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On Selberg's Lemma For Algebraic Fields

Published online by Cambridge University Press:  20 November 2018

R. G. Ayoub
R. G. Ayoub*
Affiliation:
Pennsylvania State College
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1. Introduction. Recently two Japanese authors (1) gave a beautifully simple proof of Selberg's fundamental lemma in the theory of distribution of primes. The proof is based on a curious twist in the Möbius inversion formula. The object of this note is to show that their proof may be extended to a proof of the result for algebraic fields corresponding to Selberg's lemma. Shapiro (2) has already derived this result using Selberg's methods and deduced as a consequence the prime ideal theorem.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 7 , 1955 , pp. 138 - 143
Copyright
Copyright © Canadian Mathematical Society 1955

References

1. Tatuzawa, T. and Iseki, K., On Selberg's elementary proof of the prime number theorem, Proc. Jap. Acad., 27 (1951), 340–342.Google Scholar
2. Shapiro, H., An elementary proof of the prime ideal theorem, Communications on Pure and Applied Mathematics, 2 (1949), 309–323.Google Scholar
3. Landau, E., Einführung in die elementare und analytische Théorie der algebraischen Zahlen (New York, 1949), 124 ff.Google Scholar