A math course explaining the Riemann zeta function
Curated by: Zeta Explained (121 videos)
This is the 84th 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. This particular video covers Perron's inversion formula, which is a way to invert the relation in a typical Dirichlet series F(s) = ∑_(n=1 to ∞) f(n)/n^s into ∑'_(n≤x) f(n) = ∫_{c-i∞ to c+i∞) F(s) x^s/s ds We explore 3 examples of actually evaluating this integral and interpreting the results. Among the key results will be that the location of the right-most pole of the analytic function dominates the growth rate of the arithmetic function. 00:00 - Intro 01:55 - Statement of the formula 04:39 - Evaluating the integral 09:37 - Example 1: ζ(s) 14:48 - Example 2: ζ(2s) 20:34 - Example 3: ζ'(s)/ζ(s) 35:53 - Proof of the formula (assuming a basic version of the formula proved in Ep 47)