The Riemann Hypothesis, Explained

5,060,613
0
Published 2021-01-04
The Riemann Hypothesis is the most notorious unsolved problem in all of mathematics. Ever since it was first proposed by Bernhard Riemann in 1859, the conjecture has maintained the status of the "Holy Grail" of mathematics. In fact, the person who solves it will win a $1 million prize from the Clay Institute of Mathematics. So, what is the Riemann hypothesis? Why is it so important? What can it tell us about the chaotic universe of prime numbers? And why is its proof so elusive? Alex Kontorovich, professor of mathematics at Rutgers University, breaks it all down in this comprehensive explainer.

00:00 A glimpse into the mystery of the Riemann Hypothesis
01:42 The world of prime numbers
02:30 Carl Friedrich Gauss looks for primes, Prime Counting Function
03:30 Logarithm Function and Gauss's Conjecture
04:39 Leonard Euler and infinite series
06:30 Euler and the Zeta Function
07:30 Bernhard Riemann enters the prime number picture
08:18 Imaginary and complex numbers
09:40 Complex Analysis and the Zeta Function
10:25 Analytic Continuation: two functions at work at once
11:14 Zeta Zeros and the critical strip
12:20 The critical line
12:51 Why the Riemann's Hypothesis has a profound consequence to number theory
13:04 Riemann's Hypothesis shows the distribution of prime numbers can be predicted
14:59 The search for a proof of the Riemann Hypothesis

Read more at Quanta Magazine: www.quantamagazine.org/how-i-learned-to-love-and-f…

- VISIT our Website: www.quantamagazine.org/
- LIKE us on Facebook: www.facebook.com/QuantaNews
- FOLLOW us Twitter: twitter.com/QuantaMagazine

Quanta Magazine is an editorially independent publication supported by the Simons Foundation: www.simonsfoundation.org/

All Comments (21)
  • @kabauny
    My math professor once said, “I’ve know the existence of these math problems for many years. And I assure you, there are a lot easier ways to make a million dollars”
  • @weimondo
    I have discovered a truly marvellous proof of this, but it's much too large for this youtube comment to contain. Therefore it is left as an exercise to the reader.
  • I really appreciate that you explain the more “basic” things (e.g. what a log function is). It makes the video feel welcoming to people who aren’t necessary very good at math (like me, lol)
  • This is like becoming an astronaut, discovering a previously unknown planet, finding a river on that planet, and at the bottom of the river is the perfectly fitting other half to a broken rock you found in a river on Earth as a kid. The Universe sees the look on your face and laughs silently.
  • @artisorak
    Proving the Riemann Hypothesis is probably one of the hardest ways to make a million dollars.
  • Whoever does these animations, massive props to you. These are literally the best math illustrations I've ever seen.
  • For the first time in my 46 years, I have truly understood what the Riemann Hypothesis actually is. Thank you!
  • @whatthepi
    I'm amazed by Riemann, Euler, Gauss and other mathematicians/physicists how their brain and curiousity for math and science managed to find these sort of algorithm and new fundamentals that we even use today. Amazing vid, love your animations!
  • @joserojas9876
    Thank you, Quanta Magazine. My understanding of the Riemann Hypothesis went from 0% to 15%. Great job (I mean it).
  • @southbayjay2540
    Literally if my math teacher had just said “logarithms are to exponents what division is to multiplication,” I would have had much less trouble with them. Thanks dude
  • @hallu6666
    When pure mathematics comes with lucid explanations, and the two are complemented by a perfect vanilla icing of aesthetic graphics. A million thanks for this amazing presentation.
  • Watched this a few months back. A few months of studying maths rigorously later, and I can finally start to appreciate how magnificent this is
  • Reimann, gauss, euler and all other guys did all this stuff without matplotlib😳
    I can't even imagine the extent of their hardwork and dedication
  • @setmason1510
    hold my beer, I got one A in math in high school, I got this
  • Hats off to Kontorovich sir. He explained such a complicated topic in a very simple manner. I just want to develop this skill.
  • I can't believe I understood this.
    I've heard about this for years, but this is the first explanation I've seen that makes sense.
    Great video.
  • @yunooooo_
    Can I just appreciate how well the animation is? Literally, WOW.
  • @JS-rt7kp
    If there was a video like this for every math concept, I would never take my eyes off the computer screen.
  • @napalm5
    This is a truly fantastic video. The gradual exposition to the very casual person interested in maths is top notch
  • @ADPuckey
    I love this video! The explanation of the hypothesis itself is great, with all the right context and friendly explanations, but more importantly I've never seen such a concise yet captivating description of why the hypothesis matters