This equation blew my mind // Euler Product Formula
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There are many beautiful math equations, but in this video I want to make the case for the Euler Product formula. In 1737 Euler proved this crazy formula that compares a sum over natural numbers with a product over all prime numbers - weird! We'll see a proof using the Sieve of Eratosthenes, see the Basel problem fall out, extend beyond the proof of infinitely many prime numbers to the sum of reciprocal primes, and talk about the Riemann-Zeta function and how it connects to the Riemann Hypothesis, one of the million dollar Millennium Problems.
0:00 Euler Product Formula
2:56 Proof of the Formula
7:52 s=1 and Reciprocal Primes
12:30 Reimann-zeta function and Riemann Hypothesis
15:48 Brilliant.org/TreforBazett
Some references:
A nice intro to analytic number theory: https://dms.umontreal.ca/~andrew/PDF/...
Contains the inequality I skipped in proof of the divergence of reciprocal primes: https://citeseerx.ist.psu.edu/documen...
Article on beautiful equations chosen by a range of mathematicians: https://www.scientificamerican.com/ar...
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