This equation blew my mind // Euler Product Formula
Dr. Trefor Bazett
@drtreforAbout
This channel is about helping you learn math. I've got full playlists for Discrete Math, Linear Algebra, Calculus I-IV and Differential equations, as well as many more videos on cool math topics or about learning effectively. I am an Associate Teaching Professor teaching mathematics at the University of Victoria, in Canada. I completed my PhD in a fun branch of math called Algebraic Topology at the University of Toronto. Many of the videos on this channel were filmed during my time as an Assistant Professor, Educator at the University of Cincinnati. Mathematics is a journey we can all participate in. My videos can help support you, give you tools, and show you some of beauty and power of mathematics. But ultimately it is a journey we must travel together, so make sure you don't JUST watch my videos. Ask questions, try problems, and do as much math as you can on your own too!
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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/PrinceComp.pdf Contains the inequality I skipped in proof of the divergence of reciprocal primes: https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=1c996c03d57462a5d9e44c4dc8898de4ccfdae8a#:~:text=His%20conclusion%20is%20that%20the,reciprocals%20of%20the%20primes%20diverges Article on beautiful equations chosen by a range of mathematicians: https://www.scientificamerican.com/article/these-are-the-most-beautiful-equations-in-mathematics/ BECOME A MEMBER: ►Join: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join MATH BOOKS I LOVE (affilliate link): ► https://www.amazon.com/shop/treforbazett COURSE PLAYLISTS: ►DISCRETE MATH: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS ►LINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6 ►CALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m ► CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n ►MULTIVARIABLE CALCULUS (Calc III): https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd ►VECTOR CALCULUS (Calc IV) https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfW0GMqeUE1bLKaYor6kbHa ►DIFFERENTIAL EQUATIONS: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxde-SlgmWlCmNHroIWtujBw ►LAPLACE TRANSFORM: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxcJXnLr08cyNaup4RDsbAl1 ►GAME THEORY: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxdzD8KpTHz6_gsw9pPxRFlX OTHER PLAYLISTS: ► Learning Math Series https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw ►Cool Math Series: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho SOCIALS: ►X/Twitter: http://X.com/treforbazett ►TikTok: http://tiktok.com/@drtrefor ►Instagram (photography based): http://instagram.com/treforphotography
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