Every Prime Number Obeys a Law That No One Has Proved in 167 Years

🔴 Support the channel and get exclusive content: Patreon: MeridianLabs The Riemann Hypothesis has been open for one hundred sixty seven years. In November 1859 Bernhard Riemann sent the Berlin Academy eight pages on prime numbers as a thank-you note for his election, admitted in print that he could not prove one sentence in them, called it very probable, and never published on the subject again. This is the story of those eight pages, and of the constraint nobody has ever been able to explain. The primes deviate from their average distribution by exactly as much as a fair coin would, and the thing holding them to that tolerance is a single vertical line in the complex plane. Ten trillion zeros have been checked. Every one of them sits on it. That proves nothing, and two conjectures in this same corner of mathematics have already survived everything computable and then broken at scales no machine will ever reach. Featuring Euler and the Basel problem, Gauss and the prime number theorem, Hardy, Turing at Manchester, Littlewood and the Skewes number, the Mertens conjecture, Montgomery and Dyson at Princeton, Hilbert, the Clay Millennium Prize, Godel, and Chaitin's argument that the primes may obey a law that can never be proved. 0:00 Intro 4:50 Eight Pages 12:20 The Housekeeper's Fire 16:15 How Rare A Prime Is 22:25 Euler's Doorway 26:30 The Line Down The Middle 32:20 The Primes Are Interference 42:30 What Would Actually Break 46:00 Ten Trillion Confirmations 55:10 Numbers Past The Horizon 59:15 The Conjecture That Was False 1:06:10 Barely True 1:11:20 One Hundred Thirty Years 1:20:20 The Men It Broke 1:33:05 A Conversation Over Tea 1:43:45 The Spectrum With No System 1:49:10 A Million Dollars In Paris 1:52:40 True And Unprovable 2:03:05 It May Not Be Anywhere Grounded in primary sources, including: Bernhard Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen Groesse (1859), Monatsberichte der Berliner Akademie Kurt Goedel, Ueber formal unentscheidbare Saetze der Principia Mathematica und verwandter Systeme I (1931) Helge von Koch, Sur la distribution des nombres premiers (1901), Acta Mathematica J. E. Littlewood, Sur la distribution des nombres premiers (1914), Comptes Rendus G. H. Hardy, Sur les zeros de la fonction zeta de Riemann (1914), Comptes Rendus Carl Ludwig Siegel, Ueber Riemanns Nachlass zur analytischen Zahlentheorie (1932) Hugh L. Montgomery, The Pair Correlation of Zeros of the Zeta Function (1973), AMS Proceedings of Symposia in Pure Mathematics XXIV Lowell Schoenfeld, Sharper Bounds for the Chebyshev Functions (1976), Mathematics of Computation Martin Davis, Yuri Matiyasevich and Julia Robinson, Hilbert's Tenth Problem: Positive Aspects of a Negative Solution (1976), AMS Guy Robin, Grandes valeurs de la fonction somme des diviseurs (1984), Journal de Mathematiques Pures et Appliquees Andrew Odlyzko and Herman te Riele, Disproof of the Mertens Conjecture (1985), Journal fuer die reine und angewandte Mathematik Michael Berry and Jonathan Keating, The Riemann Zeros and Eigenvalue Asymptotics (1999), SIAM Review Enrico Bombieri, Problems of the Millennium: The Riemann Hypothesis (2000), Clay Mathematics Institute Jeffrey C. Lagarias, An Elementary Problem Equivalent to the Riemann Hypothesis (2002), American Mathematical Monthly David Platt and Timothy Trudgian, The Riemann Hypothesis is True up to Three Times Ten to the Twelfth (2021), Bulletin of the London Mathematical Society #RiemannHypothesis #PrimeNumbers #NumberTheory #Mathematics #MathDocumentary