Zeta Explained #27: The Prime Number Theorem (1/2) скачать в хорошем качестве

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 the Riemann-von Mangoldt equation estimating the number of zeros in the critical strip between 0 and T as T/(2π)log(T/(2π)) - T/(2π). 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 book "The Riemann Zeta Function: Theory and Applications" by Alexandar Ivić. 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