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    • Why is pi here?  And why is it square...  

Taylor series    • Taylor series | Chapter 11, Essence o...  

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Further viewing from Mathologer:

Euler's real identity NOT e to the i pi = -1 https:/   • Euler's real identity NOT e to the i ...  

Euler's Pi Prime Product and Riemann's Zeta Function    • Euler’s Pi Prime Product  and Riemann...  

Ramanujan: Making sense of 1+2+3+... = -1/12 and Co.    • Ramanujan: Making sense of 1+2+3+... ...  

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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