Abstract
The sixth chapter covers the last quarter of the century. We present first the large sieve and its applications (Bombieri’s density theorem and the Bombieri–Vinogradov theorem), treating also problems dealing with zeros of the zeta-function and L-functions (in particular the Pair Correlation Conjecture of H.L.M. Montgomery), questions connected with primes, and Selberg’s definition of the Selberg class as well as the related conjectures. Then we describe Baker’s method of evaluating linear forms of logarithms with applications to Diophantine equations, present the solution of the class-number one problem of Gauss and finally we turn to the progress in the theory of elliptic curves and modular forms.
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Notes
- 1.
Note however that in Vinogradov’s result the summation covered the range k≤x 1/2−ε with ε>0.
- 2.
- 3.
A simpler elementary proof was found in 1986 by A. Hildebrand [2798].
- 4.
“What has been said for τ possibly also holds for coefficients of every cusp form of weight k \(\varPhi(X)=\sum _{n=1}^{\infty}a_{n}X^{n}\), a 1=1, which is an eigenfunction of Hecke operators and has coefficients in Z.”
- 5.
A polynomial f of degree n is said to be reciprocal if f(X)=±X n f(1/X).
- 6.
- 7.
The paper by Orde earned a hostile review in Math. Reviews (80a:10036), amended later by the editors.
- 8.
- 9.
Dennis Ray Estes (1941–1999), professor at the University of South California.
- 10.
Eugène Catalan (1814–1894), professor at l’École Polytechnique in Paris.
- 11.
Kuusta Adolf Inkeri (1908–1997), professor in Turku. See [4273].
- 12.
Julia Robinson (1919–1985), sister of C. Reid, professor at Berkeley. See [5152].
- 13.
Hans Hermes (1912–2003), professor in Münster and Freiburg. See [4647].
- 14.
Vladimir Igorevič Arnold (1937–2010), professor in Moscow.
- 15.
It is one of the Millennium conjectures, with a prize of $106 for its solution.
- 16.
Tate noted in [6061] that the computer calculations performed by M. Sato led him to formulate this conjecture.
- 17.
They tried, without success, to publish their paper under the name Anne Arbor.
- 18.
See also Čudnovskiĭ, G.V.
- 19.
See also Chudnovsky, G.V.
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Narkiewicz, W. (2012). The Last Period. In: Rational Number Theory in the 20th Century. Springer Monographs in Mathematics. Springer, London. https://doi.org/10.1007/978-0-85729-532-3_6
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