Zeta Explained Course

Currently Playing: Zeta Explained #52: Prime Gaps

This is the 52nd 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 introduces the theory of prime gaps. We cover the historical result of Bertrand-Chebyshev which looks far too weak, and also of asymptotics from assuming the Riemann Hypothesis. We cover Schoenfeld's bound of 1/(4pi)*sqrt(x)*(log^2 x), and then see how even this seems like too large of a bound, and show graphically that the bound is likely O(x^ε). Link to the paper referenced in the video: Schoenfeld, "Sharper bounds for the Chebyshev functions θ(x) and ψ(x) II" (1976): https://www.ams.org/journals/mcom/1976-30-134/S0025-5718-1976-0457374-X/S0025-5718-1976-0457374-X.pdf 00:00 - Intro and history of prime gaps 06:22 - The Chebyshev functions θ(x) and ψ(x) 09:56 - Plotting θ(x) and visualizing prime gaps 15:22 - Plotting prime gaps 20:05 - Comparing bounds and the Riemann Hypothesis strength 26:30 - Zeta function zeros and limitations of the Riemann Hypothesis