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Une démonstration élémentaire du théorème des idéaux premiers “via une inégalité du type grand crible”. (An elementary proof of the prime ideal theorem “via an inequality of the large sieve type”). (French) Zbl 0729.11059

The author gives a new elementary proof of the Prime Ideal Theorem, based on an idea from a recent proof of the Prime Number Theorem due to A. Hildebrand (preprint, per bib.). The main tool is an analogue of the Turán-Kubilius inequality for algebraic number fields which implies a kind of the large sieve inequality. The first elementary proof of this theorem has been obtained by H. N. Shapiro [Commun. Pure Appl. Math. 2, 309-323 (1949; Zbl 0036.307)] who used an analogue of Selberg’s lemma. Other elementary proofs were given by Y. Eda and N. Nakagoshi [Sci. Rep. Kanazawa Univ. 12, 1-12 (1967)] and the author and H. S. Zargoumi [Colloq. Math. 57, 157-172 (1989; Zbl 0696.12001)].

MSC:

11R44 Distribution of prime ideals
11N36 Applications of sieve methods
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References:

[1] Elliott, P.D.T.A., Probabilistic Number Theory, SpringerN.Y. I (1979). · Zbl 0431.10029
[2] Hildebrand, A., The Prime Number Theorem via the large sieve, Preprint. · Zbl 0575.10036
[3] Shapiro, H., An elementary proof of the prime ideal theorem, Communication on Pure and Applied Mathematics (1949), 309-323. · Zbl 0036.30701
[4] Touibi, C., Zargouni, H., Smida, Une démonstration élémentaire du théorème des idéaux premiers, Colloquium Math.LVII. (1989), p. 157. Fas.1 · Zbl 0696.12001
[5] Weyl, H., Algebraic Number Theory, Princeton University Press. · JFM 66.1210.02
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