Zeta Explained Course

Currently Playing: Zeta Explained #98: The Dirichlet Divisor Problem

This is the 98th 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. Sources Graham & Kolesnik, "Van der Corput's Method of Exponential Sums" Ivić, "The Riemann Zeta-Function: Theory and Applications" This particular video discusses the Dirichlet Divisor Problem. Let d(n) be the number of divisors of n, and let D(x) = ∑_(n ≤ x) d(n). Then D(x) = x log x + (2γ-1)x + Δ(x) where γ is the Euler-Mascheroni constant. The question is what the growth rate of Δ(x) is. Dirichlet originally proved the formula where Δ(x) = O(x^(1/2)), but the exponent has been decreased to O(x^(131/416)) 00:00 - Intro 01:36 - Graphs of D(x) 02:57 - Dirichlet's formula for D(x) and the hyperbola method 13:02 - Plot twist 14:10 - Graphs of Δ(x) 20:39 - Voronoi's summation formula 24:55 - van der Corput's results using the method of exponent pairs 35:50 - Connection with the Riemann zeta function and the Lindelöf Hypothesis