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The Riemann Hypothesis, Explained

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. 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 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: https://www.quantamagazine.org/how-i-... - VISIT our Website: https://www.quantamagazine.org - LIKE us on Facebook:   / quantanews   - FOLLOW us Twitter:   / quantamagazine   Quanta Magazine is an editorially independent publication supported by the Simons Foundation: https://www.simonsfoundation.org/