Zeta Explained Course

Currently Playing: Zeta Explained #77: Square-Free and k-Free Numbers

This is the 77th 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 square-free numbers, which are positive integers that are not divisible by any square number. We also discuss the generalization, k-free numbers, which are not divisible by any k-th power. We show that the probability that a random number is square-free is 6/π^2. We also cover that the Dirichlet series is ∑_(n=1 to ∞) Is_square_free(n)/n^s = ζ(s)/ζ(2s). And the generalized form for k-free is ∑_(n=1 to ∞) Is_k_free(n)/n^s = ζ(s)/ζ(ks). Finally, we relate this to the Riemann Hypothesis. 00:00 - Intro 04:19 - Theorem 1: P(square-free) = 1/ζ(2) = 6/π^2 08:35 - k-free numbers 16:28 - Theorem 2: Dirichlet series for square-free and k-free indicator functions 24:29 - Lemma 1: μ^2(n) = ∑_(d|n) μ(d) 26:38 - Theorem 3: Q(x) = x/ζ(2) + O(√x) 32:22 - Relation to Riemann hypothesis 34:41 - Graphs of Q(x) and error term 36:21 - Complex analysis on the Dirichlet series using Perron's inversion formula 45:54 - Cosine wave sum for Q(x) 49:06 - Summary of results