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The sieve method in algebraic fields. Lower bounds. (Die Siebmethode in algebraischen Zahlkörpern. Abschätzung von unten.) (Russian) Zbl 0281.12003

The author extends the method for “lower bounds” due to A. Selberg, and proves the following theorem.
Let \(K\) be an algebraic number field over the rationals. Then every large integer \(\zeta\in K\) of sufficiently high norm can be represented in the form \[ \zeta=\alpha+\beta,\tag{\text{A}} \] where \(\alpha,\beta\) are integers of \(K\) such that the ideal \((\alpha)\) is divisible by at most 2 prime ideals, and the the ideal \((\beta)\) by at most 3. In fact, the number of solutions of (A) exceeds \[ cS_K(\zeta)\frac{| N\zeta|}{\log^2| N\zeta|}\,, \] where \(c>0\) is a constant depending only on \(K\), and \(S_K(\zeta)\) is the “singular” series \[ S_K(\zeta)=\prod_{P\mid \zeta}\left(1-\frac 1{NP}\right)^{-1}\prod_{P\nmid \zeta}\left(1-\frac 2{NP}\right)\cdot\left(1-\frac 1{NP}\right)^{-2} \] and \(P\) stands for a prime ideal (of \(K\)).
The method of proof also requires estimates for the number of zeros of the Dedekind zeta-function in portions of the critical strip (see the proof of Lemma 4, pp. 57–61).
See also G. J. Rieger [J. Reine Angew. Math. 208, 79–90 (1961; Zbl 0103.02703)]; T. Tatuzawa [J. Math. Soc. Japan 7, 409–423 (1955; Zbl 0067.27403)]; R. Brauer [Am. J. Math. 69, 243–250 (1947; Zbl 0029.01502)].

MSC:

11N35 Sieves
11R47 Other analytic theory
11R42 Zeta functions and \(L\)-functions of number fields
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