of the International Mathematical Olympiad. https://www.imo-official.org . The problem I am considering in this video is Problem 1 of the 1972 olympiad. You can download all the problems from here. https://www.imo-official.org/problems.aspx Check out this very nice article about the Fitch Cheney five-card trick by Colm Mulcahy https://tinyurl.com/wttkfdwe Todays music is English Country Garden (and as usual Morning Mandolin at the end) from the free YouTube music library. Todays t-shirt I got from here https://www.theshirtlist.com/pizzibonacci-t-shirt/ Enjoy! Burkard$

$How did Ramanujan solve the STRAND puzzle? Todays video is about making sense of an infinite fraction that pops up in an anecdote about the mathematical genius Srinivasa Ramanujan. 00:00 Intro 04:31 Chapter 1: Getting a feel for the puzzle 08:27 Chapter 2: Algebra autopilot 12:37 Chapter 3: Infinite fraction 17:51 Chapter 4: Root 2 21:19 Chapter 5: Euclidean algorithm 30:15 Chapter 6: The best of the best: 17/12 36:34 Chapter 7: Outramanujing Ramanujan This was supposed to be a short video but in the end turned out to be quite a tricky to sort out. Anyway, as it sometimes happens, I got carried away and now the video really covers a lot of ground : Pell equations, visualising continued fractions by dissecting rectangles into squares, the relationship between continued fractions and the Euclidean algorithm, the irrationality of root 2. Overall quite a few things that you wont find anywhere else :) The way I tell the anecdote in this video is based on the following account by Ramanujans friend Prasanta Mahalanobis: Current Science, Vol. 9 (3), pp. 74-75. On another occasion, I went to his room to have lunch with him. The First World War had started some time ago. I had in my hand a copy of the monthly Strand Magazine which at that time used to publish a number of puzzles to be solved by the readers. Ramanujan was stirring something in a pan over the fire for our lunch. I was sitting near the table, turning over the pages of the Strand Magazine. I got interested in a problem involving a relation between two numbers. I have forgotten the details but I remember the type of the problem. Two British officers had been billeted in Paris in two different houses in a long street; the two numbers of these houses were related in a special way; the problem was to find out the two numbers. It was not at all difficult; I got the solution in a few minutes by trial and error. In a joking way, I told Ramanujan, Now here is a problem for you. He said, What problem, tell me, and went on stirring the pan. I read out the question from the Strand Magazine. He promptly answered Please take down the solution and dictated a continued fraction. The first term was the solution which I had obtained. Each successive term represented successive solutions for the same type of relation between two numbers, as the number of houses in the street would increase indefinitely. I was amazed and I asked him how he got the solution in a flash. He said, Immediately I heard the problem it was clear that the solution should obviously be a continued fraction; I then thought, which continued fraction? And the answer came to my mind. It was just as simple as this. There is a complete digital archive of The Strand magazine. You can find the page with the puzzle here: https://tinyurl.com/y2lnb8xf (page 790) If you read the puzzle in the Strand youll find that the problem is actually phrased somewhat differently to what Mahalanobis remembers and Mahalanobis also does not spell out the infinite fraction that Ramanujan came up with. And if you do the math(s) some of the other things he says also dont quite sound right. What I am presenting in this video is my best guess for what really happened. In particular, the continued fraction that I am talking about in video is probably the most natural candidate for Ramanujans infinite fraction, but others have argued that it could have been a different continued fraction (which I dont buy :) You can find these other infinite fractions here: Ramanujans Continued Fraction for a Puzzle by Poo-Sung Park https://tinyurl.com/yyfdscgr and here On Ramanujan, continued fractions and an interesting street number by John Butcher https://tinyurl.com/yy6nv2yg Solution to the red cross puzzle from Dudeneys book Amusements in Mathematics p. 168 :) https://imgur.com/a/bBuLOZN Another interesting way to systematically search for solutions to the Strand puzzle is this: The equation we want to solve is 2 x^2=y^2+y. You can rewrite this as x^2 = y(y+1)/2. The formula on the right is just the formula for 1+2+3+...+y. So just keep adding 1+2+3+... and at every step check whether the number you get is a square ... :) Other short formulas: 1) Expanding (1+√ 2)^n gives a number a+b√2. Then a/b is the nth partial fraction. 2) Play with powers of the matrix {{2, 1}, {1, 0}} Some number Easter eggs are hiding on this slide https://youtu.be/V2BybLCmUzs?t=1402 :) Link to the unlisted Marching Squares video: https://youtu.be/f1yDExNAEMg Here is a version of the t-shirt I am wearing: https://tinyurl.com/y5vgo7zb This one is about that other famous Ramanujan anecdote: https://tinyurl.com/y626c86x actually features prominently in another one of my videos. The music in this video is by Chris Haugen, Fresh Fallen Snow (playing in the video) and Morning Mandolin (for the credits) and Nate Blaze Tis the season, all from the free YouTube music library Enjoy! Burkard 14.9.2021: Thank you very much Michael Didenko for your Russian subtitles.$