Zeta Explained Course

Currently Playing: Zeta Explained #57: The Duality of Primes and Zeta Zeros

This is the 57th 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. We follow the texts "The Theory of the Riemann Zeta-Function" by E.C. Titchmarsh, "The Riemann Zeta-Function: Theory and Applications" by Aleksandar Ivić, and "Riemann's Zeta Function" by H.M. Edwards. This particular video covers the duality between prime numbers and zeta zeros, from the perspective of the explicit formulas for the prime counting function π(x) and the zero-counting function N(T). The formulas are the "Riemann–von Mangoldt explicit formula" for π(x) and Riemann–von Mangoldt formula for N(T). Using results from prior videos in the series, we note that while π(x) counts primes, it is partly composed of waves involving zeta zeros, and while N(T) counts zeta zeros, it is partly composed of waves involving primes and prime powers. 00:00 - Primes and zeta zeros 03:57 - Counting functions 06:49 - Formulas 12:36 - Sawtooths 14:45 - Primes from zeros, zeros from primes