turing and the computational tradition in pure mathematics
TRANSCRIPT
Turing and the Computational
Tradition in Pure Mathematics
Leo Corry Tel Aviv University
Turing (1953): “Some calculations of the Riemann zeta function,” Proc. London Math. Soc.
Turing (1953): “Some calculations of the Riemann zeta function,” Proc. London Math. Soc.
The calculations had been planned some time in
advance, but had in fact to be carried out in great haste.
If it had not been for the fact that the computer
remained in serviceable condition for an unusually long
period from 3 p.m. one afternoon to 8 a.m. the
following morning it is probable that the calculations
would never have been done at all.
Turing (1939): Application for grant support from the Royal Society for the engineering of a special machine to calculate approximate values for the Riemann zeta-function on its critical line
Turing (1939): … a special machine to calculate approximate values for the Riemann zeta-function on its critical line
Turing (1953): Calculation in the Manchester University Mark I Electronic Computer “concerned with the distribution of the zeros of the Riemann zeta-function”.
Once he had written up his proof of Hilbert’s decision problem for publication, Turing looked around for another big problem to attack. Cracking the Decision Problem would be a hard act to follow. But if you were going to go for another big problem, why not go for the ultimate prize, the Riemann Hypothesis?
By 1950 he had his new machine up and running and ready to start navigating the zeta landscape. … Titchmarsch had confirmed that the first 1,041 points … fulfilled RH. Turing went further and managed to make his machine check as far as the 1,104 zeros.
The Riemann zeta-function and the distribution of prime numbers
Carl Friedrich Gauss (1777-1855)
The riddle of prime numbers
2 3 5 7 11 13 17 19 23 29 31
37 41 43 47 53 59 61 67 71 73 79
83 89 97 101 103 107 109 113 127 131
137 139 149 151 157 163 167 173 179
181 191 193 197 199 211 223 227 229
233 239 241 251 257 263 269 271 277
281 283 293 307 311 313 317 331 337
347 349 353 359 367 373 379 383 389
397 401 409 419 421 431 433 439 443
449 457 461 463 467 479 487 491 499
503 509 521 523 541 547 557 563 569 571
577 587 593 599 601 607 613 617 619
631 641 643 647 653 659 661 673 677
683 691 701 709 719 727 733 739 743
751 757 761 769 773 787 797 809 811
821 823 827 829 839 853 857 859 863
877 881 883 887 907 911 919 929 937
941 947 953 967 971 977 983 991 997
The riddle of prime numbers
2 3 5 7 11 13 17 19 23 29 31
37 41 43 47 53 59 61 67 71 73 79
83 89 97 101 103 107 109 113 127 131
137 139 149 151 157 163 167 173 179
181 191 193 197 199 211 223 227 229
233 239 241 251 257 263 269 271 277
281 283 293 307 311 313 317 331 337
347 349 353 359 367 373 379 383 389
397 401 409 419 421 431 433 439 443
449 457 461 463 467 479 487 491 499
503 509 521 523 541 547 557 563 569 571
577 587 593 599 601 607 613 617 619
631 641 643 647 653 659 661 673 677
683 691 701 709 719 727 733 739 743
751 757 761 769 773 787 797 809 811
821 823 827 829 839 853 857 859 863
877 881 883 887 907 911 919 929 937
941 947 953 967 971 977 983 991 997
Carl Friedrich Gauss (1777-1855)
The riddle of prime numbers
2 3 5 7 11 13 17 19 23 29 31
37 41 43 47 53 59 61 67 71 73 79
83 89 97 101 103 107 109 113 127 131
137 139 149 151 157 163 167 173 179
181 191 193 197 199 211 223 227 229
233 239 241 251 257 263 269 271 277
281 283 293 307 311 313 317 331 337
347 349 353 359 367 373 379 383 389
397 401 409 419 421 431 433 439 443
449 457 461 463 467 479 487 491 499
503 509 521 523 541 547 557 563 569 571
577 587 593 599 601 607 613 617 619
631 641 643 647 653 659 661 673 677
683 691 701 709 719 727 733 739 743
751 757 761 769 773 787 797 809 811
821 823 827 829 839 853 857 859 863
877 881 883 887 907 911 919 929 937
941 947 953 967 971 977 983 991 997
Carl Friedrich Gauss (1777-1855)
The distribution of prime numbers
1792-93: How many primes are smaller than a given number x?
The number of primes under x Carl Friedrich Gauss (1777-1855) )(x
dtt
xLi
x
2
ln
1)(
The number of primes less than x
Carl Friedrich Gauss (1777-1855)
)(x
x
txLi
2ln
1)(
Li(x) (x) x
168 168 1000
78,627 78,498 106
37,607,950,280 37,607,912,018 1012
18,435,599,767,366,347,775,143 18,435,599,767,349,200,867,866 1024
The distribution of prime numbers
1792-93: How many primes are smaller than a given number x?
Bernhard Riemann
(1826-1866)
The distribution of prime numbers
(1859): for complex numbers ...
The key issue in understanding this function:
where does ζ(s)=0?
* Trivial fact: ζ(-2)=0, ζ(-4)=0, ζ(-6)=0, …
* And in addition (Riemann’s Hypothesis):
all other zeros are of the form ½ + iz
Jaques Hadamard
(1865-1963)
The Prime Number Theorem (1896)
Charles de la Vallée-Poussin
(1866-1962)
... and (von Koch 1901) RH is correct iff
David Hilbert
(1862-1943)
List of 23 Problems for the New Century (1900)
Problem 8: The Riemann Hypothesis
Edmund Landau
(1877-1938)
„Gelöste und ungelöste Probleme aus der Theorie der Primzahlverteilung.“
ICM - Cambridge 1912
David Hilbert
(1862-1943)
List of 23 Problems for the New Century (1900)
Problem 8: The Riemann Hypothesis
Edmund Landau
(1877-1938)
„Gelöste und ungelöste Probleme aus der Theorie der Primzahlverteilung.“
ICM - Cambridge 1912
Godfrey Harold Hardy (1877-1947)
Intensive Work on RH
Littlewood (1912): for infinite values of x, we have
(x) > Li(x)
John E. Littlewood
(1885-1977)
From Gauss it was know that
(x) < Li(x) for x< 105,
and Riemann thought that always
(x) < Li(x)
Littlewood (1962): “I believe RH to be false. There is no evidence whatever for it. … there is no imaginable reason why it should be true …”
Edward Ch. Titchmarsch
(1899-1963)
“The zeros of the Riemann zeta-function,” Proc. Royal Soc. London (1936)
Leslie John Comrie
(1893-1950)
“Titchmarsch had confirmed that the first 1,041 points … fulfilled RH.”
John Irwin Hutchinson
(1867-1935)
“On the Roots of the Riemann Zeta Function”, Trans. AMS (1925)
“I am indebted to Dr. Jesse Osborne for carrying out most of these calculations. A Monroe calculating machine and the Smithsonian Mathematical Tables by Becker and Van Orstrand with the trigonometric functions of angles expressed in radian measure were indispensable adjuncts.”
Confirmed that the first 138 points … fulfilled RH.
Edward Ch. Titchmarsch
(1899-1963)
“The zeros of the Riemann zeta-function,” Proc. Royal Soc. London (1936)
Leslie John Comrie
(1893-1950)
Edward Ch. Titchmarsch
(1899-1963)
“The zeros of the Riemann zeta-function,” Proc. Royal Soc. London (1936)
Carl Ludwig Siegel
(1896-1981)
Riemann-Siegel Formula
(1932)
Titchmarsch -1930
Hutchinson - 1925
Turing - 1953
138 cases - Monroe calculating machine
1,041 cases – Comrie, tables, and team + Riemann-Siegel Formula
1,104 cases – Manchester Mark I (+ a failed analogic attempt in 1939)
RH: The ultimate prize?
The Holy Grail?
Turing (1953): “Some calculations of the Riemann zeta function,” Proc. London Math. Soc.
The calculations had been planned some time in
advance, but had in fact to be carried out in great
haste. If it had not been for the fact that the computer
remained in serviceable condition for an unusually long
period from 3 p.m. one afternoon to 8 a.m. the
following morning it is probable that the calculations
would never have been done at all.
Computing particular cases of prime numbers
Derrick Henry Lehmer (1905-1991)
The case of FLT – 1930s
Emma Lehmer (1906-2007)
Harry Shultz Vandiver (1882-1973)
Back in America for the war … Dick was recruited to help
design and work on ENIAC. Some weekends the Lehmers
used it to solve certain number theory problems using the
sieve methods that they were working on, in particular to
research Fermat's Last Theorem. Emma was pleased that
ENIAC (when it was not broken down) could search a
million or so numbers in only three minutes.
The case of FLT – 1930s
Computing particular cases of prime numbers
February 10, 1936 (DHL to HSV):
I had tried the Annals but received an immediate rejection from
Lefschetz on the grounds that it is against the policy of the Annals
to publish tables. He suggested that the tables be deposited with
the AMS library or else published in some obscure journal. So I
tried the Duke journal.
Archibald suggested that the A.P.S. pay to have it printed in
Scripta Mathematica. I have never been paid for an article and
wouldn’t like to have journal accept anything that it would not
pay for itself. Anyway, I didn’t like the idea.
“On Bernoulli numbers and FLT,”
Duke Math. Journal, Vol. 3 (1937).
The case of FLT – 1930s
Computing particular cases of prime numbers
October 1903, NYC Meeting of the AMS
Frank Nelson Cole (1861-1926)
“On the Factorization of Large Numbers”
M67 is not prime
Edouard Lucas (1842-1891)
Mersenne Primes:
Mn = 2n -1
M127 is prime
(1876)
267 – 1 =
147,573,952,589,676,412,927
Frank Nelson Cole (1861-1926)
“On the Factorization of Large Numbers”
761,838,257,287 ×
193,707,721 =
147,573,952,589,676,412,927
October 1903, NYC Meeting of the AMS
Frank Nelson Cole (1861-1926)
Eric T. Bell, Mathematics; Queen and Servant of Sciences (1951)
When I asked Cole in 1911 how long it had taken to crack M67, Cole answered:
October 1903, NYC Meeting of the AMS
Frank Nelson Cole (1861-1926)
Eric T. Bell, Mathematics; Queen and Servant of Sciences (1951)
When I asked Cole in 1911 how long it had taken to crack M67, Cole answered: “three years of Sundays.”
Number Theory in the XIXth Century The Theory of Algebraic Number Fields
Carl Friedrich Gauss (1777-1855):
Disquisitiones Arithmeticae (1800)
Number Theory in the XIXth Century The theory of Algebraic Number Fields
DA (1800)
Kummer:
Theory of Ideal Complex Numbers
(1850)
Number Theory in the XIXth Century The theory of Algebraic Number Fields
DA (1800)
ICN (1850) Kronecker/Dedekind
Divisor Theory/Ideal Theory (1865-75)
Number Theory in the XIXth Century The theory of Algebraic Number Fields
DA (1800)
ICN (1850)
DT/IT (1865-75)
David Hilbert (1862-1943) Zahlbericht (1896)
Number Theory in the XIXth Century The theory of Algebraic Number Fields
Hilbert: Zahlbericht (1896)
“It is clear that the theory of these Kummer fields represents the highest
peak reached on the mountain of today’s knowledge of arithmetic; … I
have tried to avoid Kummer’s elaborate computational machinery, so that here too Riemann’s principle
may be realized and the proof completed not by calculations but
purely by ideas.”
Computations with Mersenne Primes
Index Discoverer Year
17, 19 Cataldi 1558
31 Euler 1772
61 Pervusin / Seelhof 1883 / 1886
89, 107 Powers 1911 - 1914
127 Lucas 1876
521, 607, 1279,
2203, 2281 Robinson 1952
• 1951 – Turing at Manchester
• No new primes were found
• No remainders saved for purposes of comparison
Mersenne Primes – 2n – 1
Lucas-Lehmer Test
Kummer (1850): 37, 59, 67, 101, 103, 131, 149, and 157 [157 index of irregularity = 2]
Kaj Løchte Jensen (1915): Infinity of irregulars
Vandiver (1930): up to 293
Vandiver, Emma and Dick Lehmer (1939): up to 619
Vandiver, Emma and Dick Lehmer (1954): up to 2000
Vandiver, Selfridge & Nicol (1955): up to 4002
…
Buhler et al. 2001 up to 12 millions
Computations with Irregular Primes
Ohm (1840): up to B31
Adams (1878): up to B62
Serebrennikov (1907): up to B92
Dick Lehmer (1936): up to B196
Emma and Dick Lehmer (1953): up to B214
…
Computations with Bernoulli Numbers
George W. Reitwiesner
Mathematical Tables and Other Aids to
Computation (1950)
Von Neumann at ENIAC – e and
First Draft of a Report on the EDVAC
November 1936
UTM
June 1945
“Dr. A. M. Turing”,
The Times, 16 June 1954
“Right from the start Turing
was interested in the
possibility of actually
building such a machine” Max Newman 1897-1984
Turing (1939): Application for grant support from the Royal Society for the engineering of a special machine to calculate approximate values for the Riemann zeta-function on its critical line
Derrick Henry Lehmer (1905-1991)
1932: Photoelectric Sieve
1927: Bicycle Chain Sieve
Derrick Henry Lehmer (1905-1991)
Turing and the Computational
Tradition in Pure Mathematics
Leo Corry Tel Aviv University