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This equation blew my mind // Euler Product Formula

Explore more beautiful equations at https://brilliant.org/TreforBazett to get started for free for 30 days, and to get 20% off an annual premium subscription! 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... BECOME A MEMBER: ►Join:    / @drtrefor   MATH BOOKS I LOVE (affilliate link): ► https://www.amazon.com/shop/treforbazett COURSE PLAYLISTS: ►DISCRETE MATH:    • Discrete Math (Full Course: Sets, Log...   ►LINEAR ALGEBRA:    • Linear Algebra (Full Course)   ►CALCULUS I:    • Calculus I (Limits, Derivative, Integ...   ► CALCULUS II:    • Calculus II (Integration Methods, Ser...   ►MULTIVARIABLE CALCULUS (Calc III):    • Calculus III: Multivariable Calculus ...   ►VECTOR CALCULUS (Calc IV)    • Calculus IV: Vector Calculus (Line In...   ►DIFFERENTIAL EQUATIONS:    • Ordinary Differential Equations (ODEs)   ►LAPLACE TRANSFORM:    • Laplace Transforms and Solving ODEs   ►GAME THEORY:    • Game Theory   OTHER PLAYLISTS: ► Learning Math Series    • 5 Tips To Make Math Practice Problems...   ►Cool Math Series:    • Cool Math Series   SOCIALS: ►X/Twitter: http://X.com/treforbazett ►TikTok:   / drtrefor   ►Instagram (photography based):   / treforphotography