Zeta Explained Course

Currently Playing: Zeta Explained #27: The Prime Number Theorem (1/2)

This is the 27th video in a series explaining the Riemann zeta function. The idea of the series is to start with basics and eventually work our way to advanced topics. The viewer is expected to understand calculus and complex numbers, whereas I will try to explain concepts from complex analysis as needed. We will follow the books "The Riemann Zeta Function: Theory and Applications" by Aleksandar Ivić and "Riemann's Zeta Function" by H. M. Edwards. This particular video introduces the Prime Number Theorem, a historical context, the prime counting function π(x), and the Chebyshev functions θ(x) and ψ(x), and expresses ψ(x) in terms of the von Mangoldt function. 00:00 - Intro 01:16 - Intro to prime numbers 02:26 - Distribution of primes and Euler's theorem that 1/p diverges 06:59 - Gauss' conjecture that the density of primes is 1/(log x) 09:19 - Intro to Chebyshev's functions θ(x) and ψ(x) 10:54 - Graphs of Gauss and Chebyshev functions 13:47 - Gauss' false conjecture about Li(x) greater than π(x) 16:40 - Deeper dive into Chebyshev's functions θ(x) and ψ(x) 21:55 - Intro to the von Mangoldt function 23:30 - The Riemann zeta function and ζ'(s)/ζ(s) 30:38 - Two formulas involving the von Mangoldt function