turing and the computational tradition in pure mathematics

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.


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


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


)(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


1792-93: How many primes are smaller than a given number x?

Bernhard Riemann

(1826-1866)


(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

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.


(1899-1963)


Leslie John Comrie

(1893-1950)


(1899-1963)


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?


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.



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).



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


“On the Factorization of Large Numbers”

761,838,257,287 ×

193,707,721 =

147,573,952,589,676,412,927



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:



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)


DA (1800)

ICN (1850) Kronecker/Dedekind

Divisor Theory/Ideal Theory (1865-75)


DA (1800)

ICN (1850)

DT/IT (1865-75)

David Hilbert (1862-1943) Zahlbericht (1896)


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


1932: Photoelectric Sieve

1927: Bicycle Chain Sieve


Turing and the Computational

Tradition in Pure Mathematics

Leo Corry Tel Aviv University

turing and the computational tradition in pure mathematics

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