In this video, I present Euler's proof that the solution to the Basel problem is pi^2/6. I discuss a surprising connection Euler discovered between a generalization of the Basel problem and the Bernoulli numbers, as well as his invention of the zeta function. I explain Euler's discovery of the connection between the zeta function and the prime numbers, and I discuss how Riemann's continuation of Euler's work led him to state the Riemann hypothesis, one of the most important conjectures in the entire history of mathematics.
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Sections of this video:
00:00 Intro
01:24 Euler's Basel proof
23:20 The zeta function and the Bernoulli numbers
32:01 Zeta and the primes
48:15 The Riemann hypothesis
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Further viewing from 3Blue1Brown:
Why is pi here? And why is it squared? A geometric answer to the Basel Problem https://youtu.be/d-o3eB9sfls
Taylor series https://youtu.be/3d6DsjIBzJ4
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Further viewing from Mathologer:
Euler's real identity NOT e to the i pi = -1 https:/youtu.be/yPl64xi_ZZA
Euler's Pi Prime Product and Riemann's Zeta Function https://youtu.be/LFwSIdLSosI
Ramanujan: Making sense of 1+2+3+... = -1/12 and Co. https://youtu.be/jcKRGpMiVTw
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Thanks go to Keith Welker for our theme music. https://www.lunarchariot.com
Some of the animations in this video were created with Manim Community. More information can be found at https://manim.community