alan turing

alan turing

Alan Turing and Number Theory

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VideoLectures.Net View the talk in context: http://videolectures.net/turing100_matiyasevich_number_theory/ View the complete Alan Turing Centenary Conference Manchester, 2012: http://videolectures.net/turing100_conference2012_manchester/ Speaker: Yuri Matiyasevich, St.Petersburg Department of Steklov Mathematical Institute, Russian Academy of Sciences License: Creative Commons CC BY-NC-ND 3.0 More information at http://videolectures.net/site/about/ More talks at http://videolectures.net/ Beside well-known revolutionary contributions, Alan Turing had a number of significant results in "traditional" mathematics. In particular he was very much interested in the famous Riemann Hypothesis. This hypothesis, stated by Berhard Riemann in 1859 and included by David Hilbert in his 8th problem in 1990, still remains open, being now one of the Millennium Problems. The Riemann Hypothesis predicts positions of zeros of so called zeta function, and Alan Turing developed a rigorous method for verifying the Hypothesis for the initial zeros. He also invented a machine for calculating the values of the zeta function. In contrast to celebrated imaginable Turing machines, Turing started to implement this machine but never finished because of the War. 0:00 Alan Turing and Number Theory 1:14 Recognition of Merits (1) 1:20 Recognition of Merits (2) 1:25 Recognition of Merits (3) 2:06 Distribution of Prime Numbers (1) 2:15 Distribution of Prime Numbers (2) 2:22 Distribution of Prime Numbers (3) 2:34 Distribution of Prime Numbers (4) 3:01 Distribution of Prime Numbers (5) 3:04 Distribution of Prime Numbers (6) 3:08 Distribution of Prime Numbers (7) 3:25 Distribution of Prime Numbers (8) 3:47 Distribution of Prime Numbers (9) 4:06 Numerical Values 4:15 Riemann's formula for (x) (1) 4:17 Riemann's formula for (x) (2) 4:54 Riemann's formula for (x) (3) 5:20 Riemann's formula for (x) (4) 5:34 Riemann's formula for (x) (5) 6:09 Ingham's Book 7:03 Riemann's zeta function (1) 7:06 Riemann's zeta function (2) 7:20 The Riemann Hypothesis (1) 7:23 The Riemann Hypothesis (2) 7:49 The Riemann Hypothesis (3) 8:40 The Riemann Hypothesis (4) 8:49 The Riemann Hypothesis (5) 8:55 Numerical checking of RH 9:19 Riemann's formula for (x) (6) 9:23 Riemann's formula for (x) (7) 10:05 Riemann's formula for (x) (8) 11:22 Application for a grant of Royal Society 12:14 "Turing Machine" 13:19 The cost 13:48 Tide-predicting Machine (1) 15:37 Tide-predicting Machine (2) 15:49 Tide-predicting Machines 16:39 March, 1939 (1) 16:47 March, 1939 (2) 17:10 Methods for the calculation of the zeta-function (1) 17:15 Methods for the calculation of the zeta-function (2) 17:18 Methods for the calculation of the zeta-function (3) 17:26 Methods for the calculation of the zeta-function (4) 17:40 Methods for the calculation of the zeta-function (5) 18:50 Proceedings of the London Mathematical Society (1) 19:13 Proceedings of the London Mathematical Society (2) 19:26 Proceedings of the London Mathematical Society (3) 19:46 Proceedings of the London Mathematical Society (4) 20:14 Proceedings of the London Mathematical Society (5) 20:29 Proceedings of the London Mathematical Society (6) 20:35 Proceedings of the London Mathematical Society (7) 20:55 Proceedings of the London Mathematical Society (8) 21:49 Cauchy integral (1) 22:04 Cauchy integral (2) 22:53 Alongside the Critical Line (1) 23:10 Alongside the Critical Line (2) 23:34 Alongside the Critical Line (3) 23:52 Alongside the Critical Line (4) 24:04 Alongside the Critical Line (5) 24:40 Alongside the Critical Line (6) 24:42 Alongside the Critical Line (7) 24:48 Classical Method for Checking Riemann's Hypothesis (1) 24:56 Classical Method for Checking Riemann's Hypothesis (2) 25:03 Classical Method for Checking Riemann's Hypothesis (3) 25:16 Classical Method for Checking Riemann's Hypothesis (4) 25:39 Gram Points (1) 25:51 Gram Points (2) 26:08 Gram Points (3) 26:14 Gra...