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The Forties and Fifties

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Rational Number Theory in the 20th Century

Part of the book series: Springer Monographs in Mathematics ((SMM))

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Abstract

The central events of this period in analytical number theory are the creation of the sieves of Yu.V. Linnik and A. Selberg, a new method of evaluating character sums invented by D.A. Burgess, the proof of the α+β-conjecture by H.B. Mann, and K.F. Roth’s theorem on approximation of algebraic numbers by rationals. We cover also the apparition of the Tate–Šafarevič group in the theory of elliptic curves, the birth of probabilistic number theory and progress in geometry of numbers (proof of Newton’s conjecture of the kissing problem by K. Schütte and B.L. van der Waerden).

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Notes

  1. 1.

    Alfréd Rényi (1921–1970), student of Linnik, professor in Budapest. See [520, 2399].

  2. 2.

    The first version of Ross’s argument can be found in the book by H. Halberstam and H.-E. Richert [2455].

  3. 3.

    Earlier [3909] he had obtained this for a part of progressions mod k.

  4. 4.

    Paul Joseph Cohen (1934–2007), professor at Stanford University. Fields Medal 1966 for proving the independence of the axiom of choice and the continuum hypothesis of the Zermelo–Fraenkel axioms of set theory.

  5. 5.

    Sergeĭ Mikhaĭlovič Voronin (1946–1997), worked at the Steklov Institute. See [117].

  6. 6.

    Hans-Heinrich Ostmann (1913–1959), professor in Berlin. See [2907, 3243].

  7. 7.

    Donald J. Newman (1930–2007), professor at MIT, Brown University, Yeshiva University and Temple University. See [6766].

  8. 8.

    Simon Sidon (1892–1941).

  9. 9.

    Chidambaran Padmanabhan Ramanujam (1938–1973), professor at the Tata Institute. See [5072].

  10. 10.

    Peter Arthur Barry Pleasants (1939–2008), worked in Cambridge, Cardiff, Macquarie University, the University of the South Pacific and Brisbane. See [1036].

  11. 11.

    The name of the letter Ш is “sha”.

  12. 12.

    Turán himself was at that time unaware that his proof can be interpreted in that way.

  13. 13.

    Patrick Billingsley (1925–2011), professor at the University of Chicago.

  14. 14.

    A generalization of this formula was obtained in 1971 by H. Delange [1432].

  15. 15.

    Beppo Levi (1875–1961), professor in Piacenza, Cagliari, Parma, Rosario and Bologna. See [5424].

  16. 16.

    His argument in the general case turned out to be insufficient. See [3295].

  17. 17.

    Bohuslav Diviš (1942–1976), professor at Ohio State University. See [5585].

  18. 18.

    Karl Zsigmondy (1867–1925), professor at the Technische Hochschule in Vienna. See [5472].

  19. 19.

    George David Birkhoff (1884–1944), professor at the University of Wisconsin, Princeton and Harvard. See [4438, 6655].

  20. 20.

    Harry Schultz Vandiver (1882–1973), professor at the University of Texas. See [1251].

  21. 21.

    See also Čudnovskiĭ, G.V.

  22. 22.

    See also Shparlinski, I.

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    Władysław Narkiewicz

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Narkiewicz, W. (2012). The Forties and Fifties. In: Rational Number Theory in the 20th Century. Springer Monographs in Mathematics. Springer, London. https://doi.org/10.1007/978-0-85729-532-3_5

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