Every Prime Number Obeys a Law That No One Has Proved in 167 Years
Ninth Axiom
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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
12 Comments
Also, we have consistent modular exponentiation, ex x^60 = 1 mod 60 for all x coprime to 60, and this is for all twice-pfimorials.
Weil proved general Riemann hypothesis for finite structures.
These patterns could help extend this result, and seem to create new geometry for L-functions and p-adic numbers.
Also, twice-primorials 4x3x5x7... correspond to the correct numbering of the roots on the unit circle (Gaussian integers correspond to 4-roots, not 2-roots).
If we think that the growing system of twice-primorials has to lead to this new geometry and incorporate all primes which are waiting, then we get exactly the boundary which you mention, and therefore Riemann hypothesis (and general Riemann hypothesis) must be "barely" true.
Every Prime Number Obeys a Law That No One Has Proved in 167 Years
Ninth Axiom
6,032 views
🔴 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